Optimal. Leaf size=431 \[ \frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.58, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4649, 4647, 4641, 4629, 4707, 4635, 4406, 12, 3305, 3351, 4677, 4723} \[ \frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
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 4649
Rule 4677
Rule 4707
Rule 4723
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx-\frac {\left (5 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^{3/2} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \int \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)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)} \, dx}{256 \sqrt {1-a^2 x^2}}+\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\sin ^{-1}(a x)}} \, dx}{512 \sqrt {1-a^2 x^2}}+\frac {\left (45 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{512 \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{128 \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt {1-a^2 x^2}}+\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{1024 \sqrt {1-a^2 x^2}}+\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{256 \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {\left (15 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 )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \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 )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 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 )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 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 )}{512 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \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 )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 c \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 = 0.40, size = 180, normalized size = 0.42 \[ \frac {c \sqrt {c-a^2 c x^2} \left (1680 \sqrt {\pi } \sqrt {\sin ^{-1}(a x)} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )+1536 \sin ^{-1}(a x)^4+3584 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)^3-3360 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)+4480 \sin ^{-1}(a x)^2 \cos \left (2 \sin ^{-1}(a x)\right )-7 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-4 i \sin ^{-1}(a x)\right )-7 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},4 i \sin ^{-1}(a x)\right )\right )}{14336 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \]
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 = 0.38, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\mathrm {asin}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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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