3.4.24 \(\int \frac {(a+b \log (c x^n))^3}{(d+e x^3)^2} \, dx\) [324]

Optimal. Leaf size=1198 \[ \frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {3 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {6 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 b^3 n^3 \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 \sqrt [3]{-1} b^3 n^3 \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b^3 n^3 \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 n^3 \text {Li}_4\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 n^3 \text {Li}_4\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {12 b^3 n^3 \text {Li}_4\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \]

[Out]

1/9*x*(a+b*ln(c*x^n))^3/d^(5/3)/(d^(1/3)+e^(1/3)*x)-(-1)^(1/3)*x*(a+b*ln(c*x^n))^3/(1+(-1)^(1/3))^4/d^(5/3)/((
-1)^(2/3)*d^(1/3)+e^(1/3)*x)+1/9*x*(a+b*ln(c*x^n))^3/d^(5/3)/(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)-1/3*b*n*(a+b*ln(c*
x^n))^2*ln(1+e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/9*(a+b*ln(c*x^n))^3*ln(1+e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+
3*(-1)^(1/3)*b*n*(a+b*ln(c*x^n))^2*ln(1-(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)+1/3*(-1
)^(1/3)*b*n*(a+b*ln(c*x^n))^2*ln(1+(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-2/3*b^2*n^2*(a+b*ln(c*x^n))*p
olylog(2,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-e^(1/3)*x/d^(1/3))/d^(5/3)/e
^(1/3)+6*(-1)^(1/3)*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e
^(1/3)+2/3*(-1)^(1/3)*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/3*b^3
*n^3*polylog(3,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-4/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-e^(1/3)*x/d^(1/3))/d
^(5/3)/e^(1/3)-6*(-1)^(1/3)*b^3*n^3*polylog(3,(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)-2
/3*(-1)^(1/3)*b^3*n^3*polylog(3,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+4/3*b^3*n^3*polylog(4,-e^(1/3)*
x/d^(1/3))/d^(5/3)/e^(1/3)-4/9*(a+b*ln(c*x^n))^3*ln(1-1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1-
I*3^(1/2))-4/3*b*n*(a+b*ln(c*x^n))^2*polylog(2,1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1-I*3^(1/
2))+8/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1-I*3^(1/2))-8
/3*b^3*n^3*polylog(4,1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1-I*3^(1/2))-4/9*(a+b*ln(c*x^n))^3*
ln(1-1/2*e^(1/3)*x*(1+I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1+I*3^(1/2))-4/3*b*n*(a+b*ln(c*x^n))^2*polylog(2,1/
2*e^(1/3)*x*(1+I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1+I*3^(1/2))+8/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,1/2*e^(
1/3)*x*(1+I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1+I*3^(1/2))-8/3*b^3*n^3*polylog(4,1/2*e^(1/3)*x*(1+I*3^(1/2))/
d^(1/3))/d^(5/3)/e^(1/3)/(1+I*3^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 1.06, antiderivative size = 1198, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2367, 2355, 2354, 2421, 6724, 2430} \begin {gather*} \frac {2 b^3 \text {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 \sqrt [3]{-1} b^3 \text {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b^3 \text {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 \text {PolyLog}\left (4,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 \text {PolyLog}\left (4,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {12 b^3 \text {PolyLog}\left (4,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {6 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {12 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {3 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {6 b \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3/(d + e*x^3)^2,x]

[Out]

(x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^3)/((1 + (-1)^(1
/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e
^(1/3)*x)) - (b*n*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n
])^3*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (3*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^
(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*(a + b*Log[c*x^n])^3*Log[1 -
((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*L
og[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^3*Log[1 + ((-1)^(2/3)*e^(1
/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)
/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (2*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e
^(1/3)) + (6*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3
))^4*d^(5/3)*e^(1/3)) - ((6*I)*Sqrt[3]*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((
1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(((-1)^(2/3)*e^(1/3)
*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (6*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]
)/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b^3*n^3*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) -
(4*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (6*(-1)^(1/3)*b^3*n^3*
PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + ((12*I)*Sqrt[3]*b^2*n^2*(a
+ b*Log[c*x^n])*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) - (2*(-1)^(1/
3)*b^3*n^3*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (12*b^2*n^2*(a + b*Log[c*x^n])
*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (4*b^3*n^3*PolyLog[4, -
((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - ((12*I)*Sqrt[3]*b^3*n^3*PolyLog[4, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3
)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (12*b^3*n^3*PolyLog[4, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-
1)^(1/3))^4*d^(5/3)*e^(1/3))

