Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{c (a+b x)^2+c}},x\right ) \]
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Rubi [A] time = 0.17908, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \tan ^{-1}(x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.312256, size = 225, normalized size = 7.26 \[ -\frac{3 \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )^{2/3} \left (\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{(a+b x)^2+1}\right )}{\left ((a+b x)^2+1\right )^2}+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{24 (a+b x) \tan ^{-1}(a+b x) \text{Hypergeometric2F1}\left (1,\frac{4}{3},\frac{11}{6},\frac{1}{(a+b x)^2+1}\right )}{\left ((a+b x)^2+1\right )^2}+\frac{90}{(a+b x)^2+1}+5 \tan ^{-1}(a+b x) \left (6 \sin \left (2 \tan ^{-1}(a+b x)\right )-4 (a+b x)\right )+15\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.3, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{2}\arctan \left ( bx+a \right ){\frac{1}{\sqrt [3]{ \left ({a}^{2}+1 \right ) c+2\,abcx+{b}^{2}c{x}^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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