3.69 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2}{f+g x} \, dx\)

Optimal. Leaf size=2989 \[ \text {result too large to display} \]

[Out]

4/9*b^2*d^2*(c^2*f^2-2*g^2)*(-c^2*d*x^2+d)^(1/2)/g^3-2*b^2*d^2*(c^2*f^2-g^2)^2*(-c^2*d*x^2+d)^(1/2)/g^5+26/675
*b^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/g-2/125*b^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/g+a^2*d^2*(c^2*
f^2-g^2)^2*(-c^2*d*x^2+d)^(1/2)/g^5+2/27*b^2*d^2*(c^2*f^2-2*g^2)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/g^3-1/15*c^
2*d^2*x^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/g+1/5*c^4*d^2*x^4*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/
g-1/3*d^2*(c^2*f^2-2*g^2)*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/g^3+b^2*d^2*(c^2*f^2-g^2)^2*ar
csin(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g^5-2/9*b*c^3*d^2*(c^2*f^2-2*g^2)*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/
g^3/(-c^2*x^2+1)^(1/2)+1/8*b*c^5*d^2*f*x^4*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)-1/6*c
*d^2*f*(c^2*f^2-2*g^2)*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/g^4/(-c^2*x^2+1)^(1/2)+1/3*c*d^2*(c^2*f^2-g^
2)^2*x*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/g^5/(-c^2*x^2+1)^(1/2)+1/3*d^2*(c^2*f^2-g^2)^3*(a+b*arcsin(c
*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/g^6/(g*x+f)/(-c^2*x^2+1)^(1/2)+I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)+1/3*d^
2*(c^2*f^2-g^2)^2*(a+b*arcsin(c*x))^3*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/b/c/g^4/(g*x+f)-I*b^2*d^2*(c^2*f
^2-g^2)^(5/2)*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*d*x^2+d)^(1/2
)/g^6/(-c^2*x^2+1)^(1/2)-2*a*b*c*d^2*(c^2*f^2-g^2)^2*x*(-c^2*d*x^2+d)^(1/2)/g^5/(-c^2*x^2+1)^(1/2)-1/4*b^2*c*d
^2*f*(c^2*f^2-2*g^2)*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-2*b^2*c*d^2*(c^2*f^2-g^2)^2*x*arc
sin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^5/(-c^2*x^2+1)^(1/2)+2/3*b*c*d^2*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x))*(-c^2*d*x
^2+d)^(1/2)/g^3/(-c^2*x^2+1)^(1/2)-1/8*b*c^3*d^2*f*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)
^(1/2)+2*I*a*b*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)-2*I*a*b*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)+1/2*b*c^3*d^2*f*(c^2*
f^2-2*g^2)*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-2*a*b*d^2*(c^2*f^2-g^2)^(5/2)*pol
ylog(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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)-2
*b^2*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)-2*I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)+1/4*b^2*c^2*d^2*f*(c^2*f^2-2*g^2)
*x*(-c^2*d*x^2+d)^(1/2)/g^4+2*I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x)
)^2*(-c^2*d*x^2+d)^(1/2)/g^4+4/15*a*b*c*d^2*x*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+1/64*b^2*c*d^2*f*arcsi
n(c*x)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)+4/15*b^2*c*d^2*x*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x
^2+1)^(1/2)+2/45*b*c^3*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)-2/25*b*c^5*d^2*x^5*
(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)-1/24*c*d^2*f*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1
/2)/b/g^2/(-c^2*x^2+1)^(1/2)+2*a*b*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)+2*b^2*d^2*(c^2*f^2-g^2)^(5/2)*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*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)+2*a*b
*d^2*(c^2*f^2-g^2)^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^5+1/8*c^2*d^2*f*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(
1/2)/g^2-1/4*c^4*d^2*f*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/g^2-a^2*d^2*(c^2*f^2-g^2)^(5/2)*arctan((c^
2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/g^6/(-c^2*x^2+1)^(1/2)-1/64*b^2*c^2*d^2*
f*x*(-c^2*d*x^2+d)^(1/2)/g^2+1/32*b^2*c^4*d^2*f*x^3*(-c^2*d*x^2+d)^(1/2)/g^2+52/225*b^2*d^2*(-c^2*d*x^2+d)^(1/
2)/g-2/15*d^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/g

