Integrand size = 10, antiderivative size = 89 \[ \int x \arcsin (a x)^{3/2} \, dx=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \]
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Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4725, 4795, 4737, 4731, 4491, 12, 3386, 3432} \[ \int x \arcsin (a x)^{3/2} \, dx=-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4725
Rule 4731
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{16} \int \frac {x}{\sqrt {\arcsin (a x)}} \, dx-\frac {3 \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{16 a^2} \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arcsin (a x)\right )}{16 a^2} \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a^2} \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a^2} \\ & = \frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int x \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},-2 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},2 i \arcsin (a x)\right )}{16 \sqrt {2} a^2 \sqrt {\arcsin (a x)}} \]
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Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {8 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )+3 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )}{32 a^{2} \sqrt {\arcsin \left (a x \right )}}\) | \(64\) |
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Exception generated. \[ \int x \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \arcsin (a x)^{3/2} \, dx=\int x \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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Exception generated. \[ \int x \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int x \arcsin (a x)^{3/2} \, dx=-\frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} + \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} \]
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Timed out. \[ \int x \arcsin (a x)^{3/2} \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]
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