Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^{3/2}},x\right ) \]
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Rubi [A]
time = 0.06, 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 = {}
\begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx &=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 5.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.42, size = 0, normalized size = 0.00 \[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {atan}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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