Optimal. Leaf size=363 \[ -\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{512 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } c \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {27 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.43, antiderivative size = 363, 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 = {4649, 4647, 4641, 4629, 4723, 3312, 3304, 3352, 4677, 4661} \[ -\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{512 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {27 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4629
Rule 4641
Rule 4647
Rule 4649
Rule 4661
Rule 4677
Rule 4723
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sqrt {\sin ^{-1}(a x)} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{\sqrt {\sin ^{-1}(a x)}} \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \sqrt {\sin ^{-1}(a x)} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {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}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {9 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=\frac {27 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{256 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a \sqrt {1-a^2 x^2}}\\ &=\frac {27 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{512 a \sqrt {1-a^2 x^2}}-\frac {3 c \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=\frac {27 c \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{256 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt {1-a^2 x^2}}-\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{512 a \sqrt {1-a^2 x^2}}-\frac {3 c \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 186, normalized size = 0.51 \[ \frac {c \sqrt {c-a^2 c x^2} \left (-240 \sqrt {\pi } \sqrt {\sin ^{-1}(a x)^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )+\sqrt {\sin ^{-1}(a x)} \left (32 \sqrt {\sin ^{-1}(a x)^2} \left (12 \sin ^{-1}(a x)^2+20 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)+15 \cos \left (2 \sin ^{-1}(a x)\right )\right )+5 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 i \sin ^{-1}(a x)\right )+5 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},4 i \sin ^{-1}(a x)\right )\right )\right )}{2560 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\mathrm {asin}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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