Optimal. Leaf size=126 \[ 2 a \text{Unintegrable}\left (\frac{x^3}{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 x^2}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.431362, 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 = {} \[ \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^2}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}+(2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2 x^2}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+(2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^2}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+(2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^2}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+(2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^3 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^2}{a c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4 \sqrt{2 \pi } \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c \sqrt{c+a^2 c x^2}}+(2 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}
Mathematica [A] time = 2.2166, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 2.893, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \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 [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{x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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