Optimal. Leaf size=2290 \[ \text {result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.18, antiderivative size = 2290, normalized size of antiderivative = 1.00, number of steps
used = 77, number of rules used = 32, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.970, Rules used = {4861,
4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1813, 1864, 4787,
4783, 4795, 14, 4777, 470, 272, 45, 1281, 4715, 267, 457, 78, 276, 1265, 911, 1167}
\begin {gather*} -\frac {2 b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^9}{81 \sqrt {1-c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^8}{32 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^7}{441 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^7}{49 \sqrt {1-c^2 x^2}}-\frac {3}{256} b^2 c^4 d^2 f g^2 \sqrt {d-c^2 d x^2} x^7+\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^6}{48 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^5}{21 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^5}{35 \sqrt {1-c^2 x^2}}+\frac {209 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2} x^5}{4608}+\frac {1}{21} d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^4+\frac {1}{9} d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^4+\frac {5}{63} d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^4-\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^4}{128 \sqrt {1-c^2 x^2}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {3}{8} d^2 f g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {5}{16} d^2 f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{189 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{7 \sqrt {1-c^2 x^2}}-\frac {1079 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} x^3}{18432}-\frac {d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^2}{63 c^2}-\frac {5 b c d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{128 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x+\frac {5}{24} d^2 f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x+\frac {4 b^2 d^2 g^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) x}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x}{7 c \sqrt {1-c^2 x^2}}-\frac {245 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} x}{1152}-\frac {359 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} x}{12288 c^2}-\frac {1}{108} b^2 d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} x-\frac {65 b^2 d^2 f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{1728}+\frac {4 a b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{128 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{63 c^4}-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{7 c^2}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{12288 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{18 c}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{48 c}-\frac {2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {160 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {80 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 45
Rule 78
Rule 200
Rule 201
Rule 222
Rule 267
Rule 272
Rule 276
Rule 327
Rule 457
Rule 470
Rule 911
Rule 1167
Rule 1265
Rule 1281
Rule 1813
Rule 1864
Rule 4715
Rule 4723
Rule 4737
Rule 4739
Rule 4741
Rule 4743
Rule 4767
Rule 4777
Rule 4783
Rule 4787
Rule 4795
Rule 4847
Rule 4861
Rubi steps
\begin {align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (6 b d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{9 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c d^2 g^3 \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 {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \sqrt {1-c^2 x^2}}-\frac {\left (6 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{21 \sqrt {1-c^2 x^2}}-\frac {\left (10 b c d^2 g^3 \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 g^3 \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 {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {11 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {8 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (6 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (15 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^3 \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 g^3 \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 g^3 \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 g^3 \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 {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{256 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (15 b d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{96 \sqrt {1-c^2 x^2}}+\frac {\left (15 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (2 d^2 g^3 \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 g^3 \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 g^3 \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 g^3 \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 g^3 \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 {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {15}{512} b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 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 {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (45 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{512 \sqrt {1-c^2 x^2}}-\frac {\left (15 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{1536 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{72 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g^3 \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 g^3 \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 g^3 \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 g^3 \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 g^3 \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 {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}+\frac {15 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 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 {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2048 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{96 \sqrt {1-c^2 x^2}}+\frac {\left (45 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{1024 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (15 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{256 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g^3 \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 g^3 \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 g^3 \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 g^3 \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 g^3 \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 {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {134 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}}{3969 c^4}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{12288 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {122 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}-\frac {15 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1024 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 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 {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4096 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{192 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g^3 \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 g^3 \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 {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {160 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}}{3969 c^4}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{12288 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {80 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{12288 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 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 {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 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 {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 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 {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 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 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 1114, normalized size = 0.49 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (333396000 a^3 \left (8 c^3 f^3+3 c f g^2\right )+3175200 a^2 b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )-10080 a b^2 c x \left (-161280 g^3-105 c^2 g \left (20736 f^2+2835 f g x+256 g^2 x^2\right )+945 c^4 x \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )-72 c^6 x^3 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )+20 c^8 x^5 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )-b^3 \sqrt {1-c^2 x^2} \left (-1257472000 g^3+c^2 g \left (-12905422848 f^2+748057275 f g x+184115200 g^2 x^2\right )+400 c^8 x^5 \left (592704 f^3+1119744 f^2 g x+750141 f g^2 x^2+175616 g^3 x^3\right )-8 c^6 x^3 \left (179663400 f^3+262020096 f^2 g x+145166175 f g^2 x^2+29363200 g^3 x^3\right )+6 c^4 x \left (1107615600 f^3+753463296 f^2 g x+249815475 f g^2 x^2+34304000 g^3 x^3\right )\right )+315 b \left (3175200 a^2 \left (8 c^3 f^3+3 c f g^2\right )+20160 a b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )+b^2 c \left (315 g^2 (7539 f+16384 g x)-30240 c^4 x^2 \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )+3360 c^2 \left (6279 f^3+20736 f^2 g x+2835 f g^2 x^2+256 g^3 x^3\right )+2304 c^6 x^4 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )-640 c^8 x^6 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )\right ) \text {ArcSin}(c x)+3175200 b^2 \left (315 a \left (8 c^3 f^3+3 c f g^2\right )+b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )\right ) \text {ArcSin}(c x)^2+333396000 b^3 c f \left (8 c^2 f^2+3 g^2\right ) \text {ArcSin}(c x)^3\right )}{25604812800 b 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 = 1.02, size = 6031, normalized size = 2.63
method | result | size |
default | \(\text {Expression too large to display}\) | \(6031\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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: RuntimeError} \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 (f+g\,x\right )}^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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