Optimal. Leaf size=216 \[ -\frac {c \text {Int}\left (\frac {a^2 c x^2+c}{\sqrt {\tan ^{-1}(a x)}},x\right )}{64 a}-\frac {c^2 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )}{24 a}-\frac {2 c^2 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )}{9 a}+\frac {c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{9 a}+\frac {c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}{12 a^2} \]
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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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]
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 )^2 \tan ^{-1}(a x)^{5/2} \, dx &=\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {5 \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx}{12 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{64 a}-\frac {c \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{3 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{12 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{9 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{64 a}-\frac {c^2 \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx}{24 a}-\frac {\left (2 c^2\right ) \int \tan ^{-1}(a x)^{3/2} \, dx}{9 a}\\ \end {align*}
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Mathematica [A] time = 1.57, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]
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 [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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maple [A] time = 2.35, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]
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 = -0.00 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]
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 \[ c^{2} \left (\int x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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