Optimal. Leaf size=297 \[ -\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^4}+\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^4}+\frac{d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{a d^2 x}{e^3}-\frac{b d^2 \log \left (c^2 x^2+1\right )}{2 c e^3}-\frac{b d \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left (c^2 x^2+1\right )}{6 c^3 e}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}+\frac{b d x}{2 c e^2}-\frac{b x^2}{6 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.270136, antiderivative size = 297, 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 = {4876, 4846, 260, 4852, 321, 203, 266, 43, 4856, 2402, 2315, 2447} \[ -\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^4}+\frac{i b d^3 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^4}+\frac{d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{a d^2 x}{e^3}-\frac{b d^2 \log \left (c^2 x^2+1\right )}{2 c e^3}-\frac{b d \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left (c^2 x^2+1\right )}{6 c^3 e}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}+\frac{b d x}{2 c e^2}-\frac{b x^2}{6 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4876
Rule 4846
Rule 260
Rule 4852
Rule 321
Rule 203
Rule 266
Rule 43
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{e^3}-\frac{d x \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{e}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d^2 \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e^3}-\frac{d^3 \int \frac{a+b \tan ^{-1}(c x)}{d+e x} \, dx}{e^3}-\frac{d \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e^2}+\frac{\int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}\\ &=\frac{a d^2 x}{e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac{\left (b c d^3\right ) \int \frac{\log \left (\frac{2}{1-i 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+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{e^4}+\frac{\left (b d^2\right ) \int \tan ^{-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 \tan ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac{i b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac{\left (i b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i 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 \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac{b d^2 \log \left (1+c^2 x^2\right )}{2 c e^3}-\frac{i b d^3 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e^4}+\frac{i b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i 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 \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}-\frac{d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i 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{i b d^3 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e^4}+\frac{i b d^3 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}\\ \end{align*}
Mathematica [A] time = 3.28679, size = 484, normalized size = 1.63 \[ -\frac{-3 i b d^3 \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+3 i b d^3 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-6 a d^2 e x+6 a d^3 \log (d+e x)+3 a d e^2 x^2-2 a e^3 x^3-\frac{3 b d^2 e \sqrt{\frac{c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac{c d}{e}\right )}}{c}+\frac{3 b d^2 e \log \left (c^2 x^2+1\right )}{c}+\frac{3}{2} \pi b d^3 \log \left (c^2 x^2+1\right )+\frac{3 b d e^2 \tan ^{-1}(c x)}{c^2}-\frac{b e^3 \log \left (c^2 x^2+1\right )}{c^3}+\frac{b e^3}{c^3}-6 i b d^3 \tan ^{-1}(c x) \tan ^{-1}\left (\frac{c d}{e}\right )+\frac{3 b d^2 e \tan ^{-1}(c x)^2}{c}-6 b d^2 e x \tan ^{-1}(c x)+6 b d^3 \tan ^{-1}\left (\frac{c d}{e}\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+6 b d^3 \tan ^{-1}(c x) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-6 b d^3 \tan ^{-1}\left (\frac{c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )\right )+3 i b d^3 \tan ^{-1}(c x)^2+3 i \pi b d^3 \tan ^{-1}(c x)+3 \pi b d^3 \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-6 b d^3 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+3 b d e^2 x^2 \tan ^{-1}(c x)-\frac{3 b d e^2 x}{c}+\frac{b e^3 x^2}{c}-2 b e^3 x^3 \tan ^{-1}(c x)}{6 e^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 394, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}a}{3\,e}}-{\frac{ad{x}^{2}}{2\,{e}^{2}}}+{\frac{a{d}^{2}x}{{e}^{3}}}-{\frac{a{d}^{3}\ln \left ( ecx+dc \right ) }{{e}^{4}}}+{\frac{b{x}^{3}\arctan \left ( cx \right ) }{3\,e}}-{\frac{\arctan \left ( cx \right ) bd{x}^{2}}{2\,{e}^{2}}}+{\frac{b{d}^{2}x\arctan \left ( cx \right ) }{{e}^{3}}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) \ln \left ( ecx+dc \right ) }{{e}^{4}}}-{\frac{b\ln \left ({c}^{2}{d}^{2}-2\, \left ( ecx+dc \right ) cd+ \left ( ecx+dc \right ) ^{2}+{e}^{2} \right ){d}^{2}}{2\,c{e}^{3}}}-{\frac{\arctan \left ( cx \right ) bd}{2\,{c}^{2}{e}^{2}}}+{\frac{b\ln \left ({c}^{2}{d}^{2}-2\, \left ( ecx+dc \right ) cd+ \left ( ecx+dc \right ) ^{2}+{e}^{2} \right ) }{6\,{c}^{3}e}}+{\frac{bdx}{2\,c{e}^{2}}}+{\frac{2\,b{d}^{2}}{3\,c{e}^{3}}}-{\frac{b{x}^{2}}{6\,ce}}-{\frac{{\frac{i}{2}}b{d}^{3}}{{e}^{4}}{\it dilog} \left ({\frac{ie-ecx}{dc+ie}} \right ) }+{\frac{{\frac{i}{2}}b{d}^{3}\ln \left ( ecx+dc \right ) }{{e}^{4}}\ln \left ({\frac{ie+ecx}{ie-dc}} \right ) }+{\frac{{\frac{i}{2}}b{d}^{3}}{{e}^{4}}{\it dilog} \left ({\frac{ie+ecx}{ie-dc}} \right ) }-{\frac{{\frac{i}{2}}b{d}^{3}\ln \left ( ecx+dc \right ) }{{e}^{4}}\ln \left ({\frac{ie-ecx}{dc+ie}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )} + 2 \, b \int \frac{x^{3} \arctan \left (c x\right )}{2 \,{\left (e x + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \arctan \left (c x\right ) + a x^{3}}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \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.
[In]
[Out]