Optimal. Leaf size=482 \[ \frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{6 c}+d x (a+b \text {ArcSin}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x))^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \]
[Out]
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Rubi [A]
time = 0.95, 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 = {4757, 4715, 4767, 4719, 3387, 3386, 3432, 3385, 3433, 4725, 4795, 4731, 4491}
\begin {gather*} -\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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{6 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{3 c^3}+d x (a+b \text {ArcSin}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x))^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4715
Rule 4719
Rule 4725
Rule 4731
Rule 4757
Rule 4767
Rule 4795
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) \text {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 ) \text {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) \text {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 ) \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 x)\right )}{12 c^3}-\frac {\left (3 b d \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 x)\right )}{4 c}-\frac {\left (3 b d \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 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 )}{48 c^3}-\frac {\left (b e \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 x)}\right )}{3 c^3}-\frac {\left (b^2 e \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 x)\right )}{48 c^3}+\frac {\left (b^2 e \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 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 8.70, size = 873, normalized size = 1.81 \begin {gather*} \frac {a b d e^{-\frac {i a}{b}} \left (\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^{\frac {2 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 )\right )}{2 c \sqrt {a+b \text {ArcSin}(c x)}}+\frac {a b e e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 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 )+9 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 )-\sqrt {3} \left (\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 )+e^{\frac {6 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 )\right )\right )}{72 c^3 \sqrt {a+b \text {ArcSin}(c x)}}+\frac {b d \left (2 \sqrt {a+b \text {ArcSin}(c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)\right )-\sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)\right )-9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(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 \text {ArcSin}(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 } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(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 \text {ArcSin}(c x)}\right ) \left (-2 a \cos \left (\frac {3 a}{b}\right )+b \sin \left (\frac {3 a}{b}\right )\right )-6 \sqrt {a+b \text {ArcSin}(c x)} (\cos (3 \text {ArcSin}(c x))+2 \text {ArcSin}(c x) \sin (3 \text {ArcSin}(c x)))\right )}{144 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs.
\(2(374)=748\).
time = 0.47, size = 850, normalized size = 1.76
method | result | size |
default | \(-\frac {108 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} c^{2} d -108 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} c^{2} d +27 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} e -27 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} e -\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b^{2} e +\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b^{2} e +144 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} c^{2} d +288 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b \,c^{2} d -216 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} c^{2} d +36 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} e -12 \arcsin \left (c x \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2} e +144 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} c^{2} d -216 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b \,c^{2} d +72 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b e -54 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} e -24 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b e +6 \arcsin \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2} e +36 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} e -54 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b e -12 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} e +6 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b e}{144 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\) | \(850\) |
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 \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )\, 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 = 1.76, size = 2814, normalized size = 5.84 \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 (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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