Optimal. Leaf size=361 \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d} \]
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Rubi [A] time = 1.03, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {4805, 12, 4629, 4707, 4677, 4623, 3306, 3305, 3351, 3304, 3352, 4635, 4406} \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4623
Rule 4629
Rule 4635
Rule 4677
Rule 4707
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}-\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b e^2\right ) \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 )}{6 d}-\frac {\left (b^2 e^2\right ) \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+d x)\right )}{12 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 d}+\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 d}-\frac {\left (b e^2 \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 )}{6 d}-\frac {\left (b e^2 \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 )}{6 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {\left (b e^2 \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 )}{3 d}-\frac {\left (b^2 e^2 \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+d x)\right )}{48 d}+\frac {\left (b^2 e^2 \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+d x)\right )}{48 d}-\frac {\left (b e^2 \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 )}{3 d}-\frac {\left (b^2 e^2 \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+d x)\right )}{48 d}+\frac {\left (b^2 e^2 \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+d x)\right )}{48 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {b^{3/2} e^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 )}{3 d}-\frac {b^{3/2} e^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 )}{3 d}-\frac {\left (b e^2 \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 )}{24 d}+\frac {\left (b e^2 \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+d x)}\right )}{24 d}-\frac {\left (b e^2 \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 )}{24 d}+\frac {\left (b e^2 \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+d x)}\right )}{24 d}\\ &=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac {3 b^{3/2} e^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 )}{8 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 b^{3/2} e^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 )}{8 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 d}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 268, normalized size = 0.74 \[ \frac {b e^2 e^{-\frac {3 i a}{b}} \sqrt {a+b \sin ^{-1}(c+d x)} \left (27 e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt {3} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{216 d \sqrt {\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]
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 [B] time = 3.03, size = 2237, normalized size = 6.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 593, normalized size = 1.64 \[ \frac {e^{2} \left (\sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+\sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}-27 \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}-27 \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}+36 \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}-12 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) b^{2}+72 \arcsin \left (d x +c \right ) \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b +54 \arcsin \left (d x +c \right ) \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b^{2}-24 \arcsin \left (d x +c \right ) \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) a b -6 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) b^{2}+36 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{2}+54 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a b -12 \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) a^{2}-6 \cos \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) a b \right )}{144 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 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ e^{2} \left (\int a c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 a c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 b c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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