Optimal. Leaf size=119 \[ \frac {15}{8} c \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )+\frac {1}{2} c \text {Int}\left (\frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {a^2 c x^2+c}},x\right )+\frac {1}{2} x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac {5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a} \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac {1}{2} c \int \frac {\tan ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} (15 c) \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.49, size = 0, normalized size = 0.00 \[ \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]
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 = 1.52, size = 0, normalized size = 0.00 \[ \int \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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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