Optimal. Leaf size=518 \[ -\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{7/2}}{3 d}+\frac {105 b^{7/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}+\frac {105 b^{7/2} e^2 \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{864 d} \]
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Rubi [A]
time = 1.11, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps
used = 35, number of rules used = 14, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used =
{4889, 12, 4725, 4795, 4767, 4715, 4719, 3387, 3386, 3432, 3385, 3433, 4731, 4491}
\begin {gather*} \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^2 \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 \sqrt {\frac {\pi }{6}} b^{7/2} e^2 \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 \sqrt {\frac {\pi }{6}} b^{7/2} e^2 \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{3/2}}{108 d}-\frac {35 b^2 e^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{7/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4715
Rule 4719
Rule 4725
Rule 4731
Rule 4767
Rule 4795
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{36 d}\\ &=-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{18 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{108 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{12 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{432 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{216 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{24 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2\right ) \text {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 )}{216 d}+\frac {\left (35 b^3 e^2\right ) \text {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 )}{24 d}+\frac {\left (35 b^4 e^2\right ) \text {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 )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{216 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{216 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2 \cos \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+d x)}\right )}{108 d}+\frac {\left (35 b^3 e^2 \cos \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+d x)}\right )}{12 d}+\frac {\left (35 b^4 e^2 \cos \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+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2 \cos \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+d x)\right )}{1728 d}+\frac {\left (35 b^3 e^2 \sin \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+d x)}\right )}{108 d}+\frac {\left (35 b^3 e^2 \sin \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+d x)}\right )}{12 d}+\frac {\left (35 b^4 e^2 \sin \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+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2 \sin \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+d x)\right )}{1728 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {175 b^{7/2} 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 )}{54 d}+\frac {175 b^{7/2} 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 )}{54 d}+\frac {\left (35 b^3 e^2 \cos \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+d x)}\right )}{864 d}-\frac {\left (35 b^3 e^2 \cos \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+d x)}\right )}{864 d}+\frac {\left (35 b^3 e^2 \sin \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+d x)}\right )}{864 d}-\frac {\left (35 b^3 e^2 \sin \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+d x)}\right )}{864 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {105 b^{7/2} 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 )}{32 d}-\frac {35 b^{7/2} 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 )}{864 d}+\frac {105 b^{7/2} 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 )}{32 d}-\frac {35 b^{7/2} 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 )}{864 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 267, normalized size = 0.52 \begin {gather*} \frac {b e^2 e^{-\frac {3 i a}{b}} (a+b \text {ArcSin}(c+d x))^{5/2} \left (-243 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )-243 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {3} \left (\sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )}{1944 d \left (\frac {(a+b \text {ArcSin}(c+d x))^2}{b^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1233\) vs.
\(2(426)=852\).
time = 0.61, size = 1234, normalized size = 2.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(1234\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 2.67, size = 8028, normalized size = 15.50 \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 {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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