Integrand size = 24, antiderivative size = 293 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 e^2 n^2}{6 d^2 x}+\frac {5 b^2 e^3 n^2}{6 d^3 \sqrt {x}}-\frac {5 b^2 e^4 n^2 \log \left (d+e \sqrt {x}\right )}{6 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4 \sqrt {x}}-\frac {b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4} \]
-1/6*b^2*e^2*n^2/d^2/x+11/12*b^2*e^4*n^2*ln(x)/d^4-5/6*b^2*e^4*n^2*ln(d+e* x^(1/2))/d^4-1/3*b*e*n*(a+b*ln(c*(d+e*x^(1/2))^n))/d/x^(3/2)+1/2*b*e^2*n*( a+b*ln(c*(d+e*x^(1/2))^n))/d^2/x-1/2*(a+b*ln(c*(d+e*x^(1/2))^n))^2/x^2-b*e ^4*n*(a+b*ln(c*(d+e*x^(1/2))^n))*ln(1-d/(d+e*x^(1/2)))/d^4+b^2*e^4*n^2*pol ylog(2,d/(d+e*x^(1/2)))/d^4+5/6*b^2*e^3*n^2/d^3/x^(1/2)-b*e^3*n*(a+b*ln(c* (d+e*x^(1/2))^n))*(d+e*x^(1/2))/d^4/x^(1/2)
Time = 0.23 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {e \sqrt {x} \left (4 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-6 b d^2 e n \sqrt {x} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+12 b d e^2 n x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-6 e^3 x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+12 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b^2 e^3 n^2 x^{3/2} \left (2 \log \left (d+e \sqrt {x}\right )-\log (x)\right )-3 b^2 e^2 n^2 x \left (2 d-2 e \sqrt {x} \log \left (d+e \sqrt {x}\right )+e \sqrt {x} \log (x)\right )+2 b^2 e n^2 \sqrt {x} \left (d \left (d-2 e \sqrt {x}\right )+2 e^2 x \log \left (d+e \sqrt {x}\right )-e^2 x \log (x)\right )+12 b^2 e^3 n^2 x^{3/2} \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )}{d^4}}{12 x^2} \]
-1/12*(6*(a + b*Log[c*(d + e*Sqrt[x])^n])^2 + (e*Sqrt[x]*(4*b*d^3*n*(a + b *Log[c*(d + e*Sqrt[x])^n]) - 6*b*d^2*e*n*Sqrt[x]*(a + b*Log[c*(d + e*Sqrt[ x])^n]) + 12*b*d*e^2*n*x*(a + b*Log[c*(d + e*Sqrt[x])^n]) - 6*e^3*x^(3/2)* (a + b*Log[c*(d + e*Sqrt[x])^n])^2 + 12*b*e^3*n*x^(3/2)*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)] + 6*b^2*e^3*n^2*x^(3/2)*(2*Log[d + e* Sqrt[x]] - Log[x]) - 3*b^2*e^2*n^2*x*(2*d - 2*e*Sqrt[x]*Log[d + e*Sqrt[x]] + e*Sqrt[x]*Log[x]) + 2*b^2*e*n^2*Sqrt[x]*(d*(d - 2*e*Sqrt[x]) + 2*e^2*x* Log[d + e*Sqrt[x]] - e^2*x*Log[x]) + 12*b^2*e^3*n^2*x^(3/2)*PolyLog[2, 1 + (e*Sqrt[x])/d]))/d^4)/x^2
Time = 1.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^{5/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 2 \left (\frac {1}{2} b e n \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{\left (d+e \sqrt {x}\right ) x^2}d\sqrt {x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 2 \left (\frac {1}{2} b n \int \frac {a+b \log \left (c x^{n/2}\right )}{x^{5/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \int \frac {a+b \log \left (c x^{n/2}\right )}{e^4 x^{5/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^4 x^2}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {-\frac {1}{3} b n \int -\frac {1}{e^3 x^2}d\left (d+e \sqrt {x}\right )-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {-\frac {1}{3} b n \int \left (-\frac {1}{d^3 e \sqrt {x}}+\frac {1}{d^3 \sqrt {x}}+\frac {1}{d^2 e^2 x}-\frac {1}{d e^3 x^{3/2}}\right )d\left (d+e \sqrt {x}\right )-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \int \frac {1}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{d^2 \sqrt {x}}+\frac {1}{d e^2 x}\right )d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {-\frac {b n \int -\frac {1}{e \sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt {x}}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/2}\right )}{3 e^3 x^{3/2}}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^3}-\frac {\log \left (-e \sqrt {x}\right )}{d^3}-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{2 d e^2 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 x^2}\right )\) |
2*(-1/4*(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^2 + (b*e^4*n*((-1/3*(b*n*(1/( 2*d*e^2*x) - 1/(d^2*e*Sqrt[x]) + Log[d + e*Sqrt[x]]/d^3 - Log[-(e*Sqrt[x]) ]/d^3)) - (a + b*Log[c*x^(n/2)])/(3*e^3*x^(3/2)))/d + ((-1/2*(b*n*(-(1/(d* e*Sqrt[x])) + Log[d + e*Sqrt[x]]/d^2 - Log[-(e*Sqrt[x])]/d^2)) + (a + b*Lo g[c*x^(n/2)])/(2*e^2*x))/d + (((b*n*Log[-(e*Sqrt[x])])/d - ((d + e*Sqrt[x] )*(a + b*Log[c*x^(n/2)]))/(d*e*Sqrt[x]))/d + (-((Log[1 - d/Sqrt[x]]*(a + b *Log[c*x^(n/2)]))/d) + (b*n*PolyLog[2, d/Sqrt[x]])/d)/d)/d)/d))/2)
3.5.13.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}}{x^{3}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-1/2*b^2*log((e*sqrt(x) + d)^n)^2/x^2 + integrate(1/2*(2*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x + (b^2*e*n*x + 4*(b^2*e*log(c) + a*b*e)*x + 4*( b^2*d*log(c) + a*b*d)*sqrt(x))*log((e*sqrt(x) + d)^n) + 2*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*sqrt(x))/(e*x^4 + d*x^(7/2)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^3} \,d x \]