Optimal. Leaf size=133 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d} \]
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Rubi [A] time = 0.27, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4803, 4619, 4723, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4619
Rule 4723
Rule 4803
Rubi steps
\begin {align*} \int \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\left (b \cos \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+d x)\right )}{2 d}+\frac {\left (b \sin \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+d x)\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d}+\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 129, normalized size = 0.97 \[ \frac {b e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{2 d \sqrt {a+b \sin ^{-1}(c+d x)}} \]
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 [B] time = 1.00, size = 579, normalized size = 4.35 \[ \frac {\sqrt {2} \sqrt {\pi } b^{2} i \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{4 \, {\left (\frac {b^{2} i}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {2} \sqrt {\pi } b^{2} i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{4 \, {\left (\frac {b^{2} i}{\sqrt {{\left | b \right |}}} - b \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{2 \, {\left (\frac {b^{2} i}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{2 \, {\left (\frac {b^{2} i}{\sqrt {{\left | b \right |}}} - b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} i e^{\left (i \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} i e^{\left (-i \arcsin \left (d x + c\right )\right )}}{2 \, d} - \frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {\pi } a \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} - \sqrt {2} \sqrt {{\left | b \right |}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 194, normalized size = 1.46 \[ \frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b +\sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) b +2 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b +2 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a}{2 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]
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 \[ \int \sqrt {b \arcsin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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