Optimal. Leaf size=199 \[ \frac {3 \sqrt {\pi } b^{3/2} e \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d}-\frac {3 \sqrt {\pi } b^{3/2} e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {3 b e \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d} \]
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Rubi [A] time = 0.49, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4805, 12, 4629, 4707, 4641, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac {3 \sqrt {\pi } b^{3/2} e \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{32 d}-\frac {3 \sqrt {\pi } b^{3/2} e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {3 b e \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4629
Rule 4635
Rule 4641
Rule 4707
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}+\frac {\left (3 b^2 e \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b e \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}+\frac {\left (3 b e \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}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {3 b^{3/2} e \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}+\frac {3 b^{3/2} e \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}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 137, normalized size = 0.69 \[ \frac {b^2 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{16 \sqrt {2} d \sqrt {a+b \sin ^{-1}(c+d x)}} \]
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 = 1.33, size = 984, normalized size = 4.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 294, normalized size = 1.48 \[ -\frac {e \left (3 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}-3 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}+16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b -6 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}+8 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{2}-6 \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b \right )}{32 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ e \left (\int a c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int b d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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