Optimal. Leaf size=29 \[ \text {Int}\left (x^m \left (a^2 c x^2+c\right )^{5/2} \sqrt {\tan ^{-1}(a x)},x\right ) \]
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Rubi [A] time = 0.11, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx &=\int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx\\ \end {align*}
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Mathematica [A] time = 1.48, size = 0, normalized size = 0.00 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} x^{m} \sqrt {\arctan \left (a x\right )}, x\right ) \]
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 = 3.42, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arctan \left (a x \right )}\, dx \]
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 = -0.03 \[ \int x^m\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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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