Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2 \tan ^{-1}(a x)^{3/2}}{x},x\right ) \]
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Rubi [A] time = 0.0503347, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx &=\int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx\\ \end{align*}
Mathematica [A] time = 1.69833, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.517, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{x} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}\, 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 \frac{\operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{x}\, dx + \int 2 a^{2} x \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{3} \operatorname{atan}^{\frac{3}{2}}{\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 \frac{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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