Integrand size = 24, antiderivative size = 404 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=-\frac {77 b^2 e^5 n^2 \sqrt {x}}{90 d^5}+\frac {47 b^2 e^4 n^2 x}{180 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{90 d^6}+\frac {2 b e^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {2 b e^6 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {2 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{3 d^6} \] Output:
-77/90*b^2*e^5*n^2*x^(1/2)/d^5+47/180*b^2*e^4*n^2*x/d^4-1/10*b^2*e^3*n^2*x ^(3/2)/d^3+1/30*b^2*e^2*n^2*x^2/d^2+77/90*b^2*e^6*n^2*ln(d+e/x^(1/2))/d^6+ 2/3*b*e^5*n*(d+e/x^(1/2))*x^(1/2)*(a+b*ln(c*(d+e/x^(1/2))^n))/d^6-1/3*b*e^ 4*n*x*(a+b*ln(c*(d+e/x^(1/2))^n))/d^4+2/9*b*e^3*n*x^(3/2)*(a+b*ln(c*(d+e/x ^(1/2))^n))/d^3-1/6*b*e^2*n*x^2*(a+b*ln(c*(d+e/x^(1/2))^n))/d^2+2/15*b*e*n *x^(5/2)*(a+b*ln(c*(d+e/x^(1/2))^n))/d+2/3*b*e^6*n*ln(1-d/(d+e/x^(1/2)))*( a+b*ln(c*(d+e/x^(1/2))^n))/d^6+1/3*x^3*(a+b*ln(c*(d+e/x^(1/2))^n))^2+137/1 80*b^2*e^6*n^2*ln(x)/d^6-2/3*b^2*e^6*n^2*polylog(2,d/(d+e/x^(1/2)))/d^6
Time = 0.40 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \left (x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e n \left (120 a d e^4 \sqrt {x}-154 b d e^4 n \sqrt {x}-60 a d^2 e^3 x+47 b d^2 e^3 n x+40 a d^3 e^2 x^{3/2}-18 b d^3 e^2 n x^{3/2}-30 a d^4 e x^2+6 b d^4 e n x^2+24 a d^5 x^{5/2}+214 b e^5 n \log \left (d+\frac {e}{\sqrt {x}}\right )+120 b d e^4 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-60 b d^2 e^3 x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+40 b d^3 e^2 x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-30 b d^4 e x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+24 b d^5 x^{5/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-120 a e^5 \log \left (e+d \sqrt {x}\right )+60 b e^5 n \log \left (e+d \sqrt {x}\right )-120 b e^5 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )+60 b e^5 n \log ^2\left (e+d \sqrt {x}\right )-120 b e^5 n \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+107 b e^5 n \log (x)-120 b e^5 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )}{60 d^6}\right ) \] Input:
Integrate[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2,x]
Output:
(x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*e*n*(120*a*d*e^4*Sqrt[x] - 15 4*b*d*e^4*n*Sqrt[x] - 60*a*d^2*e^3*x + 47*b*d^2*e^3*n*x + 40*a*d^3*e^2*x^( 3/2) - 18*b*d^3*e^2*n*x^(3/2) - 30*a*d^4*e*x^2 + 6*b*d^4*e*n*x^2 + 24*a*d^ 5*x^(5/2) + 214*b*e^5*n*Log[d + e/Sqrt[x]] + 120*b*d*e^4*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] - 60*b*d^2*e^3*x*Log[c*(d + e/Sqrt[x])^n] + 40*b*d^3*e^2*x ^(3/2)*Log[c*(d + e/Sqrt[x])^n] - 30*b*d^4*e*x^2*Log[c*(d + e/Sqrt[x])^n] + 24*b*d^5*x^(5/2)*Log[c*(d + e/Sqrt[x])^n] - 120*a*e^5*Log[e + d*Sqrt[x]] + 60*b*e^5*n*Log[e + d*Sqrt[x]] - 120*b*e^5*Log[c*(d + e/Sqrt[x])^n]*Log[ e + d*Sqrt[x]] + 60*b*e^5*n*Log[e + d*Sqrt[x]]^2 - 120*b*e^5*n*Log[e + d*S qrt[x]]*Log[-((d*Sqrt[x])/e)] + 107*b*e^5*n*Log[x] - 120*b*e^5*n*PolyLog[2 , 1 + (d*Sqrt[x])/e]))/(60*d^6))/3
Time = 3.91 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.37, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.042, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -2 \int x^{7/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -2 \left (\frac {1}{3} b e n \int \frac {x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -2 \left (\frac {1}{3} b n \int x^{7/2} \left (a+b \log \left (c x^{-n/2}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \int \frac {x^{7/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\int \frac {x^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {x^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int -\frac {x^3}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int \left (-\frac {x^{5/2}}{d e^5}+\frac {x^2}{d^2 e^4}-\frac {x^{3/2}}{d^3 e^3}+\frac {x}{d^4 e^2}-\frac {\sqrt {x}}{d^5 e}+\frac {\sqrt {x}}{d^5}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\int -\frac {x^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int -\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int \frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \frac {x^{5/2}}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \left (\frac {x^2}{d e^4}-\frac {x^{3/2}}{d^2 e^3}+\frac {x}{d^3 e^2}-\frac {\sqrt {x}}{d^4 e}+\frac {\sqrt {x}}{d^4}\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int \frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {-\frac {1}{3} b n \int -\frac {x^2}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {x^{3/2}}{d e^3}+\frac {x}{d^2 e^2}-\frac {\sqrt {x}}{d^3 e}+\frac {\sqrt {x}}{d^3}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int -\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^{3/2}}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x}{d e^2}-\frac {\sqrt {x}}{d^2 e}+\frac {\sqrt {x}}{d^2}\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int \frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int \frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {-\frac {b n \int -\frac {\sqrt {x}}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}+\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\frac {b n \int \sqrt {x} \log \left (1-d \sqrt {x}\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-d \sqrt {x}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\sqrt {x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt {x}\right )}{d}-\frac {\log \left (1-d \sqrt {x}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )}{d}}{d}}{d}}{d}+\frac {-\frac {x^{3/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\sqrt {x}}{d^2 e}+\frac {x}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\sqrt {x}}{d^3 e}+\frac {x}{2 d^2 e^2}-\frac {x^{3/2}}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/2} \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt {x}}\right )}{d^5}-\frac {\sqrt {x}}{d^4 e}+\frac {x}{2 d^3 e^2}-\frac {x^{3/2}}{3 d^2 e^3}+\frac {x^2}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2,x]
Output:
-2*(-1/6*(x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2) + (b*e^6*n*((-1/5*(b*n*( -(Sqrt[x]/(d^4*e)) + x/(2*d^3*e^2) - x^(3/2)/(3*d^2*e^3) + x^2/(4*d*e^4) + Log[d + e/Sqrt[x]]/d^5 - Log[-(e/Sqrt[x])]/d^5)) - (x^(5/2)*(a + b*Log[c/ x^(n/2)]))/(5*e^5))/d + ((-1/4*(b*n*(-(Sqrt[x]/(d^3*e)) + x/(2*d^2*e^2) - x^(3/2)/(3*d*e^3) + Log[d + e/Sqrt[x]]/d^4 - Log[-(e/Sqrt[x])]/d^4)) + (x^ 2*(a + b*Log[c/x^(n/2)]))/(4*e^4))/d + ((-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*(-(Sqrt[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/Sqrt[x])*Sqrt[x]*(a + b*L og[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)/d)/d))/3)
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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}d x\]
Input:
int(x^2*(a+b*ln(c*(d+e/x^(1/2))^n))^2,x)
Output:
int(x^2*(a+b*ln(c*(d+e/x^(1/2))^n))^2,x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n))^2,x, algorithm="fricas")
Output:
integral(b^2*x^2*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 2*a*b*x^2*log(c*((d*x + e*sqrt(x))/x)^n) + a^2*x^2, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \] Input:
