Optimal. Leaf size=115 \[ -\frac{5 c \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )}{64 a}-\frac{5 c \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )}{12 a}+\frac{c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{5 c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{5 c \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{32 a^2} \]
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Rubi [A] time = 0.0551784, 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 = {} \[ \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx &=\frac{c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{5 \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{8 a}\\ &=\frac{5 c \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{5 c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{(5 c) \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx}{64 a}-\frac{(5 c) \int \tan ^{-1}(a x)^{3/2} \, dx}{12 a}\\ \end{align*}
Mathematica [A] time = 1.92466, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.377, size = 0, normalized size = 0. \begin{align*} \int 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 [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{\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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