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

Optimal. Leaf size=1533 \[ \text{result too large to display} \]

[Out]

(64*b^2*d^2*f*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f^2*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*g
^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*g^
2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (b^2*c^4*d^2*g^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (32*b^2*d^2*f*g*(1 - c^2*x^
2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (24*b^2*d^2*f*
g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (b^2*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 +
(4*b^2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*
x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2
]) + (4*b*d^2*f*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^2*x^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 -
c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (12*b*c^3*d
^2*f*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^
2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (4*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin
[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*
x^2]) + (5*b*d^2*f^2*(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^2*(1 - c^2
*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c
*x])^2)/16 - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) + (5*d^2*g^2*x^3*Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (
5*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/48 + (d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 + (d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2
)/8 - (2*d^2*f*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^2*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*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^
3)/(384*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.0527, antiderivative size = 1533, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 24, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, 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} \[ -\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^8}{32 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^7}{49 \sqrt{1-c^2 x^2}}-\frac{1}{256} b^2 c^4 d^2 g^2 \sqrt{d-c^2 d x^2} x^7+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^6}{144 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{35 \sqrt{1-c^2 x^2}}+\frac{209 b^2 c^2 d^2 g^2 \sqrt{d-c^2 d x^2} x^5}{13824}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{384 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac{1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac{5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3-\frac{4 b c d^2 f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{7 \sqrt{1-c^2 x^2}}-\frac{1079 b^2 d^2 g^2 \sqrt{d-c^2 d x^2} x^3}{55296}-\frac{5 b c d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{16 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{128 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{128 c^2}+\frac{1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{4 b d^2 f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{7 c \sqrt{1-c^2 x^2}}-\frac{245 b^2 d^2 f^2 \sqrt{d-c^2 d x^2} x}{1152}-\frac{359 b^2 d^2 g^2 \sqrt{d-c^2 d x^2} x}{36864 c^2}-\frac{1}{108} b^2 d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} x-\frac{65 b^2 d^2 f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x}{1728}+\frac{5 d^2 f^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 f 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{115 b^2 d^2 f^2 \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 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^2 \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 b d^2 f^2 \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{4 b^2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{24 b^2 d^2 f g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{64 b^2 d^2 f g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*b^2*d^2*f*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f^2*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*g
^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*g^
2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (b^2*c^4*d^2*g^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (32*b^2*d^2*f*g*(1 - c^2*x^
2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (24*b^2*d^2*f*
g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (b^2*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 +
(4*b^2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*
x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2
]) + (4*b*d^2*f*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^2*x^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 -
c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (12*b*c^3*d
^2*f*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^
2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (4*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin
[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*
x^2]) + (5*b*d^2*f^2*(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^2*(1 - c^2
*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c
*x])^2)/16 - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) + (5*d^2*g^2*x^3*Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (
5*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/48 + (d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 + (d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2
)/8 - (2*d^2*f*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^2*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*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^
3)/(384*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]

