Optimal. Leaf size=754 \[ d^2 x \sqrt {a+b \text {ArcSin}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {ArcSin}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {ArcSin}(c x)}-\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{16 c^5}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^5}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{6 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 c^5}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{80 c^5} \]
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Rubi [A]
time = 1.50, antiderivative size = 754, normalized size of antiderivative = 1.00, number of steps
used = 42, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4757, 4715,
4809, 3387, 3386, 3432, 3385, 3433, 4725, 3393} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^5}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{16 c^5}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \sin \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{80 c^5}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{16 c^5}-\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{6 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{c}+d^2 x \sqrt {a+b \text {ArcSin}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {ArcSin}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {ArcSin}(c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4715
Rule 4725
Rule 4757
Rule 4809
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \sqrt {a+b \sin ^{-1}(c x)} \, dx &=\int \left (d^2 \sqrt {a+b \sin ^{-1}(c x)}+2 d e x^2 \sqrt {a+b \sin ^{-1}(c x)}+e^2 x^4 \sqrt {a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \sqrt {a+b \sin ^{-1}(c x)} \, dx+(2 d e) \int x^2 \sqrt {a+b \sin ^{-1}(c x)} \, dx+e^2 \int x^4 \sqrt {a+b \sin ^{-1}(c x)} \, dx\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {1}{2} \left (b c d^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{3} (b c d e) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{10} \left (b c e^2\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}-\frac {(b d e) \text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sin ^5(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {(b d e) \text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {a+b x}}-\frac {\sin (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \left (\frac {5 \sin (x)}{8 \sqrt {a+b x}}-\frac {5 \sin (3 x)}{16 \sqrt {a+b x}}+\frac {\sin (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{10 c^5}-\frac {\left (b d^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}+\frac {\left (b d^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}+\frac {(b d e) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {(b d e) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}-\frac {\left (d^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{c}+\frac {\left (d^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{c}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}-\frac {\left (b d e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {\left (b e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}+\frac {\left (b d e \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}+\frac {\left (b e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}+\frac {\left (b d e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}+\frac {\left (b e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}-\frac {\left (b d e \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {\left (b e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}+\frac {\left (b e^2 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}-\frac {\left (d e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 c^3}-\frac {\left (e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{8 c^5}+\frac {\left (d e \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{6 c^3}+\frac {\left (e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{16 c^5}-\frac {\left (e^2 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{80 c^5}+\frac {\left (d e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 c^3}+\frac {\left (e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{8 c^5}-\frac {\left (d e \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{6 c^3}-\frac {\left (e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{16 c^5}+\frac {\left (e^2 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{80 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^5}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^5}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{6 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 c^5}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{80 c^5}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.96, size = 400, normalized size = 0.53 \begin {gather*} \frac {b e^{-\frac {5 i a}{b}} \left (450 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+450 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c x))}{b}\right )-e \left (25 \sqrt {3} \left (8 c^2 d+3 e\right ) e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+25 \sqrt {3} \left (8 c^2 d+3 e\right ) e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )-9 \sqrt {5} e \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )+e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )\right )\right )\right )}{7200 c^5 \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 1155, normalized size = 1.53
method | result | size |
default | \(\text {Expression too large to display}\) | \(1155\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.66, size = 3216, normalized size = 4.27 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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