Optimal. Leaf size=66 \[ \frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{5 a^2 c}-\frac {3 \text {Int}\left (\left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)},x\right )}{10 a} \]
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Rubi [A] time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx &=\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{5 a^2 c}-\frac {3 \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \, dx}{10 a}\\ \end {align*}
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Mathematica [A] time = 2.56, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.70, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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