Optimal. Leaf size=237 \[ \frac{i b d^2 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^3}-\frac{i b d^2 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^3}-\frac{d^2 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{a d x}{e^2}+\frac{b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b d x \tan ^{-1}(c x)}{e^2}-\frac{b x}{2 c e} \]
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Rubi [A] time = 0.208499, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {4876, 4846, 260, 4852, 321, 203, 4856, 2402, 2315, 2447} \[ \frac{i b d^2 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^3}-\frac{i b d^2 \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^3}-\frac{d^2 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{a d x}{e^2}+\frac{b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b d x \tan ^{-1}(c x)}{e^2}-\frac{b x}{2 c e} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 260
Rule 4852
Rule 321
Rule 203
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{e}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{d \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e^2}+\frac{d^2 \int \frac{a+b \tan ^{-1}(c x)}{d+e x} \, dx}{e^2}+\frac{\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac{a d x}{e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^3}+\frac{d^2 \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^3}+\frac{\left (b c d^2\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{e^3}-\frac{\left (b c d^2\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^3}-\frac{(b d) \int \tan ^{-1}(c x) \, dx}{e^2}-\frac{(b c) \int \frac{x^2}{1+c^2 x^2} \, dx}{2 e}\\ &=-\frac{a d x}{e^2}-\frac{b x}{2 c e}-\frac{b d x \tan ^{-1}(c x)}{e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^3}+\frac{d^2 \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^3}-\frac{i b d^2 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}+\frac{\left (i b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{e^3}+\frac{(b c d) \int \frac{x}{1+c^2 x^2} \, dx}{e^2}+\frac{b \int \frac{1}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac{a d x}{e^2}-\frac{b x}{2 c e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b d x \tan ^{-1}(c x)}{e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^3}+\frac{d^2 \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^3}+\frac{b d \log \left (1+c^2 x^2\right )}{2 c e^2}+\frac{i b d^2 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e^3}-\frac{i b d^2 \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}\\ \end{align*}
Mathematica [A] time = 1.52991, size = 404, normalized size = 1.7 \[ \frac{-i b d^2 \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+i b d^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+2 a d^2 \log (d+e x)-2 a d e x+a e^2 x^2-\frac{b d e \sqrt{\frac{c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac{c d}{e}\right )}}{c}+\frac{1}{2} \pi b d^2 \log \left (c^2 x^2+1\right )+\frac{b d e \log \left (c^2 x^2+1\right )}{c}+\frac{b e^2 \tan ^{-1}(c x)}{c^2}-2 i b d^2 \tan ^{-1}(c x) \tan ^{-1}\left (\frac{c d}{e}\right )+2 b d^2 \tan ^{-1}\left (\frac{c d}{e}\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 b d^2 \tan ^{-1}(c x) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-2 b d^2 \tan ^{-1}\left (\frac{c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac{c d}{e}\right )+\tan ^{-1}(c x)\right )\right )+i b d^2 \tan ^{-1}(c x)^2+i \pi b d^2 \tan ^{-1}(c x)+\pi b d^2 \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-2 b d^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\frac{b d e \tan ^{-1}(c x)^2}{c}-2 b d e x \tan ^{-1}(c x)+b e^2 x^2 \tan ^{-1}(c x)-\frac{b e^2 x}{c}}{2 e^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.05, size = 305, normalized size = 1.3 \begin{align*}{\frac{a{x}^{2}}{2\,e}}-{\frac{adx}{{e}^{2}}}+{\frac{a{d}^{2}\ln \left ( ecx+dc \right ) }{{e}^{3}}}+{\frac{b{x}^{2}\arctan \left ( cx \right ) }{2\,e}}-{\frac{bdx\arctan \left ( cx \right ) }{{e}^{2}}}+{\frac{b{d}^{2}\arctan \left ( cx \right ) \ln \left ( ecx+dc \right ) }{{e}^{3}}}+{\frac{{\frac{i}{2}}b{d}^{2}\ln \left ( ecx+dc \right ) }{{e}^{3}}\ln \left ({\frac{ie-ecx}{dc+ie}} \right ) }-{\frac{{\frac{i}{2}}b{d}^{2}\ln \left ( ecx+dc \right ) }{{e}^{3}}\ln \left ({\frac{ie+ecx}{ie-dc}} \right ) }+{\frac{{\frac{i}{2}}b{d}^{2}}{{e}^{3}}{\it dilog} \left ({\frac{ie-ecx}{dc+ie}} \right ) }-{\frac{{\frac{i}{2}}b{d}^{2}}{{e}^{3}}{\it dilog} \left ({\frac{ie+ecx}{ie-dc}} \right ) }+{\frac{bd\ln \left ({c}^{2}{d}^{2}-2\, \left ( ecx+dc \right ) cd+ \left ( ecx+dc \right ) ^{2}+{e}^{2} \right ) }{2\,c{e}^{2}}}+{\frac{b\arctan \left ( cx \right ) }{2\,{c}^{2}e}}-{\frac{bx}{2\,ce}}-{\frac{bd}{2\,c{e}^{2}}} \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}{2} \, a{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} + 2 \, b \int \frac{x^{2} \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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{2} \arctan \left (c x\right ) + a x^{2}}{e x + d}, x\right ) \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*} \int \frac{x^{2} \left (a + b \operatorname{atan}{\left (c x \right )}\right )}{d + e x}\, dx \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 \arctan \left (c x\right ) + a\right )} x^{2}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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