Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{x \left (a^2 c x^2+c\right )^{3/2}}{\tan ^{-1}(a x)^3},x\right ) \]
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Rubi [A] time = 0.0800006, 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 = {} \[ \int \frac{x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx &=\int \frac{x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx\\ \end{align*}
Mathematica [A] time = 6.04173, size = 0, normalized size = 0. \[ \int \frac{x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.696, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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 (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}{\arctan \left (a x\right )^{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 (a^{2} c x^{3} + c x\right )} \sqrt{a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{3}}, x\right ) \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*} \int \frac{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}{\operatorname{atan}^{3}{\left (a x \right )}}\, dx \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 (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}{\arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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