Optimal. Leaf size=343 \[ \frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} c \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} c \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d^2}-\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}+\frac {3 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}}{16 d^2}-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]
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Rubi [A] time = 1.04, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4805, 4747, 6741, 6742, 3386, 3385, 3354, 3352, 3351, 3353} \[ \frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{32 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} c \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} c \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d^2}-\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}+\frac {3 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}}{16 d^2}-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3385
Rule 3386
Rule 4747
Rule 4805
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int x \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x)^{3/2} \cos (x) \left (-\frac {c}{d}+\frac {\sin (x)}{d}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^4 \cos \left (\frac {a-x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^4 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int \left (c x^4 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^4 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {\operatorname {Subst}\left (\int x^4 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^2}-\frac {(2 c) \operatorname {Subst}\left (\int x^4 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac {3 \operatorname {Subst}\left (\int x^2 \cos \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 d^2}-\frac {(3 c) \operatorname {Subst}\left (\int x^2 \sin \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d^2}\\ &=-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac {3 b \sqrt {a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{16 d^2}+\frac {(3 b c) \operatorname {Subst}\left (\int \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 d^2}\\ &=-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac {3 b \sqrt {a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}+\frac {\left (3 b c \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^2}-\frac {\left (3 b \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{16 d^2}+\frac {\left (3 b c \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^2}+\frac {\left (3 b \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{16 d^2}\\ &=-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac {3 b^{3/2} c \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^2}-\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}+\frac {3 b^{3/2} c \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^2}+\frac {3 b^{3/2} \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 d^2}+\frac {3 b \sqrt {a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}\\ \end {align*}
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Mathematica [C] time = 7.63, size = 635, normalized size = 1.85 \[ -\frac {b c \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)}\right )}{4 d^2}+\frac {a \left (\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+\sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )-2 \sqrt {\frac {1}{b}} \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 \sqrt {\frac {1}{b}} d^2}+\frac {b \left (\sqrt {\pi } \sqrt {\frac {1}{b}} \left (3 b \sin \left (\frac {2 a}{b}\right )-4 a \cos \left (\frac {2 a}{b}\right )\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+\sqrt {\pi } \left (-\sqrt {\frac {1}{b}}\right ) \left (4 a \sin \left (\frac {2 a}{b}\right )+3 b \cos \left (\frac {2 a}{b}\right )\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+2 \left (3 \sin \left (2 \sin ^{-1}(c+d x)\right )-4 \sin ^{-1}(c+d x) \cos \left (2 \sin ^{-1}(c+d x)\right )\right ) \sqrt {a+b \sin ^{-1}(c+d x)}\right )}{32 d^2}-\frac {a b c 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^2 \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 = 2.98, size = 2058, normalized size = 6.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 577, normalized size = 1.68 \[ -\frac {-24 \sqrt {2}\, \sqrt {\frac {1}{b}}\, \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} c -24 \sqrt {2}\, \sqrt {\frac {1}{b}}\, \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} c +3 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {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 {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+32 \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} c +8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}+64 \arcsin \left (d x +c \right ) \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b c +48 \arcsin \left (d x +c \right ) \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b^{2} c +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b -6 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}+32 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{2} c +48 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b c +8 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{2}-6 \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b}{32 d^{2} \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 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int x \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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