Integral of (4x+1)/(sqrt(2x-3)+7) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sqrt{2 x - 3}$.

      Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$:

      $$\int \frac{4 u \left(\frac{u^{2}}{2} + \frac{3}{2}\right) + u}{u + 7}\, du$$

      1. Rewrite the integrand:

         $$\frac{4 u \left(\frac{u^{2}}{2} + \frac{3}{2}\right) + u}{u + 7} = 2 u^{2} - 14 u + 105 - \frac{735}{u + 7}$$

      2. Integrate term-by-term:

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int 2 u^{2}\, du = 2 \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $\frac{2 u^{3}}{3}$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 14 u\right)\, du = - 14 \int u\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u\, du = \frac{u^{2}}{2}$$

            So, the result is: $- 7 u^{2}$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 105\, du = 105 u$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- \frac{735}{u + 7}\right)\, du = - 735 \int \frac{1}{u + 7}\, du$$

            1. Let $u = u + 7$.

               Then let $du = du$ and substitute $du$:

               $$\int \frac{1}{u}\, du$$

               1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

               Now substitute $u$ back in:

               $$\log{\left(u + 7 \right)}$$

            So, the result is: $- 735 \log{\left(u + 7 \right)}$

         The result is: $\frac{2 u^{3}}{3} - 7 u^{2} + 105 u - 735 \log{\left(u + 7 \right)}$

      Now substitute $u$ back in:

      $$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$$

   Method #2

   1. Rewrite the integrand:

      $$\frac{4 x + 1}{\sqrt{2 x - 3} + 7} = \frac{4 x}{\sqrt{2 x - 3} + 7} + \frac{1}{\sqrt{2 x - 3} + 7}$$

   2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{4 x}{\sqrt{2 x - 3} + 7}\, dx = 4 \int \frac{x}{\sqrt{2 x - 3} + 7}\, dx$$

         1. Let $u = \sqrt{2 x - 3}$.

            Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$:

            $$\int \frac{u \left(\frac{u^{2}}{2} + \frac{3}{2}\right)}{u + 7}\, du$$

            1. Rewrite the integrand:

               $$\frac{u \left(\frac{u^{2}}{2} + \frac{3}{2}\right)}{u + 7} = \frac{u^{2}}{2} - \frac{7 u}{2} + 26 - \frac{182}{u + 7}$$

            2. Integrate term-by-term:

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                  So, the result is: $\frac{u^{3}}{6}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- \frac{7 u}{2}\right)\, du = - \frac{7 \int u\, du}{2}$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u\, du = \frac{u^{2}}{2}$$

                  So, the result is: $- \frac{7 u^{2}}{4}$

               1. The integral of a constant is the constant times the variable of integration:

                  $$\int 26\, du = 26 u$$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- \frac{182}{u + 7}\right)\, du = - 182 \int \frac{1}{u + 7}\, du$$

                  1. Let $u = u + 7$.

                     Then let $du = du$ and substitute $du$:

                     $$\int \frac{1}{u}\, du$$

                     1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

                     Now substitute $u$ back in:

                     $$\log{\left(u + 7 \right)}$$

                  So, the result is: $- 182 \log{\left(u + 7 \right)}$

               The result is: $\frac{u^{3}}{6} - \frac{7 u^{2}}{4} + 26 u - 182 \log{\left(u + 7 \right)}$

            Now substitute $u$ back in:

            $$- \frac{7 x}{2} + \frac{\left(2 x - 3\right)^{\frac{3}{2}}}{6} + 26 \sqrt{2 x - 3} - 182 \log{\left(\sqrt{2 x - 3} + 7 \right)} + \frac{21}{4}$$

         So, the result is: $- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 104 \sqrt{2 x - 3} - 728 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$

      1. Let $u = \sqrt{2 x - 3}$.

         Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$:

         $$\int \frac{u}{u + 7}\, du$$

         1. Rewrite the integrand:

            $$\frac{u}{u + 7} = 1 - \frac{7}{u + 7}$$

         2. Integrate term-by-term:

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 1\, du = u$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \left(- \frac{7}{u + 7}\right)\, du = - 7 \int \frac{1}{u + 7}\, du$$

               1. Let $u = u + 7$.

                  Then let $du = du$ and substitute $du$:

                  $$\int \frac{1}{u}\, du$$

                  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

                  Now substitute $u$ back in:

                  $$\log{\left(u + 7 \right)}$$

               So, the result is: $- 7 \log{\left(u + 7 \right)}$

            The result is: $u - 7 \log{\left(u + 7 \right)}$

         Now substitute $u$ back in:

         $$\sqrt{2 x - 3} - 7 \log{\left(\sqrt{2 x - 3} + 7 \right)}$$

      The result is: $- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$

2. Now simplify:

   $$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$$

3. Add the constant of integration:

   $$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21+ \mathrm{constant}$$

The answer is: