Optimal. Leaf size=175 \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}+\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d} \]
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Rubi [A] time = 0.27, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4803, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352} \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}+\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4619
Rule 4623
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b \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+d x)\right )}{4 d}-\frac {\left (3 b \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+d x)\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b \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+d x)}\right )}{2 d}-\frac {\left (3 b \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+d x)}\right )}{2 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 3.34, size = 313, normalized size = 1.79 \[ \frac {b \left (-\sqrt {2 \pi } \sqrt {\frac {1}{b}} \left (2 a \sin \left (\frac {a}{b}\right )+3 b \cos \left (\frac {a}{b}\right )\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+\sqrt {2 \pi } \sqrt {\frac {1}{b}} \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+2 \left (3 \sqrt {1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}+\frac {2 a e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{\sqrt {a+b \sin ^{-1}(c+d x)}}\right )}{4 d} \]
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 = 3.87, size = 1091, normalized size = 6.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 296, normalized size = 1.69 \[ \frac {-3 \sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}-3 \sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+4 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b +6 \arcsin \left (d x +c \right ) \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b^{2}+4 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{2}+6 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b}{4 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]
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 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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