Optimal. Leaf size=118 \[ \frac{\text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{2} x^2 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
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Rubi [A] time = 0.0650263, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2532, 2531, 2282, 6589} \[ \frac{\text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{2} x^2 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2532
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac{1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{2} x^2 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac{1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{2} x^2 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{\int \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac{1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{2} x^2 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^2 c^2 n \log ^2(f)}\\ &=\frac{1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{2} x^2 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{\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)}\\ \end{align*}
Mathematica [A] time = 0.0054364, size = 118, normalized size = 1. \[ \frac{\text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{2} x^2 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 598, normalized size = 5.1 \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}{6} \, b c n x^{3} \log \left (f\right ) + b c d n \int \frac{x^{2}}{2 \,{\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}{2} \, x^{2} \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 = 2.08905, size = 382, normalized size = 3.24 \begin{align*} -\frac{2 \, b c n x{\rm Li}_2\left (-\frac{e f^{b c n x + a c n} + d}{d} + 1\right ) \log \left (f\right ) -{\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \left (f\right )^{2} +{\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \left (f\right )^{2} \log \left (\frac{e f^{b c n x + a c n} + d}{d}\right ) - 2 \,{\rm polylog}\left (3, -\frac{e f^{b c n x + a c n}}{d}\right )}{2 \, b^{2} c^{2} n^{2} \log \left (f\right )^{2}} \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^{2} 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}{2} + \frac{x^{2} \log{\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{2} \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 \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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