Optimal. Leaf size=482 \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 1.42, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {4667, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352, 4629, 4707, 4635, 4406} \[ -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4619
Rule 4623
Rule 4629
Rule 4635
Rule 4667
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \sin ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (3 b c d) \int \frac {x \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} (b c e) \int \frac {x^3 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{4} \left (3 b^2 d\right ) \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{12} \left (b^2 e\right ) \int \frac {x^2}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {(b e) \int \frac {x \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx}{3 c}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {(3 b d) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {\left (b^2 e\right ) \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{6 c^2}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac {\left (b^2 e\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 x)\right )}{12 c^3}-\frac {\left (3 b d \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 x)\right )}{4 c}-\frac {\left (3 b d \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 x)\right )}{4 c}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac {\left (3 b d \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 x)}\right )}{2 c}-\frac {\left (b e \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 x)\right )}{6 c^3}-\frac {\left (3 b d \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 x)}\right )}{2 c}-\frac {\left (b e \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 x)\right )}{6 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {\left (b e \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 x)}\right )}{3 c^3}-\frac {\left (b^2 e \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 x)\right )}{48 c^3}+\frac {\left (b^2 e \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 x)\right )}{48 c^3}-\frac {\left (b e \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 x)}\right )}{3 c^3}-\frac {\left (b^2 e \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 x)\right )}{48 c^3}+\frac {\left (b^2 e \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 x)\right )}{48 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{3 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c^3}-\frac {\left (b e \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 x)}\right )}{24 c^3}+\frac {\left (b e \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 x)}\right )}{24 c^3}-\frac {\left (b e \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 x)}\right )}{24 c^3}+\frac {\left (b e \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 x)}\right )}{24 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3}\\ \end {align*}
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Mathematica [C] time = 10.28, size = 873, normalized size = 1.81 \[ \frac {a b d e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {a b e e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt {3} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {b d \left (2 \sqrt {a+b \sin ^{-1}(c x)} \left (2 c x \sin ^{-1}(c x)+3 \sqrt {1-c^2 x^2}\right )-\sqrt {\frac {1}{b}} \sqrt {2 \pi } C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c}+\frac {b e \left (18 \sqrt {a+b \sin ^{-1}(c x)} \left (2 c x \sin ^{-1}(c x)+3 \sqrt {1-c^2 x^2}\right )-9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {6 \pi } C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {6 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}\right ) \left (b \sin \left (\frac {3 a}{b}\right )-2 a \cos \left (\frac {3 a}{b}\right )\right )-6 \sqrt {a+b \sin ^{-1}(c x)} \left (\cos \left (3 \sin ^{-1}(c x)\right )+2 \sin ^{-1}(c x) \sin \left (3 \sin ^{-1}(c x)\right )\right )\right )}{144 c^3} \]
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 [C] time = 5.68, size = 3039, normalized size = 6.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 835, normalized size = 1.73 \[ \text {result too large to display} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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