Rubi steps

\begin{align*} \int (f+g x)^2 \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)^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 \sqrt{d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+2 f g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^2 x^2 \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^2 \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 (2 d^2 f 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 (d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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^2 \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 (4 b d^2 f 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 (5 d^2 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 (b c d^2 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{4 b d^2 f 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{4 b c d^2 f 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{b c d^2 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{12 b c^3 d^2 f 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{b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{b d^2 f^2 \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^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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^2 \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^2 \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 (4 b^2 d^2 f 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 (5 d^2 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 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}{24 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 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{1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b d^2 f 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{96 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^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}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^2 \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^2 \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^2 \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 (4 b^2 d^2 f 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 (5 d^2 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 (5 b c d^2 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 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}{96 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 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}{24 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt{d-c^2 d x^2}-\frac{65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b d^2 f 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^2 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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^2 \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^2 \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^2 \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 (2 b^2 d^2 f 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 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}{768 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 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 (5 b d^2 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 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}{288 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 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{245 b^2 d^2 f^2 x \sqrt{d-c^2 d x^2}}{1152}-\frac{5}{512} b^2 d^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{209 b^2 c^2 d^2 g^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt{d-c^2 d x^2}-\frac{65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b d^2 f 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^2 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{5 b d^2 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 f^2 \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^2 \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^2 \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 (2 b^2 d^2 f 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 (15 b^2 d^2 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 (5 b^2 d^2 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 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{4608 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{216 \sqrt{1-c^2 x^2}}\\ &=\frac{64 b^2 d^2 f g \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{245 b^2 d^2 f^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{5 b^2 d^2 g^2 x \sqrt{d-c^2 d x^2}}{1024 c^2}-\frac{1079 b^2 d^2 g^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 g^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2}-\frac{65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{24 b^2 d^2 f 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^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b^2 d^2 f 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^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{4 b d^2 f 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^2 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{5 b d^2 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{6144 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{288 \sqrt{1-c^2 x^2}}+\frac{\left (15 b^2 d^2 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 (5 b^2 d^2 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{64 b^2 d^2 f g \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{245 b^2 d^2 f^2 x \sqrt{d-c^2 d x^2}}{1152}-\frac{359 b^2 d^2 g^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}-\frac{1079 b^2 d^2 g^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 g^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2}-\frac{65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{24 b^2 d^2 f 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^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b^2 d^2 f 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^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}-\frac{5 b^2 d^2 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 d^2 f 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^2 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{5 b d^2 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{12288 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{576 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{64 b^2 d^2 f g \sqrt{d-c^2 d x^2}}{245 c^2}-\frac{245 b^2 d^2 f^2 x \sqrt{d-c^2 d x^2}}{1152}-\frac{359 b^2 d^2 g^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}-\frac{1079 b^2 d^2 g^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 g^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt{d-c^2 d x^2}+\frac{32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2}-\frac{65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{1728}+\frac{24 b^2 d^2 f 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^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}+\frac{4 b^2 d^2 f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 f 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^2 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{5 b d^2 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{4 b c d^2 f 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 g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f 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{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f 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{b c^5 d^2 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{5 b d^2 f^2 \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^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{24} d^2 f^2 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}{48} d^2 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{1}{6} d^2 f^2 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{1}{8} d^2 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{2 d^2 f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.42679, size = 742, normalized size = 0.48 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (105 b \sin ^{-1}(c x) \left (352800 a^2 \left (8 c^2 f^2+g^2\right )+6720 a b c \sqrt{1-c^2 x^2} \left (56 c^2 f^2 x \left (8 c^4 x^4-26 c^2 x^2+33\right )+768 f g \left (c^2 x^2-1\right )^3+7 g^2 x \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )+b^2 \left (-640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )-3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )+87955 g^2\right )\right )+352800 a^2 b c \sqrt{1-c^2 x^2} \left (56 c^2 f^2 x \left (8 c^4 x^4-26 c^2 x^2+33\right )+768 f g \left (c^2 x^2-1\right )^3+7 g^2 x \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )[phone] a^3 \left (8 c^2 f^2+g^2\right )-3360 a b^2 c^2 x \left (1960 c^2 f^2 x \left (8 c^4 x^4-39 c^2 x^2+99\right )+4608 f g \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+245 g^2 x \left (36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right )\right )+352800 b^2 \sin ^{-1}(c x)^2 \left (105 a \left (8 c^2 f^2+g^2\right )+b c \sqrt{1-c^2 x^2} \left (56 c^2 f^2 x \left (8 c^4 x^4-26 c^2 x^2+33\right )+768 f g \left (c^2 x^2-1\right )^3+7 g^2 x \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )\right )-b^3 c \sqrt{1-c^2 x^2} \left (274400 c^2 f^2 x \left (32 c^4 x^4-194 c^2 x^2+897\right )+147456 f g \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right )+8575 g^2 x \left (432 c^6 x^6-1672 c^4 x^4+2158 c^2 x^2+1077\right )\right )[phone] b^3 \left (8 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^3\right )}{948326400 b c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^2*(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]*(12348000*a^3*(8*c^2*f^2 + g^2) - 3360*a*b^2*c^2*x*(1960*c^2*f^2*x*(99 - 39*c^2*x^2 +
8*c^4*x^4) + 4608*f*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^
4 + 36*c^6*x^6)) + 352800*a^2*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2
+ 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)) - b^3*c*Sqrt[1 - c^2*x^2]*(274400*c^2*f
^2*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) + 147456*f*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 8575*g^2
*x*(1077 + 2158*c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6)) + 105*b*(352800*a^2*(8*c^2*f^2 + g^2) + b^2*(87955*g^2
+ 1120*c^2*(2093*f^2 + 4608*f*g*x + 315*g^2*x^2) - 3360*c^4*x^2*(1848*f^2 + 1536*f*g*x + 413*g^2*x^2) - 640*c^
8*x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728*f*g*x + 595*g^2*x^2)) + 6720*a*b*c*
Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*
c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x] + 352800*b^2*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2
]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^
4*x^4 + 48*c^6*x^6)))*ArcSin[c*x]^2 + 12348000*b^3*(8*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(948326400*b*c^3*Sqrt[1 -
c^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.702, size = 3750, normalized size = 2.5 \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)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x)

[Out]

