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

Optimal. Leaf size=1154 \[ \text{result too large to display} \]

[Out]

(4*b^2*f^2*g*Sqrt[d - c^2*d*x^2])/(3*c^2) + (52*b^2*g^3*Sqrt[d - c^2*d*x^2])/(225*c^4) - (b^2*f^3*x*Sqrt[d - c
^2*d*x^2])/4 + (3*b^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) - (3*b^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2])/32 + (4*a*
b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (2*b^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(9*c
^2) + (26*b^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(675*c^4) - (2*b^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2
])/(125*c^4) + (b^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (3*b^2*f*g^2*Sqrt[d - c^2*d
*x^2]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*g^3*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(15*c^3*Sqrt[1 -
c^2*x^2]) + (2*b*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[1 - c^2*x^2]) - (b*c*f^3*x^2*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x]))/(8*c*Sqrt[1 - c^2*x^2]) - (2*b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2])
+ (2*b*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (2*b*c*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])
)/(25*Sqrt[1 - c^2*x^2]) - (2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^4) + (f^3*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) - (g^3*x^2*Sqrt[
d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 + (
g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x])^2)/c^2 + (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (f*g^2*Sqrt[d - c^2
*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.5548, antiderivative size = 1154, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.485, Rules used = {4777, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707, 4619, 261, 266} \[ -\frac{2 b c g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt{1-c^2 x^2}}+\frac{1}{5} g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^4-\frac{3 b c f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{8 \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac{2 b g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{45 c \sqrt{1-c^2 x^2}}-\frac{2 b c f^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{3 \sqrt{1-c^2 x^2}}-\frac{3}{32} b^2 f g^2 \sqrt{d-c^2 d x^2} x^3-\frac{g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^2}{15 c^2}-\frac{b c f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{2 \sqrt{1-c^2 x^2}}+\frac{3 b f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 c \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{8 c^2}+\frac{4 b^2 g^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) x}{15 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b f^2 g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f^3 \sqrt{d-c^2 d x^2} x+\frac{3 b^2 f g^2 \sqrt{d-c^2 d x^2} x}{64 c^2}+\frac{4 a b g^3 \sqrt{d-c^2 d x^2} x}{15 c^3 \sqrt{1-c^2 x^2}}+\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4}-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}+\frac{b^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{3 b^2 f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}}+\frac{52 b^2 g^3 \sqrt{d-c^2 d x^2}}{225 c^4}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^4}+\frac{4 b^2 f^2 g \sqrt{d-c^2 d x^2}}{3 c^2}+\frac{26 b^2 g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{675 c^4}+\frac{2 b^2 f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{9 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^2*f^2*g*Sqrt[d - c^2*d*x^2])/(3*c^2) + (52*b^2*g^3*Sqrt[d - c^2*d*x^2])/(225*c^4) - (b^2*f^3*x*Sqrt[d - c
^2*d*x^2])/4 + (3*b^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) - (3*b^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2])/32 + (4*a*
b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (2*b^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(9*c
^2) + (26*b^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(675*c^4) - (2*b^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2
])/(125*c^4) + (b^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (3*b^2*f*g^2*Sqrt[d - c^2*d
*x^2]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*g^3*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(15*c^3*Sqrt[1 -
c^2*x^2]) + (2*b*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[1 - c^2*x^2]) - (b*c*f^3*x^2*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x]))/(8*c*Sqrt[1 - c^2*x^2]) - (2*b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2])
+ (2*b*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (2*b*c*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])
)/(25*Sqrt[1 - c^2*x^2]) - (2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^4) + (f^3*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) - (g^3*x^2*Sqrt[
d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 + (
g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x])^2)/c^2 + (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (f*g^2*Sqrt[d - c^2
*d*x^2]*(a + b*ArcSin[c*x])^3)/(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 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 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 444

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

Rule 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 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 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 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]]

Rubi steps

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

Mathematica [A]  time = 1.1789, size = 696, normalized size = 0.6 \[ \frac{\sqrt{d-c^2 d x^2} \left (-\frac{f g^2 \left (-3 b^2 \left (c x \left (2 a c x+b \sqrt{1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )+6 b c x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 \left (a+b \sin ^{-1}(c x)\right )^3\right )}{16 b c^3}-\frac{f^2 g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2}-\frac{2 b f^2 g \left (3 a c x \left (c^2 x^2-3\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2-7\right )+3 b c x \left (c^2 x^2-3\right ) \sin ^{-1}(c x)\right )}{9 c^2}+\frac{1}{2} f^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b f^3 \left (c x \left (2 a c x+b \sqrt{1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{4 c}+\frac{3}{4} f g^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 b f g^2 \left (8 a c^4 x^4+b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4-3\right ) \sin ^{-1}(c x)\right )}{64 c^3}+\frac{1}{5} g^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 b g^3 \left (15 a c^5 x^5+b \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )+15 b c^5 x^5 \sin ^{-1}(c x)\right )}{375 c^4}-\frac{g^3 \left (9 c^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (3 c^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )\right )+18 \left (\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )\right )}{135 c^4}+\frac{f^3 \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c}\right )}{\sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*((f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])^2)/4 + (g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)^(3/2)*(a + b*
ArcSin[c*x])^2)/c^2 + (f^3*(a + b*ArcSin[c*x])^3)/(6*b*c) - (2*b*g^3*(15*a*c^5*x^5 + b*Sqrt[1 - c^2*x^2]*(8 +
4*c^2*x^2 + 3*c^4*x^4) + 15*b*c^5*x^5*ArcSin[c*x]))/(375*c^4) - (2*b*f^2*g*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2)
+ 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x^2)*ArcSin[c*x]))/(9*c^2) - (b*f^3*(c*x*(2*a*c*x + b*Sqrt[1 - c
^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*c) - (3*b*f*g^2*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c
^2*x^2) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*c^3) - (f*g^2*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2
- 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(16
*b*c^3) - (g^3*(9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3
*c^3*x^3*(a + b*ArcSin[c*x])) + 18*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b*Sqrt[1 - c^2*x^2]
+ b*c*x*ArcSin[c*x]))))/(135*c^4)))/Sqrt[1 - c^2*x^2]

