Optimal. Leaf size=72 \[ -\frac{i a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{x (a x+i)},x\right )}{c}+\frac{\text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{x^3},x\right )}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{5/2}}{5 c} \]
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Rubi [A] time = 0.192333, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^{3/2}}{x \left (c+a^2 c x^2\right )} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3} \, dx}{c}\\ &=\frac{2 i a^2 \tan ^{-1}(a x)^{5/2}}{5 c}+\frac{\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3} \, dx}{c}-\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{x (i+a x)} \, dx}{c}\\ \end{align*}
Mathematica [A] time = 1.71906, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({a}^{2}c{x}^{2}+c \right ) } \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{\operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, 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{\arctan \left (a x\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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