\(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^3}{x^3} \, dx\) [420]

Optimal result

Integrand size = 24, antiderivative size = 573 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^4} \] Output:

-1/2*b^3*e^3*n^3/d^3/x^(1/2)+1/2*b^3*e^4*n^3*ln(d+e*x^(1/2))/d^4-1/2*b^2*e 
^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))/d^2/x+5/2*b^2*e^3*n^2*(d+e*x^(1/2))*(a+ 
b*ln(c*(d+e*x^(1/2))^n))/d^4/x^(1/2)+5/2*b^2*e^4*n^2*ln(1-d/(d+e*x^(1/2))) 
*(a+b*ln(c*(d+e*x^(1/2))^n))/d^4-1/2*b*e*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d 
/x^(3/2)+3/4*b*e^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d^2/x-3/2*b*e^3*n*(d+e* 
x^(1/2))*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d^4/x^(1/2)-3/2*b*e^4*n*ln(1-d/(d+e 
*x^(1/2)))*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d^4-1/2*(a+b*ln(c*(d+e*x^(1/2))^n 
))^3/x^2+3*b^2*e^4*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*ln(-e*x^(1/2)/d)/d^4-3/ 
2*b^3*e^4*n^3*ln(x)/d^4-5/2*b^3*e^4*n^3*polylog(2,d/(d+e*x^(1/2)))/d^4+3*b 
^2*e^4*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*polylog(2,d/(d+e*x^(1/2)))/d^4+3*b^ 
3*e^4*n^3*polylog(2,1+e*x^(1/2)/d)/d^4+3*b^3*e^4*n^3*polylog(3,d/(d+e*x^(1 
/2)))/d^4
 

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^3,x]
 

Output:

-1/4*(2*b*d^3*e*n*Sqrt[x]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqr 
t[x])^n])^2 - 3*b*d^2*e^2*n*x*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e 
*Sqrt[x])^n])^2 + 6*b*d*e^3*n*x^(3/2)*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[ 
c*(d + e*Sqrt[x])^n])^2 + 6*b*d^4*n*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e* 
Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 - 6*b*e^4*n*x^2*Log[d + e*Sqrt[x] 
]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 + 2*d^4*(a - 
 b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^3 + 3*b*e^4*n*x^2*(a 
 - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2*Log[x] - 2*b^2*n 
^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])*(-3*(d^4 - e^ 
4*x^2)*Log[d + e*Sqrt[x]]^2 + e^2*x*(-d^2 + 5*d*e*Sqrt[x] + 11*e^2*x*Log[- 
((e*Sqrt[x])/d)]) - Log[d + e*Sqrt[x]]*(2*d^3*e*Sqrt[x] - 3*d^2*e^2*x + 6* 
d*e^3*x^(3/2) + 11*e^4*x^2 + 6*e^4*x^2*Log[-((e*Sqrt[x])/d)]) - 6*e^4*x^2* 
PolyLog[2, 1 + (e*Sqrt[x])/d]) + b^3*n^3*(d^2*e^2*x*(2 - 3*Log[d + e*Sqrt[ 
x]])*Log[d + e*Sqrt[x]] + 2*d^3*e*Sqrt[x]*Log[d + e*Sqrt[x]]^2 + 2*d^4*Log 
[d + e*Sqrt[x]]^3 + 2*d*e^3*x^(3/2)*(1 - 5*Log[d + e*Sqrt[x]] + 3*Log[d + 
e*Sqrt[x]]^2) + 12*e^4*x^2*(-Log[d + e*Sqrt[x]] + Log[-((e*Sqrt[x])/d)]) + 
 11*e^4*x^2*(Log[d + e*Sqrt[x]]*(Log[d + e*Sqrt[x]] - 2*Log[-((e*Sqrt[x])/ 
d)]) - 2*PolyLog[2, 1 + (e*Sqrt[x])/d]) - 2*e^4*x^2*(Log[d + e*Sqrt[x]]^2* 
(Log[d + e*Sqrt[x]] - 3*Log[-((e*Sqrt[x])/d)]) - 6*Log[d + e*Sqrt[x]]*Poly 
Log[2, 1 + (e*Sqrt[x])/d] + 6*PolyLog[3, 1 + (e*Sqrt[x])/d])))/(d^4*x^2...
 

