Optimal. Leaf size=137 \[ \frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}+\frac {\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)} \]
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Rubi [A] time = 0.45, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4629, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{8 c^2}+\frac {\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3351
Rule 3352
Rule 4629
Rule 4723
Rubi steps
\begin {align*} \int x \sqrt {a+b \sin ^{-1}(c x)} \, dx &=\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}-\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx\\ &=\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2}\\ &=\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2}\\ &=-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {b \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^2}\\ &=-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {\left (b \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (b \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^2}\\ &=-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 c^2}\\ &=-\frac {\sqrt {a+b \sin ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 c^2}+\frac {\sqrt {b} \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 c^2}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 141, normalized size = 1.03 \[ -\frac {e^{-\frac {2 i a}{b}} \sqrt {a+b \sin ^{-1}(c x)} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{8 \sqrt {2} c^2 \sqrt {\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}}} \]
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 [C] time = 1.46, size = 448, normalized size = 3.27 \[ \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}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 173, normalized size = 1.26 \[ -\frac {-\cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, b -\sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, b +2 \arcsin \left (c x \right ) \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b +2 \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a}{8 c^{2} \sqrt {a +b \arcsin \left (c x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \arcsin \left (c x\right ) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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