Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 (a+b \text {ArcSin}(c x))^{3/2}},x\right ) \]
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Rubi [A]
time = 0.04, 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 (d+e x^2\right )^2 (a+b \text {ArcSin}(c x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=\int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {ArcSin}(c x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.56, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \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 {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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