Optimal. Leaf size=1685 \[ \frac {16 b^2 d f^2 g \sqrt {d-c^2 d x^2}}{25 c^2}+\frac {304 b^2 d g^3 \sqrt {d-c^2 d x^2}}{3675 c^4}-\frac {15}{64} b^2 d f^3 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d f g^2 x \sqrt {d-c^2 d x^2}}{384 c^2}-\frac {43}{576} b^2 d f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{36} b^2 c^2 d f g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {4 a b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {152 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4}-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {9 b^2 d f^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d f g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{384 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d g^3 x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{5 \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{105 c \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{16 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{49 \sqrt {1-c^2 x^2}}+\frac {b d f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 c}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^4}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{7} d g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{16 b c^3 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.56, antiderivative size = 1685, normalized size of antiderivative = 1.00, number of steps
used = 56, number of rules used = 27, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {4861, 4847,
4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1261, 712, 4787, 4783, 4795, 14, 4777,
470, 4715, 267, 272, 45, 457, 78} \begin {gather*} \frac {2 b c^3 d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^7}{49 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^6}{6 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^5}{175 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{36} b^2 c^2 d f g^2 \sqrt {d-c^2 d x^2} x^5+\frac {3}{35} d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^4+\frac {1}{7} d 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 {7 b c d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^4}{16 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {1}{2} d 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 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{105 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{5 \sqrt {1-c^2 x^2}}-\frac {43}{576} b^2 d f g^2 \sqrt {d-c^2 d x^2} x^3-\frac {d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^2}{35 c^2}-\frac {3 b c d f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{16 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x}{16 c^2}+\frac {1}{4} d 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 g^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) x}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x}{5 c \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d f^3 \sqrt {d-c^2 d x^2} x-\frac {7 b^2 d f g^2 \sqrt {d-c^2 d x^2} x}{384 c^2}-\frac {1}{32} b^2 d f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x+\frac {4 a b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {d f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{16 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^4}-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{5 c^2}+\frac {9 b^2 d f^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d f g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{384 c^3 \sqrt {1-c^2 x^2}}+\frac {b d f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {304 b^2 d g^3 \sqrt {d-c^2 d x^2}}{3675 c^4}-\frac {2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}+\frac {6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {16 b^2 d f^2 g \sqrt {d-c^2 d x^2}}{25 c^2}+\frac {152 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4}+\frac {8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 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 327
Rule 457
Rule 470
Rule 712
Rule 1261
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 )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f^2 g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d 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}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{\sqrt {1-c^2 x^2}}+\frac {\left (d 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}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (3 d 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}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c d 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}{2 \sqrt {1-c^2 x^2}}+\frac {\left (6 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d 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}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{7 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c d 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}{7 \sqrt {1-c^2 x^2}}\\ &=\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}-\frac {b c d f g^2 x^4 \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^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {2 b c d g^3 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 {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (3 d 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}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (6 b^2 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d 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}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{35 \sqrt {1-c^2 x^2}}-\frac {\left (6 b c d g^3 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d 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}{7 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}-\frac {7 b c d 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 {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{25 \sqrt {1-c^2 x^2}}+\frac {\left (3 d 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}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d 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}{12 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (2 d 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}{35 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d g^3 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d 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}{245 \sqrt {1-c^2 x^2}}+\frac {\left (6 b^2 c^2 d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{175 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f^3 x \sqrt {d-c^2 d x^2}-\frac {3}{64} b^2 d f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{36} b^2 c^2 d f g^2 x^5 \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {7 b c d 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 {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {1-c^2 x^2}}+\frac {\left (9 b^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f g^2 \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 (b^2 c^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{9 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{105 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d 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 )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d 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 )}{175 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f^3 x \sqrt {d-c^2 d x^2}+\frac {3 b^2 d f g^2 x \sqrt {d-c^2 d x^2}}{128 c^2}-\frac {43}{576} b^2 d f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{36} b^2 c^2 d f g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {4 a b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {7 b c d 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 {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f^2 g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{25 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{12 \sqrt {1-c^2 x^2}}+\frac {\left (9 b^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d 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 )}{105 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 d g^3 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d 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 )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d 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 )}{175 \sqrt {1-c^2 x^2}}\\ &=\frac {16 b^2 d f^2 g \sqrt {d-c^2 d x^2}}{25 c^2}-\frac {62 b^2 d g^3 \sqrt {d-c^2 d x^2}}{1225 c^4}-\frac {15}{64} b^2 d f^3 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d f g^2 x \sqrt {d-c^2 d x^2}}{384 c^2}-\frac {43}{576} b^2 d f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{36} b^2 c^2 d f g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {4 a b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {74 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3675 c^4}-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {9 b^2 d f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 d f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {7 b c d 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 {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{24 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d 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 )}{105 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{35 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {16 b^2 d f^2 g \sqrt {d-c^2 d x^2}}{25 c^2}+\frac {304 b^2 d g^3 \sqrt {d-c^2 d x^2}}{3675 c^4}-\frac {15}{64} b^2 d f^3 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d f g^2 x \sqrt {d-c^2 d x^2}}{384 c^2}-\frac {43}{576} b^2 d f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{36} b^2 c^2 d f g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {4 a b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {152 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4}-\frac {1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {9 b^2 d f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{384 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {4 b c d f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {7 b c d 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 {6 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {16 b c d g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g^3 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 {b d 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 )}{8 c}-\frac {2 d g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {d g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{35} d g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d 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 {1}{2} d 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 {1}{7} d 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 {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.70, size = 872, normalized size = 0.52 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (3087000 a^3 c f \left (2 c^2 f^2+g^2\right )-88200 a^2 b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+840 a b^2 c x \left (6720 g^3+35 c^2 g \left (2016 f^2+315 f g x+32 g^2 x^2\right )-21 c^4 x \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+2 c^6 x^3 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (4785152 g^3+c^2 g \left (39250176 f^2-900375 f g x-429824 g^2 x^2\right )+4 c^6 x^3 \left (385875 f^3+592704 f^2 g x+343000 f g^2 x^2+72000 g^3 x^3\right )-2 c^4 x \left (6559875 f^3+5005056 f^2 g x+1843625 f g^2 x^2+278784 g^3 x^3\right )\right )+105 b \left (88200 a^2 c f \left (2 c^2 f^2+g^2\right )-1680 a b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+b^2 c \left (35 g^2 (245 f+1536 g x)+70 c^2 \left (1785 f^3+8064 f^2 g x+1260 f g^2 x^2+128 g^3 x^3\right )-168 c^4 x^2 \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+16 c^6 x^4 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )\right ) \text {ArcSin}(c x)-88200 b^2 \left (-105 a c f \left (2 c^2 f^2+g^2\right )+b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )\right ) \text {ArcSin}(c x)^2+3087000 b^3 c f \left (2 c^2 f^2+g^2\right ) \text {ArcSin}(c x)^3\right )}{49392000 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 = 0.93, size = 4216, normalized size = 2.50
method | result | size |
default | \(\text {Expression too large to display}\) | \(4216\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{3}\, dx \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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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