3.66 \(\int (f+g x)^3 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=2290 \[ \text{result too large to display} \]

[Out]

(96*b^2*d^2*f^2*g*Sqrt[d - c^2*d*x^2])/(245*c^2) + (160*b^2*d^2*g^3*Sqrt[d - c^2*d*x^2])/(3969*c^4) - (245*b^2
*d^2*f^3*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(12288*c^2) - (1079*b^2*d^2*f
*g^2*x^3*Sqrt[d - c^2*d*x^2])/18432 + (209*b^2*c^2*d^2*f*g^2*x^5*Sqrt[d - c^2*d*x^2])/4608 - (3*b^2*c^4*d^2*f*
g^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (4*a*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (16*b^2*
d^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2) + (80*b^2*d^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(1
1907*c^4) - (65*b^2*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (36*b^2*d^2*f^2*g*(1 - c^2*x^2)^2*Sqrt
[d - c^2*d*x^2])/(1225*c^2) + (4*b^2*d^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1323*c^4) - (b^2*d^2*f^3*x*
(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 + (6*b^2*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (
50*b^2*d^2*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(27783*c^4) - (2*b^2*d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*
d*x^2])/(729*c^4) + (115*b^2*d^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^
2*f*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12288*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*d^2*g^3*x*Sqrt[d - c^2*d*x^2]*
ArcSin[c*x])/(63*c^3*Sqrt[1 - c^2*x^2]) + (6*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[
1 - c^2*x^2]) - (5*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (15*b*d^2
*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (6*b*c*d^2*f^2*g*x^3*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x
]))/(189*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*Sqrt[1 - c
^2*x^2]) + (18*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c*d^
2*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(21*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c
^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) - (6*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x]))/(49*Sqrt[1 - c^2*x^2]) + (38*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(441*Sqrt[1
- c^2*x^2]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (2*b*c^
5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(81*Sqrt[1 - c^2*x^2]) + (5*b*d^2*f^3*(1 - c^2*x^2)^(3/
2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b
*ArcSin[c*x]))/(18*c) - (2*d^2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^4) + (5*d^2*f^3*x*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/16 - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) -
(d^2*g^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^2) + (15*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a +
b*ArcSin[c*x])^2)/64 + (d^2*g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/21 + (5*d^2*f^3*x*(1 - c^2*x^2
)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (5*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cSin[c*x])^2)/16 + (5*d^2*g^3*x^4*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/63 + (d^2*f^3*x*(1
- c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 + (3*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x])^2)/8 + (d^2*g^3*x^4*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/9 - (3*d
^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(
a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(12
8*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 3.28675, antiderivative size = 2290, normalized size of antiderivative = 1., number of steps used = 77, number of rules used = 32, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.97, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1799, 1850, 4699, 4697, 4707, 14, 4687, 459, 266, 43, 1267, 4619, 261, 446, 77, 270, 1251, 897, 1153} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(96*b^2*d^2*f^2*g*Sqrt[d - c^2*d*x^2])/(245*c^2) + (160*b^2*d^2*g^3*Sqrt[d - c^2*d*x^2])/(3969*c^4) - (245*b^2
*d^2*f^3*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(12288*c^2) - (1079*b^2*d^2*f
*g^2*x^3*Sqrt[d - c^2*d*x^2])/18432 + (209*b^2*c^2*d^2*f*g^2*x^5*Sqrt[d - c^2*d*x^2])/4608 - (3*b^2*c^4*d^2*f*
g^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (4*a*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (16*b^2*
d^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2) + (80*b^2*d^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(1
1907*c^4) - (65*b^2*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (36*b^2*d^2*f^2*g*(1 - c^2*x^2)^2*Sqrt
[d - c^2*d*x^2])/(1225*c^2) + (4*b^2*d^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1323*c^4) - (b^2*d^2*f^3*x*
(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 + (6*b^2*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (
50*b^2*d^2*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(27783*c^4) - (2*b^2*d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*
d*x^2])/(729*c^4) + (115*b^2*d^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^
2*f*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12288*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*d^2*g^3*x*Sqrt[d - c^2*d*x^2]*
ArcSin[c*x])/(63*c^3*Sqrt[1 - c^2*x^2]) + (6*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[
1 - c^2*x^2]) - (5*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (15*b*d^2
*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (6*b*c*d^2*f^2*g*x^3*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x
]))/(189*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*Sqrt[1 - c
^2*x^2]) + (18*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c*d^
2*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(21*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c
^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) - (6*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x]))/(49*Sqrt[1 - c^2*x^2]) + (38*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(441*Sqrt[1
- c^2*x^2]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (2*b*c^
5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(81*Sqrt[1 - c^2*x^2]) + (5*b*d^2*f^3*(1 - c^2*x^2)^(3/
2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b
*ArcSin[c*x]))/(18*c) - (2*d^2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^4) + (5*d^2*f^3*x*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/16 - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) -
(d^2*g^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^2) + (15*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a +
b*ArcSin[c*x])^2)/64 + (d^2*g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/21 + (5*d^2*f^3*x*(1 - c^2*x^2
)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (5*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cSin[c*x])^2)/16 + (5*d^2*g^3*x^4*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/63 + (d^2*f^3*x*(1
- c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 + (3*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x])^2)/8 + (d^2*g^3*x^4*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/9 - (3*d
^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(
a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(12
8*b*c^3*Sqrt[1 - c^2*x^2])

