Optimal. Leaf size=297 \[ \frac {d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 e}+\frac {a d^2 x}{e^3}-\frac {b d^2 \log \left (c^2 x^2+1\right )}{2 c e^3}-\frac {b d \tan ^{-1}(c x)}{2 c^2 e^2}+\frac {b \log \left (c^2 x^2+1\right )}{6 c^3 e}-\frac {i b d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e^4}+\frac {i b d^3 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {b d^2 x \tan ^{-1}(c x)}{e^3}+\frac {b d x}{2 c e^2}-\frac {b x^2}{6 c e} \]
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Rubi [A] time = 0.27, antiderivative size = 297, normalized size of antiderivative = 1.00, 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.
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Rule 43
Rule 203
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 2447
Rule 4846
Rule 4852
Rule 4856
Rule 4876
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*}
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Mathematica [A] time = 3.57, size = 484, normalized size = 1.63 \[ -\frac {6 a d^3 \log (d+e x)-6 a d^2 e x+3 a d e^2 x^2-2 a e^3 x^3+\frac {b e^3}{c^3}+\frac {3}{2} \pi b d^3 \log \left (c^2 x^2+1\right )-\frac {3 b d^2 e \sqrt {\frac {c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}}{c}+\frac {3 b d^2 e \log \left (c^2 x^2+1\right )}{c}+\frac {3 b d e^2 \tan ^{-1}(c x)}{c^2}-\frac {b e^3 \log \left (c^2 x^2+1\right )}{c^3}-3 i b d^3 \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-6 i b d^3 \tan ^{-1}(c x) \tan ^{-1}\left (\frac {c d}{e}\right )+6 b d^3 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+6 b d^3 \tan ^{-1}(c x) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-6 b d^3 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )+3 i b d^3 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+3 i b d^3 \tan ^{-1}(c x)^2+3 i \pi b d^3 \tan ^{-1}(c x)+3 \pi b d^3 \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-6 b d^3 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\frac {3 b d^2 e \tan ^{-1}(c x)^2}{c}-6 b d^2 e x \tan ^{-1}(c x)+3 b d e^2 x^2 \tan ^{-1}(c x)-\frac {3 b d e^2 x}{c}-2 b e^3 x^3 \tan ^{-1}(c x)+\frac {b e^3 x^2}{c}}{6 e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \arctan \left (c x\right ) + a x^{3}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 394, normalized size = 1.33 \[ \frac {a \,x^{3}}{3 e}-\frac {a \,x^{2} d}{2 e^{2}}+\frac {a \,d^{2} x}{e^{3}}-\frac {a \,d^{3} \ln \left (c e x +d c \right )}{e^{4}}+\frac {b \arctan \left (c x \right ) x^{3}}{3 e}-\frac {b \arctan \left (c x \right ) x^{2} d}{2 e^{2}}+\frac {b \,d^{2} x \arctan \left (c x \right )}{e^{3}}-\frac {b \arctan \left (c x \right ) d^{3} \ln \left (c e x +d c \right )}{e^{4}}-\frac {b \ln \left (c^{2} d^{2}-2 \left (c e x +d c \right ) c d +\left (c e x +d c \right )^{2}+e^{2}\right ) d^{2}}{2 c \,e^{3}}-\frac {b d \arctan \left (c x \right )}{2 c^{2} e^{2}}+\frac {b \ln \left (c^{2} d^{2}-2 \left (c e x +d c \right ) c d +\left (c e x +d c \right )^{2}+e^{2}\right )}{6 c^{3} e}+\frac {b d x}{2 c \,e^{2}}+\frac {2 b \,d^{2}}{3 c \,e^{3}}-\frac {b \,x^{2}}{6 c e}-\frac {i b \,d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e^{4}}-\frac {i b \,d^{3} \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e^{4}}+\frac {i b \,d^{3} \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e^{4}}+\frac {i b \,d^{3} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + 2 \, b \int \frac {x^{3} \arctan \left (c x\right )}{2 \, {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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