Optimal. Leaf size=61 \[ \frac {2 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )}{3 a^3 c}+\frac {\text {Int}\left (x \sqrt {\tan ^{-1}(a x)},x\right )}{a^2 c}-\frac {2 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c} \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {x \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \sqrt {\tan ^{-1}(a x)} \, dx}{a^2 c}\\ &=-\frac {2 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c}+\frac {2 \int \tan ^{-1}(a x)^{3/2} \, dx}{3 a^3 c}+\frac {\int x \sqrt {\tan ^{-1}(a x)} \, dx}{a^2 c}\\ \end {align*}
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Mathematica [A] time = 2.87, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx \]
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 = 3.70, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sqrt {\arctan \left (a x \right )}}{a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \frac {x^3\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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