Integrand size = 33, antiderivative size = 1442 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \arcsin (c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \arcsin (c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b c (f+g x)}-\frac {a^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}} \]
a^2*(-c^2*d*x^2+d)^(1/2)/g-2*b^2*(-c^2*d*x^2+d)^(1/2)/g+2*a*b*arcsin(c*x)* (-c^2*d*x^2+d)^(1/2)/g+b^2*arcsin(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g-2*a*b*c*x* (-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)-2*b^2*c*x*arcsin(c*x)*(-c^2*d*x^ 2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+1/3*c*x*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d) ^(1/2)/b/g/(-c^2*x^2+1)^(1/2)-1/3*(1-c^2*f^2/g^2)*(a+b*arcsin(c*x))^3*(-c^ 2*d*x^2+d)^(1/2)/b/c/(g*x+f)/(-c^2*x^2+1)^(1/2)-a^2*arctan((c^2*f*x+g)/(c^ 2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1 /2)/g^2/(-c^2*x^2+1)^(1/2)+I*b^2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^ (1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/ 2)/g^2/(-c^2*x^2+1)^(1/2)-2*I*a*b*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^( 1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2 )/g^2/(-c^2*x^2+1)^(1/2)+2*I*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/ (c*f-(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(- c^2*x^2+1)^(1/2)+2*I*a*b*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/( c*f-(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c ^2*x^2+1)^(1/2)+2*a*b*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f ^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^ (1/2)+2*b^2*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2 *f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1 )^(1/2)-2*a*b*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^...
Time = 0.97 (sec) , antiderivative size = 516, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^3+c^2 g x (f+g x) (a+b \arcsin (c x))^3+g^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+3 b c (f+g x) \left (g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b g \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )+i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{3 b c g^2 (f+g x) \sqrt {1-c^2 x^2}} \]
(Sqrt[d - c^2*d*x^2]*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^3 + g^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^3 + 3* b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*A rcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2 ])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^ 2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x ])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2 , (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I *E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(3*b*c*g^2*(f + g*x )*Sqrt[1 - c^2*x^2])
Time = 3.19 (sec) , antiderivative size = 1018, normalized size of antiderivative = 0.71, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {5276, 5264, 25, 5256, 25, 5298, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{f+g x}dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5264 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}-\frac {\int -\frac {\left (g x^2 c^2+2 f x c^2+g\right ) (a+b \arcsin (c x))^3}{(f+g x)^2}dx}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {\left (g x^2 c^2+2 f x c^2+g\right ) (a+b \arcsin (c x))^3}{(f+g x)^2}dx}{3 b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5256 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {-3 b c \int -\frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (\frac {f^2}{f+g x}+g x\right )}{g^2}\right ) (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arcsin (c x))^3}{f+g x}+\frac {c^2 x (a+b \arcsin (c x))^3}{g}}{3 b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {3 b c \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (\frac {f^2}{f+g x}+g x\right )}{g^2}\right ) (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arcsin (c x))^3}{f+g x}+\frac {c^2 x (a+b \arcsin (c x))^3}{g}}{3 b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5298 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {3 b c \int \left (-\frac {\left (f^2 c^2+g^2 x^2 c^2+f g x c^2-g^2\right ) a^2}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {2 b \left (f^2 c^2+g^2 x^2 c^2+f g x c^2-g^2\right ) \arcsin (c x) a}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b^2 \left (f^2 c^2+g^2 x^2 c^2+f g x c^2-g^2\right ) \arcsin (c x)^2}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right )dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arcsin (c x))^3}{f+g x}+\frac {c^2 x (a+b \arcsin (c x))^3}{g}}{3 b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}+\frac {\frac {c^2 x (a+b \arcsin (c x))^3}{g}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arcsin (c x))^3}{f+g x}+3 b c \left (-\frac {\sqrt {c^2 f^2-g^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right ) a^2}{g^2}+\frac {\sqrt {1-c^2 x^2} a^2}{g}-\frac {2 b c x a}{g}+\frac {2 b \sqrt {1-c^2 x^2} \arcsin (c x) a}{g}+\frac {2 i b \sqrt {c^2 f^2-g^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}-\frac {2 i b \sqrt {c^2 f^2-g^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}+\frac {2 b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}-\frac {2 b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}+\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{g}-\frac {2 b^2 c x \arcsin (c x)}{g}+\frac {i b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {i b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {2 b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 b^2 \sqrt {1-c^2 x^2}}{g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g* x)) + ((c^2*x*(a + b*ArcSin[c*x])^3)/g - ((1 - (c^2*f^2)/g^2)*(a + b*ArcSi n[c*x])^3)/(f + g*x) + 3*b*c*((-2*a*b*c*x)/g + (a^2*Sqrt[1 - c^2*x^2])/g - (2*b^2*Sqrt[1 - c^2*x^2])/g - (2*b^2*c*x*ArcSin[c*x])/g + (2*a*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/g + (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/g - (a^2* Sqrt[c^2*f^2 - g^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2 *x^2])])/g^2 + ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I* ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 + (I*b^2*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g ^2])])/g^2 - ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I*Ar cSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 - (I*b^2*Sqrt[c^2*f^2 - g^ 2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2 ])])/g^2 + (2*a*b*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/( c*f - Sqrt[c^2*f^2 - g^2])])/g^2 + (2*b^2*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]* PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 - (2* a*b*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2 *f^2 - g^2])])/g^2 - (2*b^2*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*PolyLog[2, (I* E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 + ((2*I)*b^2*Sqrt[c ^2*f^2 - g^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2 ])])/g^2 - ((2*I)*b^2*Sqrt[c^2*f^2 - g^2]*PolyLog[3, (I*E^(I*ArcSin[c*x...
3.1.61.3.1 Defintions of rubi rules used
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x _)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcSin[c*x])^n u, x] - Simp[b*c *n Int[SimplifyIntegrand[u*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2] ), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[ p, 0] && EqQ[e*g - 2*d*h, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[ (d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f + g*x)^m*(d + e*x^2)*((a + b*Arc Sin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[1/(b*c*Sqrt[d]*(n + 1)) Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[ c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p _), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) ^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c^{2} d \,x^{2}+d}}{g x +f}d x\]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{g x + f} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor e details)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \]