Optimal. Leaf size=81 \[ \frac {5}{8} c \text {Int}\left (\sqrt {\tan ^{-1}(a x)},x\right )+\frac {2}{3} c \text {Int}\left (\tan ^{-1}(a x)^{5/2},x\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{5/2}-\frac {5 c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{12 a} \]
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Rubi [A] time = 0.02, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{5/2}+\frac {1}{8} (5 c) \int \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{3} (2 c) \int \tan ^{-1}(a x)^{5/2} \, dx\\ \end {align*}
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Mathematica [A] time = 3.99, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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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 [A] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.60, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]
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 = -0.01 \[ \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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