________________________________________________________________________________________

Rubi [A]  time = 4.94, antiderivative size = 2989, normalized size of antiderivative = 1.00, number of steps used = 74, number of rules used = 35, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.061, Rules used = {4777, 4767, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707, 4619, 261, 266, 4765, 683, 4757, 4799, 1654, 12, 725, 204, 4797, 8, 4773, 3323, 2264, 2190, 2279, 2391, 2531, 2282, 6589} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]

[Out]

(52*b^2*d^2*Sqrt[d - c^2*d*x^2])/(225*g) + (4*b^2*d^2*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2])/(9*g^3) + (a^2*d^
2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (2*b^2*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (b^2*c^
2*d^2*f*x*Sqrt[d - c^2*d*x^2])/(64*g^2) + (b^2*c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(4*g^4) + (b
^2*c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2])/(32*g^2) + (4*a*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2])
- (2*a*b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (26*b^2*d^2*(1 - c^2*x^2)*Sq
rt[d - c^2*d*x^2])/(675*g) + (2*b^2*d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*g^3) - (2*b^2
*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*g) + (2*a*b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*
x])/g^5 + (b^2*c*d^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*g^2*Sqrt[1 - c^2*x^2]) - (b^2*c*d^2*f*(c^2*f^2 - 2
*g^2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*g^4*Sqrt[1 - c^2*x^2]) + (4*b^2*c*d^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c
*x])/(15*g*Sqrt[1 - c^2*x^2]) - (2*b^2*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^5*Sqrt[1
- c^2*x^2]) + (b^2*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g^5 + (2*b*c*d^2*(c^2*f^2 - 2*g^2)
*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g^3*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcSin[c*x]))/(8*g^2*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(2*g^4*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*g*Sqrt
[1 - c^2*x^2]) - (2*b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*g^3*Sqrt[1 - c
^2*x^2]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*
x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*g*Sqrt[1 - c^2*x^2]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x])^2)/(15*g) + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2
*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*g^4) - (c^2*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x])^2)/(15*g) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*g^2) + (c^4*d^2*x^4*Sqrt[d - c^2*
d*x^2]*(a + b*ArcSin[c*x])^2)/(5*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c
*x])^2)/(3*g^3) - (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(24*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d^2*f*
(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*g^4*Sqrt[1 - c^2*x^2]) + (c*d^2*(c^2*f^2 - g
^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g^5*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^3*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 -
c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)) - (a^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sq
rt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) + ((2
*I)*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[
c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2
*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - ((2*I)*a*b*d^2*(c^2*f
^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/
(g^6*Sqrt[1 - c^2*x^2]) - (I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*A
rcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d
- c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*b^
2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2
*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^
(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sq
rt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c
^2*x^2]) + ((2*I)*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f -
Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - ((2*I)*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*Poly
Log[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4697

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4757

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]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist
[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]

Rule 4765

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*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[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]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[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] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rule 4797

