Optimal. Leaf size=59 \[ \frac{\text{Unintegrable}\left (x \tan ^{-1}(a x)^{3/2},x\right )}{a^2 c}+\frac{2 \text{Unintegrable}\left (\tan ^{-1}(a x)^{5/2},x\right )}{5 a^3 c}-\frac{2 x \tan ^{-1}(a x)^{5/2}}{5 a^3 c} \]
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Rubi [A] time = 0.122881, 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 = {} \[ \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{x \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int x \tan ^{-1}(a x)^{3/2} \, dx}{a^2 c}\\ &=-\frac{2 x \tan ^{-1}(a x)^{5/2}}{5 a^3 c}+\frac{2 \int \tan ^{-1}(a x)^{5/2} \, dx}{5 a^3 c}+\frac{\int x \tan ^{-1}(a x)^{3/2} \, dx}{a^2 c}\\ \end{align*}
Mathematica [A] time = 3.95321, size = 0, normalized size = 0. \[ \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.562, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{a}^{2}c{x}^{2}+c} \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*} \frac{\int \frac{x^{3} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \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{x^{3} \arctan \left (a x\right )^{\frac{3}{2}}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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