Optimal. Leaf size=63 \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{a^2 c}-\frac {3 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )}{2 a} \]
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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 = {} \[ \int \frac {x \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{a^2 c}-\frac {3 \int \frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 0, normalized size = 0.00 \[ \int \frac {x \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
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 [A] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.37, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \left (a x \right )^{\frac {3}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 = -0.02 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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