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 4763

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

Rule 4649

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcSin[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[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 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 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 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 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 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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 12

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

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 4699

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(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] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 459

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

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 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 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 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 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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 270

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

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

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

Rubi steps

\begin{align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (5 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{6 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt{1-c^2 x^2}}+\frac{\left (6 b d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{9 \sqrt{1-c^2 x^2}}-\frac{\left (2 b c d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{9 \sqrt{1-c^2 x^2}}\\ &=\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{63 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (5 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \sqrt{1-c^2 x^2}}-\frac{\left (6 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt{1-c^2 x^2}} \, dx}{7 \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{21 \sqrt{1-c^2 x^2}}-\frac{\left (10 b c d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt{1-c^2 x^2}} \, dx}{9 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}-\frac{11 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{8 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{\left (5 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (6 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx}{245 \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (15 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{32 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{21 \sqrt{1-c^2 x^2}}-\frac{\left (2 b c d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{21 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{2835 \sqrt{1-c^2 x^2}}+\frac{\left (10 b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt{1-c^2 x^2}} \, dx}{63 \sqrt{1-c^2 x^2}}\\ &=-\frac{3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt{d-c^2 d x^2}-\frac{65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac{d^2 g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{144 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{64 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{256 \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (15 b d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{64 c \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{96 \sqrt{1-c^2 x^2}}+\frac{\left (15 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{128 \sqrt{1-c^2 x^2}}+\frac{\left (2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{63 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2835 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (7-5 c^2 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{441 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{1-c^2 