Rubi [A] (warning: unable to verify)

Time = 5.23 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 2789, 2756, 54, 2009, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}

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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle 2 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^{5/2}}d\sqrt {x}\)

\(\Big \downarrow \) 2845

\(\displaystyle 2 \left (\frac {3}{4} b e n \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{\left (d+e \sqrt {x}\right ) x^2}d\sqrt {x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2858

\(\displaystyle 2 \left (\frac {3}{4} b n \int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{x^{5/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^4 x^{5/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^4 x^2}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^2}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^3 x^2}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e^3 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^3 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \int \frac {1}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 54

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt {x}}+\frac {1}{d^2 \sqrt {x}}+\frac {1}{d e^2 x}\right )d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {\int \frac {a+b \log \left (c x^{n/2}\right )}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2751

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {-\frac {b n \int -\frac {1}{e \sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {-\frac {b n \int -\frac {1}{e \sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2755

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \int -\frac {a+b \log \left (c x^{n/2}\right )}{e \sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/2}\right )}{e x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\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}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\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}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\frac {2 b n \int \frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\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 )^2}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\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}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\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}\right )}{d}+\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}}{d}+\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\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}\right )}{d}}{d}+\frac {-\frac {2}{3} b n \left (\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}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 2 \left (\frac {3}{4} b e^4 n \left (\frac {-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/2}\right )}{2 e^2 x}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt {x}\right )}{d^2}-\frac {\log \left (-e \sqrt {x}\right )}{d^2}-\frac {1}{d e \sqrt {x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{3 e^3 x^{3/2}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{2 e^2 x}-b n \left (\frac {\frac {b n \log \left (-e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e \sqrt {x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )+b n \operatorname {PolyLog}\left (3,\frac {d}{\sqrt {x}}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 x^2}\right )\)

Input:

Int[(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^3,x]
 

Output:

2*(-1/4*(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^2 + (3*b*e^4*n*((-1/3*(a + b* 
Log[c*x^(n/2)])^2/(e^3*x^(3/2)) - (2*b*n*((-1/2*(b*n*(-(1/(d*e*Sqrt[x])) + 
 Log[d + e*Sqrt[x]]/d^2 - Log[-(e*Sqrt[x])]/d^2)) + (a + b*Log[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))/3)/d + (((a + b*Log[c*x^(n 
/2)])^2/(2*e^2*x) - b*n*(((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 + e*Sqrt[ 
x])*(a + b*Log[c*x^(n/2)])^2)/(d*e*Sqrt[x])) - (2*b*n*(-(Log[1 - (d + e*Sq 
rt[x])/d]*(a + b*Log[c*x^(n/2)])) - b*n*PolyLog[2, (d + e*Sqrt[x])/d]))/d) 
/d + (-((Log[1 - d/Sqrt[x]]*(a + b*Log[c*x^(n/2)])^2)/d) + (2*b*n*((a + b* 
Log[c*x^(n/2)])*PolyLog[2, d/Sqrt[x]] + b*n*PolyLog[3, d/Sqrt[x]]))/d)/d)/ 
d)/d))/4)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

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]
 

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

rule 2756

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]))
 

rule 2779

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]
 

rule 2789

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]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2838

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]
 

rule 2845

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]))
 

rule 2858

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]
 

rule 2904

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])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]

Input:

int((a+b*ln(c*(d+e*x^(1/2))^n))^3/x^3,x)
 

Output:

int((a+b*ln(c*(d+e*x^(1/2))^n))^3/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/2))^n))^3/x^3,x, algorithm="fricas")
 

Output:

integral((b^3*log((e*sqrt(x) + d)^n*c)^3 + 3*a*b^2*log((e*sqrt(x) + d)^n*c 
)^2 + 3*a^2*b*log((e*sqrt(x) + d)^n*c) + a^3)/x^3, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \] Input:

integrate((a+b*ln(c*(d+e*x**(1/2))**n))**3/x**3,x)
 

Output:

