Integrand size = 16, antiderivative size = 250 \[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {\sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{3/2} c^3} \]
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Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4727, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {3 \pi }{2}} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {\sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4727
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3}+\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b^2 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^3}+\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^3}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {\sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{3/2} c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {e^{-\frac {3 i (a+b \arcsin (c x))}{b}} \left (e^{\frac {3 i a}{b}}-e^{\frac {3 i a}{b}+2 i \arcsin (c x)}-e^{\frac {3 i a}{b}+4 i \arcsin (c x)}+e^{\frac {3 i (a+2 b \arcsin (c x))}{b}}+e^{\frac {2 i a}{b}+3 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}+3 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} e^{3 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} e^{3 i \left (\frac {2 a}{b}+\arcsin (c x)\right )} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )}{4 b c^3 \sqrt {a+b \arcsin (c x)}} \]
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Time = 0.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }+\sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }-\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }-\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }+\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )-\cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )}{2 c^{3} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(301\) |
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Exception generated. \[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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