Integrand size = 14, antiderivative size = 137 \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=-\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}+\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}+\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \]
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Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4725, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}+\frac {\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}-\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4725
Rule 4809
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}-\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx \\ & = \frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}-\frac {\text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{4 c^2} \\ & = \frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{4 c^2} \\ & = -\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}+\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c^2} \\ & = -\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c^2} \\ & = -\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{4 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{4 c^2} \\ & = -\frac {\sqrt {a+b \arcsin (c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \arcsin (c x)}+\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}+\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\frac {i b e^{-\frac {2 i a}{b}} \left (-\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{8 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \]
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Time = 0.06 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {-\sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +\sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +2 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b +2 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a}{8 c^{2} \sqrt {a +b \arcsin \left (c x \right )}}\) | \(186\) |
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Exception generated. \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\int x \sqrt {a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
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\[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\int { \sqrt {b \arcsin \left (c x\right ) + a} x \,d x } \]
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Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.27 \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\frac {i \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} - \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{16 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} - \frac {i \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} - \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{16 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, c^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {\sqrt {b \arcsin \left (c x\right ) + a} e^{\left (2 i \, \arcsin \left (c x\right )\right )}}{8 \, c^{2}} - \frac {\sqrt {b \arcsin \left (c x\right ) + a} e^{\left (-2 i \, \arcsin \left (c x\right )\right )}}{8 \, c^{2}} \]
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Timed out. \[ \int x \sqrt {a+b \arcsin (c x)} \, dx=\int x\,\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
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