x^2}} \, dx}{105 \sqrt{1-c^2 x^2}}\\ &=-\frac{245 b^2 d^2 f^3 x \sqrt{d-c^2 d x^2}}{1152}-\frac{15}{512} b^2 d^2 f g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{209 b^2 c^2 d^2 f g^2 x^5 \sqrt{d-c^2 d x^2}}{4608}-\frac{3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt{d-c^2 d x^2}-\frac{65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{15 b d^2 f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{2 d^2 g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac{d^2 g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{288 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{128 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{1-c^2 x}}+8 \sqrt{1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}+\frac{\left (45 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{512 \sqrt{1-c^2 x^2}}-\frac{\left (15 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{128 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{1536 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{72 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{2835 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx}{189 \sqrt{1-c^2 x^2}}+\frac{\left (4 b d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (7-5 c^2 x\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{441 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 g^3 \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 )}{105 \sqrt{1-c^2 x^2}}\\ &=\frac{96 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{245 b^2 d^2 f^3 x \sqrt{d-c^2 d x^2}}{1152}+\frac{15 b^2 d^2 f g^2 x \sqrt{d-c^2 d x^2}}{1024 c^2}-\frac{1079 b^2 d^2 f g^2 x^3 \sqrt{d-c^2 d x^2}}{18432}+\frac{209 b^2 c^2 d^2 f g^2 x^5 \sqrt{d-c^2 d x^2}}{4608}-\frac{3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{4 a b d^2 g^3 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{115 b^2 d^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{15 b d^2 f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{2 d^2 g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac{d^2 g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2048 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{96 \sqrt{1-c^2 x^2}}+\frac{\left (45 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{1024 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (15 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{256 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^4}+\frac{4 x^2}{c^4}+\frac{3 x^4}{c^4}-\frac{50 x^6}{c^4}+\frac{35 x^8}{c^4}\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{2835 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{189 \sqrt{1-c^2 x^2}}+\frac{\left (4 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{c^4 \sqrt{1-c^2 x}}+\frac{\sqrt{1-c^2 x}}{c^4}-\frac{8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{441 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 g^3 \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 )}{105 \sqrt{1-c^2 x^2}}\\ &=\frac{96 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{134 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}}{3969 c^4}-\frac{245 b^2 d^2 f^3 x \sqrt{d-c^2 d x^2}}{1152}-\frac{359 b^2 d^2 f g^2 x \sqrt{d-c^2 d x^2}}{12288 c^2}-\frac{1079 b^2 d^2 f g^2 x^3 \sqrt{d-c^2 d x^2}}{18432}+\frac{209 b^2 c^2 d^2 f g^2 x^5 \sqrt{d-c^2 d x^2}}{4608}-\frac{3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{4 a b d^2 g^3 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{122 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11907 c^4}-\frac{65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1323 c^4}-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{27783 c^4}-\frac{2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{729 c^4}+\frac{115 b^2 d^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}-\frac{15 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1024 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b^2 d^2 g^3 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{15 