Integrand size = 33, antiderivative size = 513 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 f g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \]
2*f*g*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+(c^2*f^2+g^2)*x*(a+b* arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)-I*(c^2*f^2+g^2)*(a+b*arcsin(c*x) )^2*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/3*g^2*(a+b*arcsin(c*x) )^3*(-c^2*x^2+1)^(1/2)/b/c^3/d/(-c^2*d*x^2+d)^(1/2)+8*I*b*f*g*(a+b*arcsin( c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^ 2+d)^(1/2)+2*b*(c^2*f^2+g^2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1 /2))^2)*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-4*I*b^2*f*g*polylog( 2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^( 1/2)+4*I*b^2*f*g*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2 )/c^2/d/(-c^2*d*x^2+d)^(1/2)-I*b^2*(c^2*f^2+g^2)*polylog(2,-(I*c*x+(-c^2*x ^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)
Time = 1.58 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.50 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\frac {2 g^2 (a+b \arcsin (c x))^3}{b}+3 (-c f+g)^2 \left (-(a+b \arcsin (c x))^2 \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+i \left ((a+b \arcsin (c x)) \left (a+b \arcsin (c x)-4 i b \log \left (1+i e^{-i \arcsin (c x)}\right )\right )+4 b^2 \operatorname {PolyLog}\left (2,-i e^{-i \arcsin (c x)}\right )\right )\right )-3 (c f+g)^2 \left (i \left ((a+b \arcsin (c x)) \left (a+b \arcsin (c x)+4 i b \log \left (1+i e^{i \arcsin (c x)}\right )\right )+4 b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{6 c^3 d \sqrt {d-c^2 d x^2}} \]
(Sqrt[1 - c^2*x^2]*((-2*g^2*(a + b*ArcSin[c*x])^3)/b + 3*(-(c*f) + g)^2*(- ((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSin[c*x])/4]) + I*((a + b*ArcSin[c*x ])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*Poly Log[2, (-I)/E^(I*ArcSin[c*x])])) - 3*(c*f + g)^2*(I*((a + b*ArcSin[c*x])*( a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[ 2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x ])/4])))/(6*c^3*d*Sqrt[d - c^2*d*x^2])
Time = 1.11 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.61, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5276, 5274, 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.
| \(\displaystyle \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5274 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (f^2 c^2+2 f g x c^2+g^2\right ) (a+b \arcsin (c x))^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {g^2 (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {8 i b f g \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^2}-\frac {g^2 (a+b \arcsin (c x))^3}{3 b c^3}+\frac {x \left (c^2 f^2+g^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}+\frac {2 f g (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}-\frac {i \left (c^2 f^2+g^2\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \left (c^2 f^2+g^2\right ) \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {4 i b^2 f g \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2}+\frac {4 i b^2 f g \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2}-\frac {i b^2 \left (c^2 f^2+g^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^3}\right )}{d \sqrt {d-c^2 d x^2}}\) |
(Sqrt[1 - c^2*x^2]*(((-I)*(c^2*f^2 + g^2)*(a + b*ArcSin[c*x])^2)/c^3 + (2* f*g*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[1 - c^2*x^2]) + ((c^2*f^2 + g^2)*x*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[1 - c^2*x^2]) - (g^2*(a + b*ArcSin[c*x])^3) /(3*b*c^3) + ((8*I)*b*f*g*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/c ^2 + (2*b*(c^2*f^2 + g^2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x] )])/c^3 - ((4*I)*b^2*f*g*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c^2 + ((4*I)* b^2*f*g*PolyLog[2, I*E^(I*ArcSin[c*x])])/c^2 - (I*b^2*(c^2*f^2 + g^2)*Poly Log[2, -E^((2*I)*ArcSin[c*x])])/c^3))/(d*Sqrt[d - c^2*d*x^2])
3.1.76.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x] )^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 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]
