Optimal. Leaf size=359 \[ \frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \text {ArcSin}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {ArcSin}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \text {ArcSin}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}} \]
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Rubi [A]
time = 0.30, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4743, 4741,
4737, 4725, 4809, 3393, 3385, 3433, 4767, 4753} \begin {gather*} \frac {3}{8} a^2 x \sqrt {a^2-x^2} \text {ArcSin}\left (\frac {x}{a}\right )^{3/2}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {ArcSin}\left (\frac {x}{a}\right )^{3/2}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {a^2-x^2} \text {ArcSin}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}+\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 4725
Rule 4737
Rule 4741
Rule 4743
Rule 4753
Rule 4767
Rule 4809
Rubi steps
\begin {align*} \int \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx &=\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} \left (3 a^2\right ) \int \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx-\frac {\left (3 a \sqrt {a^2-x^2}\right ) \int x \left (1-\frac {x^2}{a^2}\right ) \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \, dx}{8 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {\left (9 a \sqrt {a^2-x^2}\right ) \int x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \, dx}{16 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{8 \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (9 \sqrt {a^2-x^2}\right ) \int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {9 a^3 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} C\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{128 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} C\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 209, normalized size = 0.58 \begin {gather*} \frac {a^3 \sqrt {a^2-x^2} \left (-240 \sqrt {\pi } \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\sqrt {\text {ArcSin}\left (\frac {x}{a}\right )} \left (5 \sqrt {i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {5}{2},-4 i \text {ArcSin}\left (\frac {x}{a}\right )\right )+5 \sqrt {-i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {5}{2},4 i \text {ArcSin}\left (\frac {x}{a}\right )\right )+32 \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )^2} \left (12 \text {ArcSin}\left (\frac {x}{a}\right )^2+15 \cos \left (2 \text {ArcSin}\left (\frac {x}{a}\right )\right )+20 \text {ArcSin}\left (\frac {x}{a}\right ) \sin \left (2 \text {ArcSin}\left (\frac {x}{a}\right )\right )\right )\right )\right )}{2560 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}\, dx\]
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 [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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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 = -0.00 \begin {gather*} \int {\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}\,{\left (a^2-x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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