Optimal. Leaf size=136 \[ \frac{3}{8} a^2 \text{Unintegrable}\left (\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{1}{2} a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right )-\frac{3 a \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{2 c x^2} \]
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Rubi [A] time = 0.422847, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{2 c x^2}+\frac{1}{4} (3 a) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{4 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{2 c x^2}+\frac{1}{8} \left (3 a^2\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}
Mathematica [A] time = 3.95466, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.401, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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