Time = 1.08 (sec) , antiderivative size = 975, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(a^{2} \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{3}}{3 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )^{2} \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 i \arcsin \left (c x \right )^{2} c^{2} f^{2}+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) \arcsin \left (c x \right ) c^{2} f^{2}-i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c^{2} f^{2}+4 \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) \arcsin \left (c x \right ) c f g -4 \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) \arcsin \left (c x \right ) c f g -4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c f g +4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c f g -2 i \arcsin \left (c x \right )^{2} g^{2}+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) \arcsin \left (c x \right ) g^{2}-i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) g^{2}\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} f^{2}+g^{2}\right ) \arcsin \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right ) \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}-2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}+2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(975\) |
| parts | \(a^{2} \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{3}}{3 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )^{2} \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 i \arcsin \left (c x \right )^{2} c^{2} f^{2}+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) \arcsin \left (c x \right ) c^{2} f^{2}-i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c^{2} f^{2}+4 \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) \arcsin \left (c x \right ) c f g -4 \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) \arcsin \left (c x \right ) c f g -4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c f g +4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c f g -2 i \arcsin \left (c x \right )^{2} g^{2}+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) \arcsin \left (c x \right ) g^{2}-i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) g^{2}\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} f^{2}+g^{2}\right ) \arcsin \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right ) \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}-2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}+2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(975\) |
a^2*(f^2/d*x/(-c^2*d*x^2+d)^(1/2)+g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/ d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+2*f*g/c^2/d/ (-c^2*d*x^2+d)^(1/2))+b^2*(1/3*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d ^2/c^3/(c^2*x^2-1)*g^2*arcsin(c*x)^3-(-d*(c^2*x^2-1))^(1/2)*(c*x+I*(-c^2*x ^2+1)^(1/2))*arcsin(c*x)^2*(c^2*f^2+g^2-2*I*(-c^2*x^2+1)^(1/2)*c*f*g+2*x*c ^2*f*g)/d^2/c^3/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(-2* I*arcsin(c*x)^2*c^2*f^2+2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*arcsin(c*x)*c ^2*f^2-I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*c^2*f^2+4*ln(1+I*(I*c*x+ (-c^2*x^2+1)^(1/2)))*arcsin(c*x)*c*f*g-4*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)) )*arcsin(c*x)*c*f*g-4*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))*c*f*g+4*I*di log(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*c*f*g-2*I*arcsin(c*x)^2*g^2+2*ln(1+(I* c*x+(-c^2*x^2+1)^(1/2))^2)*arcsin(c*x)*g^2-I*polylog(2,-(I*c*x+(-c^2*x^2+1 )^(1/2))^2)*g^2)/d^2/c^3/(c^2*x^2-1))+2*a*b*(1/2*(-d*(c^2*x^2-1))^(1/2)*(- c^2*x^2+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*g^2*arcsin(c*x)^2+2*I*(-c^2*x^2+1)^(1 /2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^3/(c^2*x^2-1)*(c^2*f^2+g^2)*arcsin(c*x)-( -d*(c^2*x^2-1))^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))*arcsin(c*x)*(c^2*f^2+g^2- 2*I*(-c^2*x^2+1)^(1/2)*c*f*g+2*x*c^2*f*g)/d^2/c^3/(c^2*x^2-1)-(-d*(c^2*x^2 -1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*(c^2*f^2-2*c*f*g+g^2)*ln (I*c*x+(-c^2*x^2+1)^(1/2)+I)-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2 /c^3/(c^2*x^2-1)*(c^2*f^2+2*c*f*g+g^2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I))
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral((a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin (c*x))*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
a^2*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + 2*a *b*f^2*x*arcsin(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a^2*f^2*x/(sqrt(-c^2*d*x^2 + d)*d) - a*b*f^2*log(x^2 - 1/c^2)/(c*d^(3/2)) - sqrt(d)*integrate(((b^2* g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1) )^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/((c^2*d^2*x^2 - d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 2*a^2*f*g/( sqrt(-c^2*d*x^2 + d)*c^2*d)
Exception generated. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]