b d^2 f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{2 d^2 g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac{d^2 g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4096 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{192 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d^2 g^3 \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 )}{189 \sqrt{1-c^2 x^2}}-\frac{\left (4 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{63 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{96 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{160 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}}{3969 c^4}-\frac{245 b^2 d^2 f^3 x \sqrt{d-c^2 d x^2}}{1152}-\frac{359 b^2 d^2 f g^2 x \sqrt{d-c^2 d x^2}}{12288 c^2}-\frac{1079 b^2 d^2 f g^2 x^3 \sqrt{d-c^2 d x^2}}{18432}+\frac{209 b^2 c^2 d^2 f g^2 x^5 \sqrt{d-c^2 d x^2}}{4608}-\frac{3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{4 a b d^2 g^3 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{80 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11907 c^4}-\frac{65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1323 c^4}-\frac{1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{27783 c^4}-\frac{2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{729 c^4}+\frac{115 b^2 d^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{359 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12288 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b^2 d^2 g^3 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d^2 f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{15 b d^2 f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{21 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{441 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{2 d^2 g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac{d^2 g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} d^2 g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.93246, size = 1114, normalized size = 0.49 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (333396000 \left (8 c^3 f^3+3 c g^2 f\right ) a^3+3175200 b \sqrt{1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right ) a^2-10080 b^2 c x \left (20 x^5 \left (7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right ) c^8-72 x^3 \left (9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right ) c^6+945 x \left (1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right ) c^4-105 g \left (20736 f^2+2835 g x f+256 g^2 x^2\right ) c^2-161280 g^3\right ) a+333396000 b^3 c f \left (8 c^2 f^2+3 g^2\right ) \sin ^{-1}(c x)^3+3175200 b^2 \left (315 a \left (8 c^3 f^3+3 c g^2 f\right )+b \sqrt{1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right )\right ) \sin ^{-1}(c x)^2-b^3 \sqrt{1-c^2 x^2} \left (400 x^5 \left (592704 f^3+1119744 g x f^2+750141 g^2 x^2 f+175616 g^3 x^3\right ) c^8-8 x^3 \left (179663400 f^3+262020096 g x f^2+145166175 g^2 x^2 f+29363200 g^3 x^3\right ) c^6+6 x \left (1107615600 f^3+753463296 g x f^2+249815475 g^2 x^2 f+34304000 g^3 x^3\right ) c^4+g \left (-12905422848 f^2+748057275 g x f+184115200 g^2 x^2\right ) c^2-1257472000 g^3\right )+315 b \left (3175200 \left (8 c^3 f^3+3 c g^2 f\right ) a^2+20160 b \sqrt{1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right ) a+b^2 c \left (-640 x^6 \left (7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right ) c^8+2304 x^4 \left (9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right ) c^6-30240 x^2 \left (1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right ) c^4+3360 \left (6279 f^3+20736 g x f^2+2835 g^2 x^2 f+256 g^3 x^3\right ) c^2+315 g^2 (7539 f+16384 g x)\right )\right ) \sin ^{-1}(c x)\right )}{25604812800 b c^4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(333396000*a^3*(8*c^3*f^3 + 3*c*f*g^2) + 3175200*a^2*b*Sqrt[1 - c^2*x^2]*(-256*g^3 -
c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) -
8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g
