Optimal. Leaf size=46 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)^{3/2}},x\right )}{3 a c}-\frac{2}{3 a c x \tan ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.0772083, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c x \tan ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{1}{x^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a c}\\ \end{align*}
Mathematica [A] time = 1.29838, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) } \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{5}{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{1}{a^{2} x^{3} \operatorname{atan}^{\frac{5}{2}}{\left (a x \right )} + x \operatorname{atan}^{\frac{5}{2}}{\left (a x \right )}}\, 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{1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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