Integrand size = 33, antiderivative size = 1108 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\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} \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} \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 \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 \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 \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 \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \]
[Out]
Time = 1.00 (sec) , antiderivative size = 1108, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 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} \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {b c^3 d g^2 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \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 \arcsin (c x))^2 x^3-\frac {8 b c d f g \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2 x-\frac {d g^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^2 x+\frac {4 b d f g \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \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} \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 \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} \]
[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*} \text {integral}& = \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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2+2 f g x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \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 \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 \arcsin (c x))^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} (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x)) \, 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 (a+b \arcsin (c x)) \, 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} (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = \frac {4 b d f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 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 \arcsin (c x))}{15 \sqrt {1-c^2 x^2}}-\frac {b c d g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{12 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, 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 (a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, 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} (a+b \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 \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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 {(a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, 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} (a+b \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 \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 \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 \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^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} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^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} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {b^2 d g^2 \sqrt {d-c^2 d x^2} \arcsin (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} (a+b \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 \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 \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 \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^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} \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} \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 \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 \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 \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 \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 \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 \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 \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 \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \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 \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 \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 \arcsin (c x))^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \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 \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 616, normalized size of antiderivative = 0.56 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (9000 a^3 \left (6 c^2 f^2+g^2\right )+120 a b^2 c^2 x \left (450 c^2 f^2 x \left (-5+c^2 x^2\right )+192 f g \left (15-10 c^2 x^2+3 c^4 x^4\right )+25 g^2 x \left (9-21 c^2 x^2+8 c^4 x^4\right )\right )-1800 a^2 b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )+b^3 c \sqrt {1-c^2 x^2} \left (6750 c^2 f^2 x \left (-17+2 c^2 x^2\right )+1536 f g \left (149-38 c^2 x^2+9 c^4 x^4\right )+125 g^2 x \left (-21-86 c^2 x^2+32 c^4 x^4\right )\right )+15 b \left (1800 a^2 \left (6 c^2 f^2+g^2\right )+b^2 \left (175 g^2+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )-120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )\right )-240 a b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)+1800 b^2 \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)^2+9000 b^3 \left (6 c^2 f^2+g^2\right ) \arcsin (c x)^3\right )}{432000 b c^3 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 3032, normalized size of antiderivative = 2.74
method | result | size |
default | \(\text {Expression too large to display}\) | \(3032\) |
parts | \(\text {Expression too large to display}\) | \(3032\) |
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\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
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\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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