Optimal. Leaf size=219 \[ -\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4647, 4641, 4629, 4723, 3312, 3304, 3352} \[ -\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4629
Rule 4641
Rule 4647
Rule 4723
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a^2 x^2}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \sqrt {\sin ^{-1}(a x)} \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 \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 )}{16 a \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 \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 )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {\left (3 \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 )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {\left (3 \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 )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {3 \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 = 0.14, size = 158, normalized size = 0.72 \[ \frac {\sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \left (32 \sin ^{-1}(a x) \sqrt {\sin ^{-1}(a x)^2} \left (5 a x \sqrt {1-a^2 x^2}+2 \sin ^{-1}(a x)\right )+15 \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}(a x)\right )+15 \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}(a x)\right )\right )}{320 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)^2}} \]
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 = 0.43, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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