Optimal. Leaf size=65 \[ \frac{\left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{5 a^2 c}-\frac{\text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{10 a} \]
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Rubi [A] time = 0.12153, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx \]
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 )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx &=\frac{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{5 a^2 c}-\frac{\int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{10 a}\\ \end{align*}
Mathematica [A] time = 7.82895, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.783, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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 [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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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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