Integral((a + b*log(c*(d + e*sqrt(x))**n))**3/x**3, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/2))^n))^3/x^3,x, algorithm="maxima")
 

Output:

-1/2*b^3*log((e*sqrt(x) + d)^n)^3/x^2 + integrate(1/4*(3*(b^3*e*n*x + 4*(b 
^3*e*log(c) + a*b^2*e)*x + 4*(b^3*d*log(c) + a*b^2*d)*sqrt(x))*log((e*sqrt 
(x) + d)^n)^2 + 4*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) 
+ a^3*e)*x + 12*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d* 
log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*sqrt(x))*log((e*sqrt(x) + d)^n) + 4 
*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*sqrt(x)) 
/(e*x^4 + d*x^(7/2)), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/2))^n))^3/x^3,x, algorithm="giac")
 

Output:

integrate((b*log((e*sqrt(x) + d)^n*c) + a)^3/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^3} \,d x \] Input:

int((a + b*log(c*(d + e*x^(1/2))^n))^3/x^3,x)
 

Output:

int((a + b*log(c*(d + e*x^(1/2))^n))^3/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*(d+e*x^(1/2))^n))^3/x^3,x)
 

Output:

( - 2*sqrt(x)*log((sqrt(x)*e + d)**n*c)**2*b**3*d**3*e*n - 6*sqrt(x)*log(( 
sqrt(x)*e + d)**n*c)**2*b**3*d*e**3*n*x - 4*sqrt(x)*log((sqrt(x)*e + d)**n 
*c)*a*b**2*d**3*e*n - 12*sqrt(x)*log((sqrt(x)*e + d)**n*c)*a*b**2*d*e**3*n 
*x + 10*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**3*d*e**3*n**2*x - 2*sqrt(x)*a 
**2*b*d**3*e*n - 6*sqrt(x)*a**2*b*d*e**3*n*x + 10*sqrt(x)*a*b**2*d*e**3*n* 
*2*x - 2*sqrt(x)*b**3*d*e**3*n**3*x - 3*int(log((sqrt(x)*e + d)**n*c)**2/( 
d**2*x - e**2*x**2),x)*b**3*d**2*e**4*n*x**2 - 6*int(log((sqrt(x)*e + d)** 
n*c)/(d**2*x - e**2*x**2),x)*a*b**2*d**2*e**4*n*x**2 + 11*int(log((sqrt(x) 
*e + d)**n*c)/(d**2*x - e**2*x**2),x)*b**3*d**2*e**4*n**2*x**2 + 3*int((sq 
rt(x)*log((sqrt(x)*e + d)**n*c)**2)/(d**2*x - e**2*x**2),x)*b**3*d*e**5*n* 
x**2 + 6*int((sqrt(x)*log((sqrt(x)*e + d)**n*c))/(d**2*x - e**2*x**2),x)*a 
*b**2*d*e**5*n*x**2 - 11*int((sqrt(x)*log((sqrt(x)*e + d)**n*c))/(d**2*x - 
 e**2*x**2),x)*b**3*d*e**5*n**2*x**2 - 6*log(sqrt(x))*a**2*b*e**4*n*x**2 + 
 22*log(sqrt(x))*a*b**2*e**4*n**2*x**2 - 12*log(sqrt(x))*b**3*e**4*n**3*x* 
*2 - 2*log((sqrt(x)*e + d)**n*c)**3*b**3*d**4 - 6*log((sqrt(x)*e + d)**n*c 
)**2*a*b**2*d**4 + 3*log((sqrt(x)*e + d)**n*c)**2*b**3*d**2*e**2*n*x - 6*l 
og((sqrt(x)*e + d)**n*c)*a**2*b*d**4 + 6*log((sqrt(x)*e + d)**n*c)*a**2*b* 
e**4*x**2 + 6*log((sqrt(x)*e + d)**n*c)*a*b**2*d**2*e**2*n*x - 22*log((sqr 
t(x)*e + d)**n*c)*a*b**2*e**4*n*x**2 - 2*log((sqrt(x)*e + d)**n*c)*b**3*d* 
*2*e**2*n**2*x + 12*log((sqrt(x)*e + d)**n*c)*b**3*e**4*n**2*x**2 - 2*a...