Int[ArcSin[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4799

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] &
& IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^4}+\frac {c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^3}-\frac {c^4 f x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^2}+\frac {c^4 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g}+\frac {\left (-c^2 f^2+g^2\right )^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^4 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\left (c^4 d^2 f \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}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^4 d^2 \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}{g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \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}{g^4 \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \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}{g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (c^4 d^2 f \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 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c^5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^4 d^2 \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 g \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \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 g^4 \sqrt {1-c^2 x^2}}+\frac {\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^4 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c d^2 \left (c^2 f^2-2 g^2\right ) \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}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3}{(f+g x)^2} \, dx}{3 b c g^4 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (c^2 d^2 f \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 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^6 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 c^2 d^2 \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 g \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 g \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{25 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \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}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (3 b^2 c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (4 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{45 g \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^6 d^2 \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 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \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 )}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {a^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {2 a b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (3 b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 c d^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \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 g \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^6 d^2 \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 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \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 )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b^2 d^2 \sqrt {d-c^2 d x^2}}{25 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (4 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \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 g \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^8 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (2 a b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {\left (2 a b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{g^5 \sqrt {1-c^2 x^2}}+\frac {\left (a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (4 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (4 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (4 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f-\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 4.98, size = 1277, normalized size = 0.43 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c^4}{5 g}-\frac {f x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c^4}{4 g^2}-\frac {f \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c^2}{2 g^4}-\frac {f \left (c^2 f^2-2 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3 c}{6 b g^4}+\frac {b f \left (c^2 f^2-2 g^2\right ) \left (c x \left (\sqrt {1-c^2 x^2} b+2 a c x\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right ) c}{4 g^4}+\frac {b f \left (8 a c^4 x^4+b c \sqrt {1-c^2 x^2} \left (2 c^2 x^2+3\right ) x+b \left (8 c^4 x^4-3\right ) \sin ^{-1}(c x)\right ) c}{64 g^2}+\frac {f \left (-2 \left (a+b \sin ^{-1}(c x)\right )^3+6 b c x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-3 b^2 \left (c x \left (\sqrt {1-c^2 x^2} b+2 a c x\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )\right ) c}{48 b g^2}-\frac {\left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}+\frac {2 b \left (c^2 f^2-2 g^2\right ) \left (-3 c^3 \left (a+b \sin ^{-1}(c x)\right ) x^3+9 c \left (a+b \sin ^{-1}(c x)\right ) x-b \sqrt {1-c^2 x^2} \left (c^2 x^2-7\right )\right )}{27 g^3}-\frac {2 b \left (15 c^5 \left (a+b \sin ^{-1}(c x)\right ) x^5+b \sqrt {1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )\right )}{375 g}-\frac {9 c^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (3 c^3 \left (a+b \sin ^{-1}(c x)\right ) x^3+b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )\right )+18 \left (\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (c x \sin ^{-1}(c x) b+\sqrt {1-c^2 x^2} b+a c x\right )\right )}{135 g}-\frac {\left (g^2-c^2 f^2\right )^2 \left (c^2 x^2-1\right ) \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^4 (f+g x) c}+\frac {\left (g^2-c^2 f^2\right )^2 \left (\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3+c^2 g x (f+g x) \left (a+b \sin ^{-1}(c x)\right )^3+3 b c (f+g x) \left (g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b g \left (c x \sin ^{-1}(c x) b+\sqrt {1-c^2 x^2} b+a c x\right )+i \sqrt {c^2 f^2-g^2} \left (2 \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b^2-2 \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b^2-2 i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b+2 i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b+\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (\frac {i e^{i \sin ^{-1}(c x)} g}{\sqrt {c^2 f^2-g^2}-c f}+1\right )-\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{3 b g^6 (f+g x) c}\right )}{\sqrt {1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/g^4 - (c^4*
f*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*g^2) + (c^4*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(5*
g) - ((c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*g^3) - (c*f*(c^2*f^2 - 2*g^2)*(a + b*Arc
Sin[c*x])^3)/(6*b*g^4) - ((-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)) + (
b*c*f*(c^2*f^2 - 2*g^2)*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*g^4) + (b*c
*f*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*g^2) + (2*b*(
c^2*f^2 - 2*g^2)*(-(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2)) + 9*c*x*(a + b*ArcSin[c*x]) - 3*c^3*x^3*(a + b*ArcSin[
c*x])))/(27*g^3) - (2*b*(b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + 15*c^5*x^5*(a + b*ArcSin[c*x])))/(3
75*g) + (c*f*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x
+ b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(48*b*g^2) - (9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*(a + b*ArcSin[c*x])) + 18*(Sqrt[1 - c^2*x^2]
*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x])))/(135*g) + ((-(c^2*f^2) + g^2)
^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 + 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*ArcSin[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^6*(f + g*x))))/Sqrt[1 - c^
2*x^2]

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fricas [F]  time = 1.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/(g*x+f),x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
in(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(g*x + f), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{g x +f}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/(g*x+f),x)

[Out]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/(g*x+f),x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for m
ore details)Is g-c*f zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/(f + g*x),x)

[Out]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/(f + g*x), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2/(g*x+f),x)

[Out]

Timed out

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