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[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 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[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 2367

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2421

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] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2}-\frac {2 (-1)^{5/6} \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}+\frac {2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=\frac {2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac {2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac {\left (2 (-1)^{5/6} \sqrt {3}\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{5/3}}+\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{3 d^{5/3}}-\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{3 d^{5/3}}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 \sqrt [3]{-1} b n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (6 i \sqrt {3} b n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (4 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 \sqrt [3]{-1} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (12 i \sqrt {3} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {4 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (4 \sqrt [3]{-1} b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} b^3 n^3\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (12 i \sqrt {3} b^3 n^3\right ) \int \frac {\text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b^3 n^3 \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 (-1)^{2/3} b^3 n^3 \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b^3 n^3 \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 n^3 \text {Li}_4\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 n^3 \text {Li}_4\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {4 \sqrt [3]{-1} b^3 n^3 \text {Li}_4\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}\\ \end {align*}

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Mathematica [A]
time = 7.21, size = 2215, normalized size = 1.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3/(d + e*x^3)^2,x]

[Out]

(x*(a + b*(-(n*Log[x]) + Log[c*x^n]))^3)/(3*d*(d + e*x^3)) + (2*ArcTan[(-d^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/
3))]*(a + b*(-(n*Log[x]) + Log[c*x^n]))^3)/(3*Sqrt[3]*d^(5/3)*e^(1/3)) + (2*(a + b*(-(n*Log[x]) + Log[c*x^n]))
^3*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) - ((a + b*(-(n*Log[x]) + Log[c*x^n]))^3*Log[d^(2/3) - d^(1/3)
*e^(1/3)*x + e^(2/3)*x^2])/(9*d^(5/3)*e^(1/3)) + 3*b*n*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2*(-1/3*((-1 + (-1)^
(1/3))*((-((-1)^(1/3)/d^(1/3)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1))*Log[x] + ((-1)^(1/3)*Log[-((-1)^(2/3)*
d^(1/3)) - e^(1/3)*x])/d^(1/3)))/((1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) + ((-1)^(1/3)*((d^(-1/3) - (d^(1/3) + e^
(1/3)*x)^(-1))*Log[x] - Log[d^(1/3) + e^(1/3)*x]/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) - (Log[x]/(e
^(1/3)*((-1)^(1/3)*d^(1/3) - e^(1/3)*x)) - (-(((-1)^(2/3)*Log[x])/d^(1/3)) + ((-1)^(2/3)*Log[d^(1/3) + (-1)^(2
/3)*e^(1/3)*x])/d^(1/3))/e^(1/3))/(3*(1 + (-1)^(1/3))^2*d^(4/3)) + (2*(-1)^(1/3)*(Log[x]*Log[1 + (e^(1/3)*x)/d
^(1/3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]*Log[1 - ((-
1)^(1/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1
/3)) - (2*(-1 + (-1)^(1/3))*(Log[x]*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, -(((-1)^(2/3)*e^(1/3)
*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3))) + 3*b^2*n^2*(a + b*(-(n*Log[x]) + Log[c*x^n]))*(((-1)^
(1/3)*(Log[x]*((e^(1/3)*x*Log[x])/(d^(1/3) + e^(1/3)*x) - 2*Log[1 + (e^(1/3)*x)/d^(1/3)]) - 2*PolyLog[2, -((e^
