Optimal. Leaf size=247 \[ \frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.25, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4647, 4641, 4629, 4707, 4635, 4406, 12, 3305, 3351} \[ \frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4629
Rule 4635
Rule 4641
Rule 4647
Rule 4707
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin ^{-1}(a x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a^2 x^2}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \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 )}{128 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 158, normalized size = 0.64 \[ \frac {\sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \left (64 \left (7 a x \sqrt {1-a^2 x^2}+2 \sin ^{-1}(a x)\right ) \left (\sin ^{-1}(a x)^2\right )^{3/2}+35 i \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 i \sin ^{-1}(a x)\right )-35 i \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},2 i \sin ^{-1}(a x)\right )\right )}{896 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)^2}} \]
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 = 0.41, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\mathrm {asin}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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