^2*x^2 + 320*g^3*x^3)) - 10080*a*b^2*c*x*(-161280*g^3 - 105*c^2*g*(20736*f^2 + 2835*f*g*x + 256*g^2*x^2) + 945
*c^4*x*(1848*f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) - 72*c^6*x^3*(9555*f^3 + 18144*f^2*g*x + 12495
*f*g^2*x^2 + 3040*g^3*x^3) + 20*c^8*x^5*(7056*f^3 + 15552*f^2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)) - b^3*Sqr
t[1 - c^2*x^2]*(-1257472000*g^3 + c^2*g*(-12905422848*f^2 + 748057275*f*g*x + 184115200*g^2*x^2) + 400*c^8*x^5
*(592704*f^3 + 1119744*f^2*g*x + 750141*f*g^2*x^2 + 175616*g^3*x^3) - 8*c^6*x^3*(179663400*f^3 + 262020096*f^2
*g*x + 145166175*f*g^2*x^2 + 29363200*g^3*x^3) + 6*c^4*x*(1107615600*f^3 + 753463296*f^2*g*x + 249815475*f*g^2
*x^2 + 34304000*g^3*x^3)) + 315*b*(3175200*a^2*(8*c^3*f^3 + 3*c*f*g^2) + 20160*a*b*Sqrt[1 - c^2*x^2]*(-256*g^3
- c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3)
- 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*
f*g^2*x^2 + 320*g^3*x^3)) + b^2*c*(315*g^2*(7539*f + 16384*g*x) - 30240*c^4*x^2*(1848*f^3 + 2304*f^2*g*x + 123
9*f*g^2*x^2 + 256*g^3*x^3) + 3360*c^2*(6279*f^3 + 20736*f^2*g*x + 2835*f*g^2*x^2 + 256*g^3*x^3) + 2304*c^6*x^4
*(9555*f^3 + 18144*f^2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) - 640*c^8*x^6*(7056*f^3 + 15552*f^2*g*x + 11907*f
*g^2*x^2 + 3136*g^3*x^3)))*ArcSin[c*x] + 3175200*b^2*(315*a*(8*c^3*f^3 + 3*c*f*g^2) + b*Sqrt[1 - c^2*x^2]*(-25
6*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3
*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x +
1239*f*g^2*x^2 + 320*g^3*x^3)))*ArcSin[c*x]^2 + 333396000*b^3*c*f*(8*c^2*f^2 + 3*g^2)*ArcSin[c*x]^3))/(2560481
2800*b*c^4*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.909, size = 5226, normalized size = 2.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} g^{3} x^{7} + 3 \, a^{2} c^{4} d^{2} f g^{2} x^{6} + 3 \, a^{2} d^{2} f^{2} g x + a^{2} d^{2} f^{3} +{\left (3 \, a^{2} c^{4} d^{2} f^{2} g - 2 \, a^{2} c^{2} d^{2} g^{3}\right )} x^{5} +{\left (a^{2} c^{4} d^{2} f^{3} - 6 \, a^{2} c^{2} d^{2} f g^{2}\right )} x^{4} -{\left (6 \, a^{2} c^{2} d^{2} f^{2} g - a^{2} d^{2} g^{3}\right )} x^{3} -{\left (2 \, a^{2} c^{2} d^{2} f^{3} - 3 \, a^{2} d^{2} f g^{2}\right )} x^{2} +{\left (b^{2} c^{4} d^{2} g^{3} x^{7} + 3 \, b^{2} c^{4} d^{2} f g^{2} x^{6} + 3 \, b^{2} d^{2} f^{2} g x + b^{2} d^{2} f^{3} +{\left (3 \, b^{2} c^{4} d^{2} f^{2} g - 2 \, b^{2} c^{2} d^{2} g^{3}\right )} x^{5} +{\left (b^{2} c^{4} d^{2} f^{3} - 6 \, b^{2} c^{2} d^{2} f g^{2}\right )} x^{4} -{\left (6 \, b^{2} c^{2} d^{2} f^{2} g - b^{2} d^{2} g^{3}\right )} x^{3} -{\left (2 \, b^{2} c^{2} d^{2} f^{3} - 3 \, b^{2} d^{2} f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} g^{3} x^{7} + 3 \, a b c^{4} d^{2} f g^{2} x^{6} + 3 \, a b d^{2} f^{2} g x + a b d^{2} f^{3} +{\left (3 \, a b c^{4} d^{2} f^{2} g - 2 \, a b c^{2} d^{2} g^{3}\right )} x^{5} +{\left (a b c^{4} d^{2} f^{3} - 6 \, a b c^{2} d^{2} f g^{2}\right )} x^{4} -{\left (6 \, a b c^{2} d^{2} f^{2} g - a b d^{2} g^{3}\right )} x^{3} -{\left (2 \, a b c^{2} d^{2} f^{3} - 3 \, a b d^{2} f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*g^3*x^7 + 3*a^2*c^4*d^2*f*g^2*x^6 + 3*a^2*d^2*f^2*g*x + a^2*d^2*f^3 + (3*a^2*c^4*d^2*f^2
*g - 2*a^2*c^2*d^2*g^3)*x^5 + (a^2*c^4*d^2*f^3 - 6*a^2*c^2*d^2*f*g^2)*x^4 - (6*a^2*c^2*d^2*f^2*g - a^2*d^2*g^3
)*x^3 - (2*a^2*c^2*d^2*f^3 - 3*a^2*d^2*f*g^2)*x^2 + (b^2*c^4*d^2*g^3*x^7 + 3*b^2*c^4*d^2*f*g^2*x^6 + 3*b^2*d^2
*f^2*g*x + b^2*d^2*f^3 + (3*b^2*c^4*d^2*f^2*g - 2*b^2*c^2*d^2*g^3)*x^5 + (b^2*c^4*d^2*f^3 - 6*b^2*c^2*d^2*f*g^
2)*x^4 - (6*b^2*c^2*d^2*f^2*g - b^2*d^2*g^3)*x^3 - (2*b^2*c^2*d^2*f^3 - 3*b^2*d^2*f*g^2)*x^2)*arcsin(c*x)^2 +
2*(a*b*c^4*d^2*g^3*x^7 + 3*a*b*c^4*d^2*f*g^2*x^6 + 3*a*b*d^2*f^2*g*x + a*b*d^2*f^3 + (3*a*b*c^4*d^2*f^2*g - 2*
a*b*c^2*d^2*g^3)*x^5 + (a*b*c^4*d^2*f^3 - 6*a*b*c^2*d^2*f*g^2)*x^4 - (6*a*b*c^2*d^2*f^2*g - a*b*d^2*g^3)*x^3 -
(2*a*b*c^2*d^2*f^3 - 3*a*b*d^2*f*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(g*x + f)^3*(b*arcsin(c*x) + a)^2, x)