Optimal. Leaf size=360 \[ -\frac {75 c^3 \text {Int}\left (\frac {1}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{896 a}-\frac {25 c^3 \text {Int}\left (\frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {a^2 c x^2+c}},x\right )}{224 a}-\frac {25 c^2 \text {Int}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{1344 a}-\frac {c \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{112 a}-\frac {25 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{224 a}+\frac {75 c^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{448 a^2}+\frac {\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {5 x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac {\left (a^2 c x^2+c\right )^{5/2} \sqrt {\tan ^{-1}(a x)}}{56 a^2}-\frac {25 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}+\frac {25 c \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{672 a^2} \]
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Rubi [A] time = 0.36, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]
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 )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx &=\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {5 \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2} \, dx}{14 a}\\ &=\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}}{56 a^2}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{112 a}-\frac {(25 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx}{84 a}\\ &=\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{672 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}}{56 a^2}-\frac {25 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{112 a}-\frac {\left (25 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{1344 a}-\frac {\left (25 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx}{112 a}\\ &=\frac {75 c^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{448 a^2}+\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{672 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}}{56 a^2}-\frac {25 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{224 a}-\frac {25 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{112 a}-\frac {\left (25 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{1344 a}-\frac {\left (75 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {\left (25 c^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{224 a}\\ \end {align*}
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Mathematica [A] time = 6.05, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]
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(-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 = 2.98, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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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