Optimal. Leaf size=64 \[ \frac{3}{2} a \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right )-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{c x} \]
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Rubi [A] time = 0.213437, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{c x}+\frac{1}{2} (3 a) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}
Mathematica [A] time = 1.24526, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.8, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, 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 \frac{\arctan \left (a x\right )^{\frac{3}{2}}}{\sqrt{a^{2} c x^{2} + c} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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