Optimal. Leaf size=156 \[ \frac{2 x \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}-\frac{x^2 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{3} x^3 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
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Rubi [A] time = 0.0930105, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2532, 2531, 6609, 2282, 6589} \[ \frac{2 x \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}-\frac{x^2 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{3} x^3 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2532
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac{1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{3} x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x^2 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac{1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{3} x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^2 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{2 \int x \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac{1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{3} x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^2 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{2 x \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \int \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^2 c^2 n^2 \log ^2(f)}\\ &=\frac{1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{3} x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^2 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{2 x \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^3 c^3 n^2 \log ^3(f)}\\ &=\frac{1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{3} x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^2 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{2 x \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \text{Li}_4\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}\\ \end{align*}
Mathematica [A] time = 0.0066497, size = 156, normalized size = 1. \[ \frac{2 x \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{2 \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}-\frac{x^2 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{3} x^3 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 980, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{12} \, b c n x^{4} \log \left (f\right ) + b c d n \int \frac{x^{3}}{3 \,{\left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d\right )}}\,{d x} \log \left (f\right ) + \frac{1}{3} \, x^{3} \log \left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.19593, size = 471, normalized size = 3.02 \begin{align*} -\frac{3 \, b^{2} c^{2} n^{2} x^{2}{\rm Li}_2\left (-\frac{e f^{b c n x + a c n} + d}{d} + 1\right ) \log \left (f\right )^{2} - 6 \, b c n x \log \left (f\right ){\rm polylog}\left (3, -\frac{e f^{b c n x + a c n}}{d}\right ) -{\left (b^{3} c^{3} n^{3} x^{3} + a^{3} c^{3} n^{3}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \left (f\right )^{3} +{\left (b^{3} c^{3} n^{3} x^{3} + a^{3} c^{3} n^{3}\right )} \log \left (f\right )^{3} \log \left (\frac{e f^{b c n x + a c n} + d}{d}\right ) + 6 \,{\rm polylog}\left (4, -\frac{e f^{b c n x + a c n}}{d}\right )}{3 \, b^{3} c^{3} n^{3} \log \left (f\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x^{3} e^{b c n x \log{\left (f \right )}}}{d + e e^{a c n \log{\left (f \right )}} e^{b c n x \log{\left (f \right )}}}\, dx}{3} + \frac{x^{3} \log{\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log \left (e{\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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