Optimal. Leaf size=1154 \[ -\frac {2 b c g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{5} g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^4-\frac {3 b c f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{8 \sqrt {1-c^2 x^2}}+\frac {3}{4} f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac {2 b g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{3 \sqrt {1-c^2 x^2}}-\frac {3}{32} b^2 f g^2 \sqrt {d-c^2 d x^2} x^3-\frac {g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^2}{15 c^2}-\frac {b c f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 c \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac {3 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{8 c^2}+\frac {4 b^2 g^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) x}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 f^3 \sqrt {d-c^2 d x^2} x+\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} x}{64 c^2}+\frac {4 a b g^3 \sqrt {d-c^2 d x^2} x}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {52 b^2 g^3 \sqrt {d-c^2 d x^2}}{225 c^4}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {4 b^2 f^2 g \sqrt {d-c^2 d x^2}}{3 c^2}+\frac {26 b^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}+\frac {2 b^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{9 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.55, antiderivative size = 1154, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 16, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {4777, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707, 4619, 261, 266} \[ -\frac {2 b c g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{5} g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^4-\frac {3 b c f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{8 \sqrt {1-c^2 x^2}}+\frac {3}{4} f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac {2 b g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{3 \sqrt {1-c^2 x^2}}-\frac {3}{32} b^2 f g^2 \sqrt {d-c^2 d x^2} x^3-\frac {g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^2}{15 c^2}-\frac {b c f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 c \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac {3 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{8 c^2}+\frac {4 b^2 g^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) x}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 f^3 \sqrt {d-c^2 d x^2} x+\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} x}{64 c^2}+\frac {4 a b g^3 \sqrt {d-c^2 d x^2} x}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {52 b^2 g^3 \sqrt {d-c^2 d x^2}}{225 c^4}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {4 b^2 f^2 g \sqrt {d-c^2 d x^2}}{3 c^2}+\frac {26 b^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}+\frac {2 b^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{9 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 216
Rule 261
Rule 266
Rule 321
Rule 444
Rule 4619
Rule 4627
Rule 4641
Rule 4645
Rule 4647
Rule 4677
Rule 4697
Rule 4707
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int (f+g x)^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\sqrt {d-c^2 d x^2} \int (f+g x)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (f^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f^2 g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (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}{\sqrt {1-c^2 x^2}}+\frac {\left (3 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 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}{\sqrt {1-c^2 x^2}}+\frac {\left (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}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {\left (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}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {1-c^2 x^2}}+\frac {\left (3 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}{4 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (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}{5 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c g^3 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 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 {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 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}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 b f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (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}{15 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b g^3 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{25 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 f^3 x \sqrt {d-c^2 d x^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 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 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (9 b^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 f g^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 (2 b^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{45 \sqrt {1-c^2 x^2}}+\frac {\left (4 b g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 f^3 x \sqrt {d-c^2 d x^2}+\frac {3 b^2 f g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {4 a b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 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 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (9 b^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{45 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 )}{25 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b^2 f^2 g \sqrt {d-c^2 d x^2}}{3 c^2}-\frac {2 b^2 g^3 \sqrt {d-c^2 d x^2}}{25 c^4}-\frac {1}{4} b^2 f^3 x \sqrt {d-c^2 d x^2}+\frac {3 b^2 f g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {4 a b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {4 b^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^4}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 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 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 )}{45 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{15 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b^2 f^2 g \sqrt {d-c^2 d x^2}}{3 c^2}+\frac {52 b^2 g^3 \sqrt {d-c^2 d x^2}}{225 c^4}-\frac {1}{4} b^2 f^3 x \sqrt {d-c^2 d x^2}+\frac {3 b^2 f g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {4 a b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {26 b^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 c^4}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^4}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 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 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 696, normalized size = 0.60 \[ \frac {\sqrt {d-c^2 d x^2} \left (-\frac {f g^2 \left (-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )+6 b c x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 \left (a+b \sin ^{-1}(c x)\right )^3\right )}{16 b c^3}+\frac {1}{2} f^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b f^3 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{4 c}-\frac {f^2 g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}-\frac {2 b f^2 g \left (3 a c x \left (c^2 x^2-3\right )+b \sqrt {1-c^2 x^2} \left (c^2 x^2-7\right )+3 b c x \left (c^2 x^2-3\right ) \sin ^{-1}(c x)\right )}{9 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} g^3 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 b g^3 \left (15 a c^5 x^5+15 b c^5 x^5 \sin ^{-1}(c x)+b \sqrt {1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )\right )}{375 c^4}-\frac {3 b f g^2 \left (8 a c^4 x^4+b \left (8 c^4 x^4-3\right ) \sin ^{-1}(c x)+b c x \sqrt {1-c^2 x^2} \left (2 c^2 x^2+3\right )\right )}{64 c^3}-\frac {g^3 \left (9 c^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+18 \left (\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )-2 b \left (3 c^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )\right )\right )}{135 c^4}+\frac {f^3 \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c}\right )}{\sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} + {\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.70, size = 2947, normalized size = 2.55 \[ \text {output too large to display} \]
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 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a^{2} f^{3} - \frac {1}{15} \, a^{2} g^{3} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} + \frac {3}{8} \, a^{2} f g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2}} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2} d} + \frac {\sqrt {d} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2} f^{2} g}{c^{2} d} + \sqrt {d} \int {\left ({\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int {\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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