Optimal. Leaf size=113 \[ \frac {6 b^2 n^2 \text {Li}_3\left (-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {3 b n \text {Li}_2\left (-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {d}{e x}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 130, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2344, 2302, 30, 2317, 2374, 2383, 6589} \[ \frac {6 b^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {3 b n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac {6 b^3 n^3 \text {PolyLog}\left (4,-\frac {e x}{d}\right )}{d}-\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2302
Rule 2317
Rule 2344
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x} \, dx}{d}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d}+\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d n}+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d}+\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {e x}{d}\right )}{d}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {e x}{d}\right )}{x} \, dx}{d}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {e x}{d}\right )}{d}-\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {e x}{d}\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.20, size = 243, normalized size = 2.15 \[ \frac {-4 b^2 n^2 \left (6 \text {Li}_3\left (-\frac {e x}{d}\right )-6 \log (x) \text {Li}_2\left (-\frac {e x}{d}\right )+\log ^2(x) \left (\log (x)-3 \log \left (\frac {e x}{d}+1\right )\right )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )+6 b n \left (\log ^2(x)-2 \left (\text {Li}_2\left (-\frac {e x}{d}\right )+\log (x) \log \left (\frac {e x}{d}+1\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-4 \log (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+b^3 n^3 \left (-24 \text {Li}_4\left (-\frac {e x}{d}\right )-12 \log ^2(x) \text {Li}_2\left (-\frac {e x}{d}\right )+24 \log (x) \text {Li}_3\left (-\frac {e x}{d}\right )-4 \log ^3(x) \log \left (\frac {e x}{d}+1\right )+\log ^4(x)\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}}{e x^{2} + d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.87, size = 9909, normalized size = 87.69 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a^{3} {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} + \int \frac {b^{3} \log \relax (c)^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + 3 \, {\left (b^{3} \log \relax (c) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} + 2 \, a b^{2} \log \relax (c) + a^{2} b\right )} \log \left (x^{n}\right )}{e x^{2} + d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________