Integrand size = 22, antiderivative size = 288 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=-\frac {5 b^2 e^3 n^2 \sqrt {x}}{6 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2}+\frac {5 b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{6 d^4}+\frac {b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}-\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4} \]
1/6*b^2*e^2*n^2*x/d^2+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^4-1/2*b*e^2*n*x*(a+b*ln(c*(d+e/x^(1/2))^n))/d^2+1/3*b*e*n*x^(3/2 )*(a+b*ln(c*(d+e/x^(1/2))^n))/d+b*e^4*n*ln(1-d/(d+e/x^(1/2)))*(a+b*ln(c*(d +e/x^(1/2))^n))/d^4+1/2*x^2*(a+b*ln(c*(d+e/x^(1/2))^n))^2-b^2*e^4*n^2*poly log(2,d/(d+e/x^(1/2)))/d^4-5/6*b^2*e^3*n^2*x^(1/2)/d^3+b*e^3*n*(a+b*ln(c*( d+e/x^(1/2))^n))*(d+e/x^(1/2))*x^(1/2)/d^4
Time = 0.15 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {1}{6} \left (3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e n \left (6 a d e^2 \sqrt {x}-5 b d e^2 n \sqrt {x}-3 a d^2 e x+b d^2 e n x+2 a d^3 x^{3/2}+8 b e^3 n \log \left (d+\frac {e}{\sqrt {x}}\right )+6 b d e^2 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-3 b d^2 e x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+2 b d^3 x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-6 a e^3 \log \left (e+d \sqrt {x}\right )+3 b e^3 n \log \left (e+d \sqrt {x}\right )-6 b e^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )+3 b e^3 n \log ^2\left (e+d \sqrt {x}\right )-6 b e^3 n \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+4 b e^3 n \log (x)-6 b e^3 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )}{d^4}\right ) \]
(3*x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*e*n*(6*a*d*e^2*Sqrt[x] - 5* b*d*e^2*n*Sqrt[x] - 3*a*d^2*e*x + b*d^2*e*n*x + 2*a*d^3*x^(3/2) + 8*b*e^3* n*Log[d + e/Sqrt[x]] + 6*b*d*e^2*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] - 3*b*d^ 2*e*x*Log[c*(d + e/Sqrt[x])^n] + 2*b*d^3*x^(3/2)*Log[c*(d + e/Sqrt[x])^n] - 6*a*e^3*Log[e + d*Sqrt[x]] + 3*b*e^3*n*Log[e + d*Sqrt[x]] - 6*b*e^3*Log[ c*(d + e/Sqrt[x])^n]*Log[e + d*Sqrt[x]] + 3*b*e^3*n*Log[e + d*Sqrt[x]]^2 - 6*b*e^3*n*Log[e + d*Sqrt[x]]*Log[-((d*Sqrt[x])/e)] + 4*b*e^3*n*Log[x] - 6 *b*e^3*n*PolyLog[2, 1 + (d*Sqrt[x])/e]))/d^4)/6
Time = 1.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, 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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -2 \int x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -2 \left (\frac {1}{2} b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -2 \left (\frac {1}{2} b n \int x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \int \frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {-\frac {1}{3} b n \int -\frac {x^2}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {-\frac {1}{3} b n \int \left (-\frac {x^{3/2}}{d e^3}+\frac {x}{d^2 e^2}-\frac {\sqrt {x}}{d^3 e}+\frac {\sqrt {x}}{d^3}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\int -\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^{3/2}}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x}{d e^2}-\frac {\sqrt {x}}{d^2 e}+\frac {\sqrt {x}}{d^2}\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\int \frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {-\frac {b n \int -\frac {\sqrt {x}}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}+\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {\frac {b n \int \sqrt {x} \log \left (1-d \sqrt {x}\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-d \sqrt {x}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 \left (\frac {1}{2} b e^4 n \left (\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt {x}\right )}{d}-\frac {\log \left (1-d \sqrt {x}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d}}{d}}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
-2*(-1/4*(x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2) + (b*e^4*n*((-1/3*(b*n*( -(Sqrt[x]/(d^2*e)) + x/(2*d*e^2) + Log[d + e/Sqrt[x]]/d^3 - Log[-(e/Sqrt[x ])]/d^3)) - (x^(3/2)*(a + b*Log[c/x^(n/2)]))/(3*e^3))/d + ((-1/2*(b*n*(-(S qrt[x]/(d*e)) + Log[d + e/Sqrt[x]]/d^2 - Log[-(e/Sqrt[x])]/d^2)) + (x*(a + b*Log[c/x^(n/2)]))/(2*e^2))/d + (((b*n*Log[-(e/Sqrt[x])])/d - ((d + e/Sqr t[x])*Sqrt[x]*(a + b*Log[c/x^(n/2)]))/(d*e))/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)
3.5.30.3.1 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])
\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}d x\]
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \]
integral(b^2*x*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 2*a*b*x*log(c*((d*x + e* sqrt(x))/x)^n) + a^2*x, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \]
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \]
1/2*b^2*x^2*log((d*sqrt(x) + e)^n)^2 - integrate(-1/2*(2*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^2 + 2*(b^2*d*x^2 + b^2*e*x^(3/2))*log(x^(1/2*n) )^2 + 2*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(3/2) - (b^2*d*n*x^2 - 4*(b^2*d*log(c) + a*b*d)*x^2 - 4*(b^2*e*log(c) + a*b*e)*x^(3/2) + 4*(b^2* d*x^2 + b^2*e*x^(3/2))*log(x^(1/2*n)))*log((d*sqrt(x) + e)^n) - 4*((b^2*d* log(c) + a*b*d)*x^2 + (b^2*e*log(c) + a*b*e)*x^(3/2))*log(x^(1/2*n)))/(d*x + e*sqrt(x)), x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \]