Optimal. Leaf size=130 \[ -\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \text {ArcSin}(a x)^{3/2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {ArcSin}(a x)}}-\frac {8 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{3 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4751, 4727,
3385, 3433} \begin {gather*} -\frac {8 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{3 a \sqrt {1-a^2 x^2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {ArcSin}(a x)}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \text {ArcSin}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4727
Rule 4751
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a^2 c x^2}}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {\left (4 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sin ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {\left (8 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {\left (16 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {8 \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.27, size = 142, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {c-a^2 c x^2} \left (-1+a^2 x^2+4 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)-\sqrt {2} (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )+\frac {\sqrt {2} \text {ArcSin}(a x)^2 \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )}{\sqrt {i \text {ArcSin}(a x)}}\right )}{3 a \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {5}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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