Optimal. Leaf size=244 \[ \frac {3 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {\frac {\pi }{3}} c^2 \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.19, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4663, 4661, 3312, 3304, 3352} \[ \frac {3 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {\frac {\pi }{3}} c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4661
Rule 4663
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\sqrt {\sin ^{-1}(a x)}} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\left (1-a^2 x^2\right )^{5/2}}{\sqrt {\sin ^{-1}(a x)}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^6(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5}{16 \sqrt {x}}+\frac {15 \cos (2 x)}{32 \sqrt {x}}+\frac {3 \cos (4 x)}{16 \sqrt {x}}+\frac {\cos (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt {1-a^2 x^2}}\\ &=\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {\left (3 c^2 \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 )}{16 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c^2 \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 {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {\left (3 c^2 \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 )}{8 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c^2 \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 {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 a \sqrt {1-a^2 x^2}}+\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}+\frac {c^2 \sqrt {\frac {\pi }{3}} \sqrt {c-a^2 c x^2} C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {15 c^2 \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 = 0.74, size = 336, normalized size = 1.38 \[ \frac {c^2 \sqrt {c-a^2 c x^2} \left (240 \sin ^{-1}(a x) \sqrt {\sin ^{-1}(a x)^2}-45 i \sqrt {2} \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )-18 i \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )-i \sqrt {6} \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},6 i \sin ^{-1}(a x)\right )+6 i \sqrt {\sin ^{-1}(a x)^2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )-i \sqrt {6} \sqrt {\sin ^{-1}(a x)^2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 i \sin ^{-1}(a x)\right )+3 i \sqrt {2} \left (16 \left (i \sin ^{-1}(a x)\right )^{3/2}+\sqrt {-i \sin ^{-1}(a x)} \sqrt {\sin ^{-1}(a x)^2}\right ) \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )+24 i \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )\right )}{384 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)} \sqrt {\sin ^{-1}(a x)^2}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\sqrt {\arcsin \left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\sqrt {\arcsin \left (a x \right )}}\, dx \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]
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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