integrate(x**2*(a+b*ln(c*(d+e/x**(1/2))**n))**2,x)
Output:
Integral(x**2*(a + b*log(c*(d + e/sqrt(x))**n))**2, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n))^2,x, algorithm="maxima")
Output:
1/3*b^2*x^3*log((d*sqrt(x) + e)^n)^2 - integrate(-1/3*(3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^3 + 3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e) *x^(5/2) + 3*(b^2*d*x^3 + b^2*e*x^(5/2))*log(x^(1/2*n))^2 - (b^2*d*n*x^3 - 6*(b^2*d*log(c) + a*b*d)*x^3 - 6*(b^2*e*log(c) + a*b*e)*x^(5/2) + 6*(b^2* d*x^3 + b^2*e*x^(5/2))*log(x^(1/2*n)))*log((d*sqrt(x) + e)^n) - 6*((b^2*d* log(c) + a*b*d)*x^3 + (b^2*e*log(c) + a*b*e)*x^(5/2))*log(x^(1/2*n)))/(d*x + e*sqrt(x)), x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n))^2,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/sqrt(x))^n) + a)^2*x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \] Input:
int(x^2*(a + b*log(c*(d + e/x^(1/2))^n))^2,x)
Output:
int(x^2*(a + b*log(c*(d + e/x^(1/2))^n))^2, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {60 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right )^{2} b^{2} d \,e^{5}+24 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{5} e n \,x^{2}+40 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{3} e^{3} n x +120 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d \,e^{5} n +24 \sqrt {x}\, a b \,d^{5} e n \,x^{2}+40 \sqrt {x}\, a b \,d^{3} e^{3} n x +120 \sqrt {x}\, a b d \,e^{5} n -18 \sqrt {x}\, b^{2} d^{3} e^{3} n^{2} x -154 \sqrt {x}\, b^{2} d \,e^{5} n^{2}-30 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right )^{2}}{x}d x \right ) b^{2} d \,e^{5}+60 \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right )^{2} b^{2} d^{6} x^{3}+120 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) a b \,d^{6} x^{3}-120 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) a b \,e^{6}-30 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{4} e^{2} n \,x^{2}-60 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{2} e^{4} n x +274 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} e^{6} n -120 \,\mathrm {log}\left (\sqrt {x}\right ) a b \,e^{6} n +274 \,\mathrm {log}\left (\sqrt {x}\right ) b^{2} e^{6} n^{2}+60 a^{2} d^{6} x^{3}-30 a b \,d^{4} e^{2} n \,x^{2}-60 a b \,d^{2} e^{4} n x +6 b^{2} d^{4} e^{2} n^{2} x^{2}+47 b^{2} d^{2} e^{4} n^{2} x}{180 d^{6}} \] Input:
int(x^2*(a+b*log(c*(d+e/x^(1/2))^n))^2,x)
Output:
(60*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**2*d*e**5 + 24*sqrt( x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**5*e*n*x**2 + 40*sqrt(x)*lo g(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**3*e**3*n*x + 120*sqrt(x)*log((( sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d*e**5*n + 24*sqrt(x)*a*b*d**5*e*n*x** 2 + 40*sqrt(x)*a*b*d**3*e**3*n*x + 120*sqrt(x)*a*b*d*e**5*n - 18*sqrt(x)*b **2*d**3*e**3*n**2*x - 154*sqrt(x)*b**2*d*e**5*n**2 - 30*int((sqrt(x)*log( ((sqrt(x)*d + e)**n*c)/x**(n/2))**2)/x,x)*b**2*d*e**5 + 60*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**2*d**6*x**3 + 120*log(((sqrt(x)*d + e)**n*c)/x **(n/2))*a*b*d**6*x**3 - 120*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*a*b*e**6 - 30*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**4*e**2*n*x**2 - 60*log( ((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**2*e**4*n*x + 274*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*e**6*n - 120*log(sqrt(x))*a*b*e**6*n + 274*log(s qrt(x))*b**2*e**6*n**2 + 60*a**2*d**6*x**3 - 30*a*b*d**4*e**2*n*x**2 - 60* a*b*d**2*e**4*n*x + 6*b**2*d**4*e**2*n**2*x**2 + 47*b**2*d**2*e**4*n**2*x) /(180*d**6)