(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - ((-1 + (-1)^(1/3))*(Log[x]*((-((-1)^(1/3)/d^(1/3
)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1))*Log[x] + (2*(-1)^(1/3)*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/d^
(1/3)) + (2*(-1)^(1/3)*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(
1/3)) - (Log[x]*((-1)^(2/3)*e^(1/3)*x*Log[x] - 2*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)*Log[1 + ((-1)^(2/3)*e^(1/3)*
x)/d^(1/3)]) - 2*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*(1 + (-1)^
(1/3))^2*d^(4/3)*(-((-1)^(1/3)*d^(2/3)*e^(1/3)) + d^(1/3)*e^(2/3)*x)) + (2*(-1)^(1/3)*(Log[x]^2*Log[1 + (e^(1/
3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, -((e^(1/3)*x)/d^(1/3))] - 2*PolyLog[3, -((e^(1/3)*x)/d^(1/3))]))/(3*(1 +
(-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, ((
-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - 2*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e
^(1/3)) - (2*(-1 + (-1)^(1/3))*(Log[x]^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, -(((-1)
^(2/3)*e^(1/3)*x)/d^(1/3))] - 2*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*
e^(1/3))) + b^3*n^3*(((-1)^(1/3)*(Log[x]^2*((d^(-1/3) - (d^(1/3) + e^(1/3)*x)^(-1))*Log[x] - (3*Log[1 + (e^(1/
3)*x)/d^(1/3)])/d^(1/3)) - (6*Log[x]*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/d^(1/3) + (6*PolyLog[3, -((e^(1/3)*x)
/d^(1/3))])/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) - ((-1 + (-1)^(1/3))*(-(((-1)^(1/3)*Log[x]^3)/d^(
1/3)) - Log[x]^3/((-1)^(2/3)*d^(1/3) + e^(1/3)*x) + (3*(-1)^(1/3)*Log[x]^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1
/3)])/d^(1/3) + (6*(-1)^(1/3)*(Log[x]*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - PolyLog[3, ((-1)^(1/3)*e^(1
/3)*x)/d^(1/3)]))/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) - (((-1)^(2/3)*Log[x]^3)/d^(1/3) + Log[x]^3
/((-1)^(1/3)*d^(1/3) - e^(1/3)*x) - (3*(-1)^(2/3)*Log[x]^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/d^(1/3) -
(6*(-1)^(2/3)*(Log[x]*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))] - PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1
/3))]))/d^(1/3))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) + (2*(-1)^(1/3)*(Log[x]^3*Log[1 + (e^(1/3)*x)/d^(1/3)]
+ 3*Log[x]^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))] - 6*Log[x]*PolyLog[3, -((e^(1/3)*x)/d^(1/3))] + 6*PolyLog[4,
-((e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]^3*Log[1 - ((-1)^(1/3)*e^(1/3)*x)
/d^(1/3)] + 3*Log[x]^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - 6*Log[x]*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)
/d^(1/3)] + 6*PolyLog[4, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(-1 + (
-1)^(1/3))*(Log[x]^3*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + 3*Log[x]^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/
d^(1/3))] - 6*Log[x]*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))] + 6*PolyLog[4, -(((-1)^(2/3)*e^(1/3)*x)/d^(
1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{\left (e \,x^{3}+d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^3*(2*sqrt(3)*arctan(-1/3*sqrt(3)*(d^(1/3)*e^(-1/3) - 2*x)*e^(1/3)/d^(1/3))*e^(-1/3)/d^(5/3) - e^(-1/3)*l
og(-d^(1/3)*x*e^(-1/3) + x^2 + d^(2/3)*e^(-2/3))/d^(5/3) + 2*e^(-1/3)*log(d^(1/3)*e^(-1/3) + x)/d^(5/3) + 3*x/
(d*x^3*e + d^2)) + integrate((b^3*log(c)^3 + b^3*log(x^n)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + 3*(b^3*log(c
) + a*b^2)*log(x^n)^2 + 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x^n))/(x^6*e^2 + 2*d*x^3*e + d^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)/(x^6*e^2 + 2*d*x^3*e + d^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (e\,x^3+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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