Optimal. Leaf size=168 \[ \frac {\text {Int}\left (\frac {x}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{8 a^2}+\frac {5 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )}{4 a^3}+\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^4 c}-\frac {x \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{4 a^3 c} \]
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Rubi [A] time = 0.45, 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 = {} \[ \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac {2 \int \frac {x \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\int \frac {x^2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{4 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^3}+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 3.73, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
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 = 9.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x \right )^{\frac {3}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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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