5/64*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x*g^2+4/7*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c
^2/(c^2*x^2-1)*arcsin(c*x)-16/7*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c^2*x^2-1)*arcsin(c*x)*x^2+2/7*b^2*(-d*(c^
2*x^2-1))^(1/2)*f*g*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x^8+11/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(
-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*f^2-13/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
arcsin(c*x)*x^4*f^2+1/18*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6*f^2
-17/144*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6+59/384*b^2*(-d*(
c^2*x^2-1))^(1/2)*g^2*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4-5/128*b^2*(-d*(c^2*x^2-1))^(1/2)*g^
2*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2+1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^5/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^8-8/7*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^6+1
2/7*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^4+59/384*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2
*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4*g^2+11/16*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2
)*x^2*f^2-5/128*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2*g^2+1/32*a*b*(-d*(c^2*x^2-
1))^(1/2)*g^2*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8-17/144*a*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^3/(c^2*x^
2-1)*(-c^2*x^2+1)^(1/2)*x^6+1/18*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6*f^2-13/
48*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4*f^2+1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*g^
2*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^9-23/24*a*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^7
+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^7*f^2-5/16*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*
x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*d^2*f^2-5/128*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*
x^2-1)*arcsin(c*x)^2*d^2*g^2-17/12*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^5*f^2+127/96*a
*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5*g^2+59/24*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^
2*x^2-1)*arcsin(c*x)*x^3*f^2+59/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3*f^2-8/7*b^
2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^2-299/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*
x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*f^2-17/24*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x
^5*f^2+1/6*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x^7*f^2-359/36864*b^2*(-d*(c^2*x^2-1))
^(1/2)*g^2*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-5/48*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/
2)/c/(c^2*x^2-1)*arcsin(c*x)^3*d^2*f^2-5/384*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arc
sin(c*x)^3*d^2*g^2+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x^9-23/48*b^2*(-d*(c^2
*x^2-1))^(1/2)*g^2*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^7+127/192*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^2/(c^2*x
^2-1)*arcsin(c*x)^2*x^5+5/128*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)^2*x-4/343*b^2*(-d
*(c^2*x^2-1))^(1/2)*f*g*d^2*c^6/(c^2*x^2-1)*x^8+5/192*a^2*g^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2*g^2/c^2*d
^2*x*(-c^2*d*x^2+d)^(1/2)+5/128*a^2*g^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/7
*a^2*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d-1/8*a^2*g^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1081/110592*b^2*(-d*(c^2*x^2-1))^
(1/2)*g^2*d^2/(c^2*x^2-1)*x^3+299/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*x*f^2+5/16*a^2*f^2*d^2*x*(-c
^2*d*x^2+d)^(1/2)+5/16*a^2*f^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-133/192*a*b*(-d*
(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(c*x)*x^3*g^2-11/8*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(
c*x)*x*f^2-299/1152*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*f^2-359/36864*a*b*(-d*(c^2
*x^2-1))^(1/2)*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*g^2+568/8575*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^4/(c^2
*x^2-1)*x^6-4432/25725*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^2/(c^2*x^2-1)*x^4+2/7*b^2*(-d*(c^2*x^2-1))^(1/2)*f
*g*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)^2-133/384*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3+
11672/25725*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c^2*x^2-1)*x^2-11/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1
)*arcsin(c*x)^2*x*f^2-1/256*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^6/(c^2*x^2-1)*x^9+263/13824*b^2*(-d*(c^2*x^2-
1))^(1/2)*g^2*d^2*c^4/(c^2*x^2-1)*x^7-1915/55296*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^2/(c^2*x^2-1)*x^5+359/36
864*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/c^2/(c^2*x^2-1)*x-1091/3456*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2
-1)*x^3*f^2+113/1728*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*x^5*f^2-1/108*b^2*(-d*(c^2*x^2-1))^(1/2)*d
^2*c^6/(c^2*x^2-1)*x^7*f^2-8644/25725*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c^2/(c^2*x^2-1)+1/6*a^2*f^2*x*(-c^2*d
*x^2+d)^(5/2)+4/7*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^8-16/7*a*b*(-d*(c^2*x^2-1))
^(1/2)*f*g*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^6+24/7*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^2/(c^2*x^2-1)*arcsin(
c*x)*x^4+4/49*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^7-12/35*b^2*
(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^5+4/7*b^2*(-d*(c^2*x^2-1))^(1/
2)*f*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3-4/7*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c/(c^2*x^2-
1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+4/49*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)
*x^7-12/35*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5+4/7*a*b*(-d*(c^2*x^2-1))^
(1/2)*f*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3-4/7*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c/(c^2*x^2-1)*(-c^2*
x^2+1)^(1/2)*x+1/48*a^2*g^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/24*a^2*f^2*d*x*(-c^2*d*x^2+d)^(3/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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 )}^{2}{\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)^2*(-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)^2*(b*arcsin(c*x) + a)^2, x)