Optimal. Leaf size=65 \[ \frac{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{\text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{6 a} \]
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Rubi [A] time = 0.103673, 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 = {} \[ \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx &=\frac{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{6 a}\\ \end{align*}
Mathematica [A] time = 6.86393, size = 0, normalized size = 0. \[ \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.974, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{{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*} \int x \sqrt{c \left (a^{2} x^{2} + 1\right )} \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx \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 \sqrt{a^{2} c x^{2} + c} x \sqrt{\arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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