Optimal. Leaf size=1025 \[ \frac {g^2 \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b g^2 \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b f g \left (a+b \sin ^{-1}(c x)\right ) x}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.31, antiderivative size = 1025, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4777, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261, 4681, 4703, 288, 216} \[ \frac {g^2 \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b g^2 \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b f g \left (a+b \sin ^{-1}(c x)\right ) x}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 216
Rule 261
Rule 288
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4181
Rule 4651
Rule 4655
Rule 4657
Rule 4675
Rule 4677
Rule 4681
Rule 4703
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {f^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {2 f g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (2 f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 f g \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b f^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 f^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b^2 f g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 g^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 6.26, size = 711, normalized size = 0.69 \[ \frac {\sqrt {1-c^2 x^2} \left (-\frac {(c f-g)^2 \left (-2 \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )+\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{24 c^3}-\frac {(c f+g)^2 \left (2 \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-4 b^2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{24 c^3}+\frac {\left (c^2 f^2-g^2\right ) \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )}{4 c^3}-\frac {\left (c^2 f^2-g^2\right ) \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )}{4 c^3}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} + {\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.21, size = 9710, normalized size = 9.47 \[ \text {output too large to display} \]
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 \[ \frac {1}{3} \, a b c f^{2} {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {2}{3} \, a b f^{2} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, a^{2} f^{2} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} - \frac {1}{3} \, a^{2} g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} + \sqrt {d} \int \frac {{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} + \frac {2 \, a^{2} f g}{3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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