Optimal. Leaf size=212 \[ \frac {45}{32} c^2 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )+\frac {3}{8} c^2 \text {Int}\left (\frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {a^2 c x^2+c}},x\right )+\frac {5}{16} c \text {Int}\left (\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)},x\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac {5 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}-\frac {15 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a} \]
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Rubi [A] time = 0.18, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac {15 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{32} \left (45 c^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.59, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]
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 = 1.50, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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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