Optimal. Leaf size=329 \[ \frac {\sqrt {\frac {\pi }{2}} e \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} e \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{2}} e \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} e \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {\sqrt {2 \pi } d \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {\sqrt {2 \pi } d \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c} \]
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Rubi [A] time = 0.64, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4667, 4623, 3306, 3305, 3351, 3304, 3352, 4635, 4406} \[ \frac {\sqrt {\frac {\pi }{2}} e \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} e \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{2}} e \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} e \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {\sqrt {2 \pi } d \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {\sqrt {2 \pi } d \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4623
Rule 4635
Rule 4667
Rubi steps
\begin {align*} \int \frac {d+e x^2}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx &=\int \left (\frac {d}{\sqrt {a+b \sin ^{-1}(c x)}}+\frac {e x^2}{\sqrt {a+b \sin ^{-1}(c x)}}\right ) \, dx\\ &=d \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx+e \int \frac {x^2}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx\\ &=\frac {d \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac {e \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^3}\\ &=\frac {e \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {a+b x}}-\frac {\cos (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3}+\frac {\left (d \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac {\left (d \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {e \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}+\frac {\left (2 d \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b c}+\frac {\left (2 d \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b c}\\ &=\frac {d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {d \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} c}+\frac {\left (e \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {\left (e \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}+\frac {\left (e \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {\left (e \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}\\ &=\frac {d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {d \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} c}+\frac {\left (e \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b c^3}-\frac {\left (e \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b c^3}+\frac {\left (e \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b c^3}-\frac {\left (e \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b c^3}\\ &=\frac {e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{\sqrt {b} c}-\frac {e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 \sqrt {b} c^3}+\frac {d \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} c}-\frac {e \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{2 \sqrt {b} c^3}\\ \end {align*}
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Mathematica [C] time = 0.68, size = 246, normalized size = 0.75 \[ -\frac {i e^{-\frac {3 i a}{b}} \left (3 e^{\frac {2 i a}{b}} \left (4 c^2 d+e\right ) \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 e^{\frac {4 i a}{b}} \left (4 c^2 d+e\right ) \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt {3} e \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-e^{\frac {6 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{24 c^3 \sqrt {a+b \sin ^{-1}(c x)}} \]
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 = 2.07, size = 485, normalized size = 1.47 \[ -\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{c {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{c {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {b}} - \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (\frac {3 i \, a}{b} + 1\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} + \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} c^{3}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b} + 1\right )}}{4 \, c^{3} {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b} + 1\right )}}{4 \, c^{3} {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {b}} + \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (-\frac {3 i \, a}{b} + 1\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} - \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 248, normalized size = 0.75 \[ \frac {\sqrt {2}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \left (12 \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c^{2} d +12 \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c^{2} d -\cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, e -\sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, e +3 \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) e +3 \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) e \right )}{12 c^{3}} \]
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 \[ \int \frac {e x^{2} + d}{\sqrt {b \arcsin \left (c x\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {e\,x^2+d}{\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}} \,d x \]
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 \[ \int \frac {d + e x^{2}}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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