Optimal. Leaf size=60 \[ \frac{c^2 \left (a^2 x^2+1\right )^3 \sqrt{\tan ^{-1}(a x)}}{6 a^2}-\frac{\text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\sqrt{\tan ^{-1}(a x)}},x\right )}{12 a} \]
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Rubi [A] time = 0.0720387, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{6 a^2}-\frac{\int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{12 a}\\ \end{align*}
Mathematica [A] time = 1.77285, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.527, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}\sqrt{\arctan \left ( ax \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx + \int 2 a^{2} x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx + \int a^{4} x^{5} \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx\right ) \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{\left (a^{2} c x^{2} + c\right )}^{2} x \sqrt{\arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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