Optimal. Leaf size=36 \[ \frac{\text{Unintegrable}\left (\sqrt{\tan ^{-1}(a x)},x\right )}{a^2 c}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^3 c} \]
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Rubi [A] time = 0.0977936, 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 = {} \[ \int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{\sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int \sqrt{\tan ^{-1}(a x)} \, dx}{a^2 c}\\ &=-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^3 c}+\frac{\int \sqrt{\tan ^{-1}(a x)} \, dx}{a^2 c}\\ \end{align*}
Mathematica [A] time = 1.06475, size = 0, normalized size = 0. \[ \int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.257, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{{a}^{2}c{x}^{2}+c}\sqrt{\arctan \left ( ax \right ) }}\, 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^{2} \sqrt{\operatorname{atan}{\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^{2} \sqrt{\arctan \left (a x\right )}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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