Integrand size = 24, antiderivative size = 227 \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}} \]
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Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4743, 4741, 4737, 4731, 4491, 12, 3386, 3432, 4809} \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \sqrt {\arcsin (a x)} \left (c-a^2 c x^2\right )^{3/2}+\frac {c \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {3}{8} c x \sqrt {\arcsin (a x)} \sqrt {c-a^2 c x^2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4731
Rule 4737
Rule 4741
Rule 4743
Rule 4809
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)} \, dx-\frac {\left (a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}} \, dx}{8 \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\arcsin (a x)}} \, dx}{16 \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{16 a \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arcsin (a x)\right )}{16 a \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a \sqrt {1-a^2 x^2}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)}+\frac {c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.73 \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (32 \arcsin (a x)^2+8 \sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-2 i \arcsin (a x)\right )+8 \sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},2 i \arcsin (a x)\right )+\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-4 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},4 i \arcsin (a x)\right )\right )}{128 a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \]
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\[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arcsin \left (a x \right )}d x\]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\arcsin (a x)} \, dx=\int \sqrt {\mathrm {asin}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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