Optimal. Leaf size=1108 \[ \frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d f 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^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}+\frac {b d 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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 {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{6} d 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 {2 d f 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^2 \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 g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.95, antiderivative size = 1108, normalized size of antiderivative = 1.00, number of steps
used = 36, number of rules used = 21, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {4861, 4847,
4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1261, 712, 4787, 4783, 4795, 14, 4777,
470} \begin {gather*} \frac {b c^3 d g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^6}{18 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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}{108} b^2 c^2 d g^2 \sqrt {d-c^2 d x^2} x^5-\frac {7 b c d g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^4}{48 \sqrt {1-c^2 x^2}}+\frac {1}{8} d g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {1}{6} d 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 {8 b c d f g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{15 \sqrt {1-c^2 x^2}}-\frac {43 b^2 d g^2 \sqrt {d-c^2 d x^2} x^3}{1728}-\frac {3 b c d f^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{8 \sqrt {1-c^2 x^2}}+\frac {b d 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^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x-\frac {d 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^2 \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 d f 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^2 \sqrt {d-c^2 d x^2} x-\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} x}{1152 c^2}-\frac {1}{32} b^2 d f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x+\frac {d f^2 \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 g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d f 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^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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 {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 200
Rule 201
Rule 222
Rule 327
Rule 470
Rule 712
Rule 1261
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)^2 \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)^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 \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+2 f g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^2 x^2 \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^2 \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 (2 d f 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 (d 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 {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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^2 \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 (4 b d f 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 (d 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 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}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b d f 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^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}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f^2 \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^2 \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 (4 b^2 d f 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 (d 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 (b c d 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 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}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b d f 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^2 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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^2 \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^2 \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 (4 b^2 d f 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}{75 \sqrt {1-c^2 x^2}}+\frac {\left (d 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 (b d 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 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}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d 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 {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {1}{64} b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b d f 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^2 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 {b d 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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 g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f^2 \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^2 \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 (2 b^2 d f 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 )}{75 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d 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 (b^2 d 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 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 d g^2 x \sqrt {d-c^2 d x^2}}{128 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {4 b d f 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^2 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 {b d 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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 g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d f 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 )}{75 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d 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 (b^2 d 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 {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {b^2 d 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 d f 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^2 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 {b d 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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 g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d f 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^2 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 {b d 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 {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f 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 {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \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^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 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}{6} d 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 {2 d f 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^2 \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 g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 616, normalized size = 0.56 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (9000 a^3 \left (6 c^2 f^2+g^2\right )+120 a b^2 c^2 x \left (450 c^2 f^2 x \left (-5+c^2 x^2\right )+192 f g \left (15-10 c^2 x^2+3 c^4 x^4\right )+25 g^2 x \left (9-21 c^2 x^2+8 c^4 x^4\right )\right )-1800 a^2 b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )+b^3 c \sqrt {1-c^2 x^2} \left (6750 c^2 f^2 x \left (-17+2 c^2 x^2\right )+1536 f g \left (149-38 c^2 x^2+9 c^4 x^4\right )+125 g^2 x \left (-21-86 c^2 x^2+32 c^4 x^4\right )\right )+15 b \left (1800 a^2 \left (6 c^2 f^2+g^2\right )+b^2 \left (175 g^2+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )-120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )\right )-240 a b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \text {ArcSin}(c x)+1800 b^2 \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \text {ArcSin}(c x)^2+9000 b^3 \left (6 c^2 f^2+g^2\right ) \text {ArcSin}(c x)^3\right )}{432000 b c^3 \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 = 0.82, size = 3063, normalized size = 2.76
method | result | size |
default | \(\text {Expression too large to display}\) | \(3063\) |
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 )^{2}\, 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 )}^2\,{\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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