Optimal. Leaf size=651 \[ \frac {160 b^2 d^2 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {80 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {50 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2 \]
[Out]
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Rubi [A]
time = 0.82, antiderivative size = 651, normalized size of antiderivative = 1.00, number of steps
used = 27, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {4787,
4783, 4795, 4767, 4715, 267, 4723, 272, 45, 14, 4777, 12, 457, 78, 276, 1265, 911, 1167}
\begin {gather*} -\frac {d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{63 c^2}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{21 \sqrt {1-c^2 x^2}}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{189 c \sqrt {1-c^2 x^2}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{63 c^4}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{441 \sqrt {1-c^2 x^2}}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 x \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {50 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}+\frac {160 b^2 d^2 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {80 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 45
Rule 78
Rule 267
Rule 272
Rule 276
Rule 457
Rule 911
Rule 1167
Rule 1265
Rule 4715
Rule 4723
Rule 4767
Rule 4777
Rule 4783
Rule 4787
Rule 4795
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} (5 d) \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{9 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{63 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} \left (5 d^2\right ) \int x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (10 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {1-c^2 x^2}} \, dx}{9 \sqrt {1-c^2 x^2}}\\ &=-\frac {8 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{21 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{21 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{2835 \sqrt {1-c^2 x^2}}+\frac {\left (10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{63 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{63 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2835 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{441 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{105 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{2835 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{189 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (7-5 c^2 x\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{441 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}\\ &=\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{c^4}+\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}-\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{2835 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{189 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1-c^2 x}}+\frac {\sqrt {1-c^2 x}}{c^4}-\frac {8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{441 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}\\ &=-\frac {134 b^2 d^2 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {122 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {50 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{189 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{63 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {160 b^2 d^2 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {80 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {50 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 270, normalized size = 0.41 \begin {gather*} -\frac {d^2 \sqrt {d-c^2 d x^2} \left (3969 a^2 \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right )+126 a b c x \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \sqrt {1-c^2 x^2} \left (-6140+899 c^2 x^2+1005 c^4 x^4-1147 c^6 x^6+343 c^8 x^8\right )+126 b \left (63 a \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right )+b c x \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )\right ) \text {ArcSin}(c x)+3969 b^2 \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{250047 c^4 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.44, size = 2146, normalized size = 3.30
method | result | size |
default | \(\text {Expression too large to display}\) | \(2146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 401, normalized size = 0.62 \begin {gather*} -\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a^{2} - \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {-c^{2} x^{2} + 1} c^{6} d^{\frac {5}{2}} x^{8} - 1147 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {5}{2}} x^{6} + 1005 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} + 899 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} - \frac {6140 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}}}{c^{2}}}{c^{2}} + \frac {63 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} a b}{3969 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.23, size = 486, normalized size = 0.75 \begin {gather*} \frac {126 \, {\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x + {\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (343 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \, {\left (51597 \, a^{2} - 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} - 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \, {\left (15876 \, a^{2} - 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} - 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \, {\left (3969 \, a^{2} - 6140 \, b^{2}\right )} d^{2} + 3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 7938 \, {\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} - c^{4}\right )}} \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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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