Optimal. Leaf size=1154 \[ \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} \text {ArcSin}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(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} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))}{c \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{25 \sqrt {1-c^2 x^2}}-\frac {2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{c^2}+\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{8 b c^3 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.02, 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 = {4861,
4847, 4741, 4737, 4723, 327, 222, 4767, 4739, 455, 45, 4783, 4795, 4715, 267, 272}
\begin {gather*} -\frac {2 b c g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{5} g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^4-\frac {3 b c f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^4}{8 \sqrt {1-c^2 x^2}}+\frac {3}{4} f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x^3+\frac {2 b g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^3}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) 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} (a+b \text {ArcSin}(c x))^2 x^2}{15 c^2}-\frac {b c f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) x^2}{8 c \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 x}{8 c^2}+\frac {4 b^2 g^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) 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} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{8 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 g^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{15 c^4}-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{c^2}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 222
Rule 267
Rule 272
Rule 327
Rule 455
Rule 4715
Rule 4723
Rule 4737
Rule 4739
Rule 4741
Rule 4767
Rule 4783
Rule 4795
Rule 4847
Rule 4861
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 )}{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 ) \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 )}{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 = 0.79, size = 696, normalized size = 0.60 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2+\frac {3}{4} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{5} g^3 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-\frac {f^2 g \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{c^2}+\frac {f^3 (a+b \text {ArcSin}(c x))^3}{6 b c}-\frac {2 b g^3 \left (15 a c^5 x^5+b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+15 b c^5 x^5 \text {ArcSin}(c x)\right )}{375 c^4}-\frac {2 b f^2 g \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \text {ArcSin}(c x)\right )}{9 c^2}-\frac {b f^3 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{4 c}-\frac {3 b f g^2 \left (8 a c^4 x^4+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b \left (-3+8 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{64 c^3}-\frac {f g^2 \left (6 b c x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 (a+b \text {ArcSin}(c x))^3-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )\right )}{16 b c^3}-\frac {g^3 \left (9 c^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 b \left (b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+3 c^3 x^3 (a+b \text {ArcSin}(c x))\right )+18 \left (\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \text {ArcSin}(c x)\right )\right )\right )}{135 c^4}\right )}{\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 = 0.88, size = 2754, normalized size = 2.39
method | result | size |
default | \(\text {Expression too large to display}\) | \(2754\) |
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 \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 \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\,\sqrt {d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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