Optimal. Leaf size=107 \[ -\frac {2 \sqrt {1-a^2 x^2}}{3 a \left (c-a^2 c x^2\right )^{5/2} \text {ArcSin}(a x)^{3/2}}+\frac {8 a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^3 \text {ArcSin}(a x)^{3/2}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}} \]
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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 {1}{\left (c-a^2 c x^2\right )^{5/2} \text {ArcSin}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2}}{3 a \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{3/2}}+\frac {\left (8 a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \text {ArcSin}(a x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arcsin \left (a x \right )^{\frac {5}{2}}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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