Optimal. Leaf size=275 \[ -\frac{b d^3 \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 e^4}+\frac{b d^3 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^4}+\frac{d^3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^4}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{a d^2 x}{e^3}+\frac{b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac{b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left (1-c^2 x^2\right )}{6 c^3 e}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{b d x}{2 c e^2}+\frac{b x^2}{6 c e} \]
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Rubi [A] time = 0.255716, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {5940, 5910, 260, 5916, 321, 206, 266, 43, 5920, 2402, 2315, 2447} \[ -\frac{b d^3 \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 e^4}+\frac{b d^3 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^4}+\frac{d^3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^4}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{a d^2 x}{e^3}+\frac{b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac{b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left (1-c^2 x^2\right )}{6 c^3 e}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{b d x}{2 c e^2}+\frac{b x^2}{6 c e} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5916
Rule 321
Rule 206
Rule 266
Rule 43
Rule 5920
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )}{e^3}-\frac{d x \left (a+b \tanh ^{-1}(c x)\right )}{e^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{e}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d^2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e^3}-\frac{d^3 \int \frac{a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{e^3}-\frac{d \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e^2}+\frac{\int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e}\\ &=\frac{a d^2 x}{e^3}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}-\frac{\left (b c d^3\right ) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{e^4}+\frac{\left (b c d^3\right ) \int \frac{\log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{e^4}+\frac{\left (b d^2\right ) \int \tanh ^{-1}(c x) \, dx}{e^3}+\frac{(b c d) \int \frac{x^2}{1-c^2 x^2} \, dx}{2 e^2}-\frac{(b c) \int \frac{x^3}{1-c^2 x^2} \, dx}{3 e}\\ &=\frac{a d^2 x}{e^3}-\frac{b d x}{2 c e^2}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac{b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}-\frac{\left (b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{e^4}-\frac{\left (b c d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{e^3}+\frac{(b d) \int \frac{1}{1-c^2 x^2} \, dx}{2 c e^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{6 e}\\ &=\frac{a d^2 x}{e^3}-\frac{b d x}{2 c e^2}+\frac{b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac{b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}-\frac{b d^3 \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 e^4}+\frac{b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}-\frac{(b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 e}\\ &=\frac{a d^2 x}{e^3}-\frac{b d x}{2 c e^2}+\frac{b x^2}{6 c e}+\frac{b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^4}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac{b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac{b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac{b d^3 \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 e^4}+\frac{b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4}\\ \end{align*}
Mathematica [C] time = 6.67452, size = 474, normalized size = 1.72 \[ \frac{3 b d^3 \text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-3 b d^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+6 a d^2 e x-6 a d^3 \log (d+e x)-3 a d e^2 x^2+2 a e^3 x^3+\frac{3 b d^2 e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac{c d}{e}\right )}}{c}+\frac{3 b d^2 e \log \left (1-c^2 x^2\right )}{c}+\frac{3}{2} i \pi b d^3 \log \left (1-c^2 x^2\right )+\frac{3 b d e^2 \tanh ^{-1}(c x)}{c^2}+\frac{b e^3 \log \left (1-c^2 x^2\right )}{c^3}-\frac{b e^3}{c^3}-6 b d^3 \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac{c d}{e}\right )-\frac{3 b d^2 e \tanh ^{-1}(c x)^2}{c}+6 b d^2 e x \tanh ^{-1}(c x)-6 b d^3 \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-6 b d^3 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+6 b d^3 \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+3 b d^3 \tanh ^{-1}(c x)^2-3 i \pi b d^3 \tanh ^{-1}(c x)+6 b d^3 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+3 i \pi b d^3 \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )-3 b d e^2 x^2 \tanh ^{-1}(c x)-\frac{3 b d e^2 x}{c}+\frac{b e^3 x^2}{c}+2 b e^3 x^3 \tanh ^{-1}(c x)}{6 e^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.125, size = 381, normalized size = 1.4 \begin{align*}{\frac{{x}^{3}a}{3\,e}}-{\frac{da{x}^{2}}{2\,{e}^{2}}}+{\frac{ax{d}^{2}}{{e}^{3}}}-{\frac{a{d}^{3}\ln \left ( cxe+cd \right ) }{{e}^{4}}}+{\frac{b{\it Artanh} \left ( cx \right ){x}^{3}}{3\,e}}-{\frac{b{\it Artanh} \left ( cx \right ){x}^{2}d}{2\,{e}^{2}}}+{\frac{b{d}^{2}x{\it Artanh} \left ( cx \right ) }{{e}^{3}}}-{\frac{b{\it Artanh} \left ( cx \right ){d}^{3}\ln \left ( cxe+cd \right ) }{{e}^{4}}}+{\frac{b{d}^{3}\ln \left ( cxe+cd \right ) }{2\,{e}^{4}}\ln \left ({\frac{cxe+e}{-cd+e}} \right ) }+{\frac{b{d}^{3}}{2\,{e}^{4}}{\it dilog} \left ({\frac{cxe+e}{-cd+e}} \right ) }-{\frac{b{d}^{3}\ln \left ( cxe+cd \right ) }{2\,{e}^{4}}\ln \left ({\frac{cxe-e}{-cd-e}} \right ) }-{\frac{b{d}^{3}}{2\,{e}^{4}}{\it dilog} \left ({\frac{cxe-e}{-cd-e}} \right ) }-{\frac{bdx}{2\,c{e}^{2}}}-{\frac{2\,b{d}^{2}}{3\,c{e}^{3}}}+{\frac{b{x}^{2}}{6\,ce}}+{\frac{b\ln \left ( cxe+e \right ){d}^{2}}{2\,c{e}^{3}}}+{\frac{b\ln \left ( cxe+e \right ) d}{4\,{c}^{2}{e}^{2}}}+{\frac{b\ln \left ( cxe+e \right ) }{6\,{c}^{3}e}}+{\frac{b\ln \left ( cxe-e \right ){d}^{2}}{2\,c{e}^{3}}}-{\frac{b\ln \left ( cxe-e \right ) d}{4\,{c}^{2}{e}^{2}}}+{\frac{b\ln \left ( cxe-e \right ) }{6\,{c}^{3}e}} \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} \, a{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac{1}{2} \, b \int \frac{x^{3}{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \operatorname{artanh}\left (c x\right ) + a x^{3}}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )} x^{3}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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