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Maple [B]  time = 0.784, size = 2947, normalized size = 2.6 \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*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2),x)

[Out]

-856/3375*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/(c^2*x^2-1)+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f^3/(c^2*x^2-1)*x+32/3
375*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/(c^2*x^2-1)*x^4-3/4*a^2*f*g^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+3/8*a^2*f*g^2/c^
2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+4/15*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/(c^2*x^
2-1)*arcsin(c*x)-8/15*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/(c^2*x^2-1)*arcsin(c*x)*x^4-1/4*a*b*(-d*(c^2*x^2-1))^(1/2
)*f^3/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-a*b*(-d*(c^2*x^2-1))^(1/2)*f^3/(c^2*x^2-1)*arcsin(c*x)*x-1/4*b^2*(-d*(c
^2*x^2-1))^(1/2)*f^3/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-1/6*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*f^3+1/5*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^6+3/8
*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/(c^2*x^2-1)*arcsin(c*x)^2*x+b^2*(-d*(c^2*x^2-1))^(1/2)*g*c^2/(c^2*x^2-1)
*arcsin(c*x)^2*x^4*f^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f^3*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2-2/4
5*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3-4/15*b^2*(-d*(c^2*x^2-1))^(1
/2)*g^3/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+3/64*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^3/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)*arcsin(c*x)+2/25*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*
x)*x^5-1/2*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*f^3+2/5*a*b*(-d*(c^2*x^2-
1))^(1/2)*g^3*c^2/(c^2*x^2-1)*arcsin(c*x)*x^6-2/15*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/(c^2*x^2-1)*arcsin(c*x)*
x^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*g/c^2/(c^2*x^2-1)*arcsin(c*x)*f^2+a*b*(-d*(c^2*x^2-1))^(1/2)*f^3*c^2/(c^2*x^2
-1)*arcsin(c*x)*x^3+2/25*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5-2/45*a*b*(-d*(c^2
*x^2-1))^(1/2)*g^3/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3-4/15*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^3/(c^2*x^2-1)*(-
c^2*x^2+1)^(1/2)*x-9/4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/(c^2*x^2-1)*arcsin(c*x)*x^3+3/64*a*b*(-d*(c^2*x^2-1))^
(1/2)*f*g^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-4*a*b*(-d*(c^2*x^2-1))^(1/2)*g/(c^2*x^2-1)*arcsin(c*x)*x^2*f^2+
1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*f^3*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/8*b^2*(-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)^3*f*g^2+3/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c^2/(c^2*x^2-1)*arcsin(c*
x)^2*x^5+1/2*a^2*f^3*x*(-c^2*d*x^2+d)^(1/2)-2/15*a^2*g^3/d/c^4*(-c^2*d*x^2+d)^(3/2)+1/2*a^2*f^3*d/(c^2*d)^(1/2
)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/15*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/(c^2*x^2-1)*arcsin(c*x)
^2*x^2-3/32*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c^2/(c^2*x^2-1)*x^5-3/64*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/(c^
2*x^2-1)*x-2/9*b^2*(-d*(c^2*x^2-1))^(1/2)*g*c^2/(c^2*x^2-1)*x^4*f^2+b^2*(-d*(c^2*x^2-1))^(1/2)*g/c^2/(c^2*x^2-
1)*arcsin(c*x)^2*f^2-9/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3-2*b^2*(-d*(c^2*x^2-1))
^(1/2)*g/(c^2*x^2-1)*arcsin(c*x)^2*x^2*f^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f^3*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^
3-3/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2+2/3*b^2*(-d*(c^2*x^2-1
))^(1/2)*g*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3*f^2-2*b^2*(-d*(c^2*x^2-1))^(1/2)*g/c/(c^2*x^2-1)*(
-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*f^2-1/5*a^2*g^3*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d+3/8*a^2*f*g^2/c^2*x*(-c^2*d*x^2
+d)^(1/2)-a^2*f^2*g/c^2/d*(-c^2*d*x^2+d)^(3/2)-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f^3/(c^2*x^2-1)*arcsin(c*x)^2*x-
2/125*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3*c^2/(c^2*x^2-1)*x^6+878/3375*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/(c^2*x^2-
1)*x^2+2/15*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/(c^2*x^2-1)*arcsin(c*x)^2-4/15*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/(
c^2*x^2-1)*arcsin(c*x)^2*x^4+9/64*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/(c^2*x^2-1)*x^3+16/9*b^2*(-d*(c^2*x^2-1))^(
1/2)*g/(c^2*x^2-1)*x^2*f^2-14/9*b^2*(-d*(c^2*x^2-1))^(1/2)*g/c^2/(c^2*x^2-1)*f^2-1/4*b^2*(-d*(c^2*x^2-1))^(1/2
)*f^3*c^2/(c^2*x^2-1)*x^3+3/8*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-3/8*a*b*(-
d*(c^2*x^2-1))^(1/2)*f*g^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*g*c/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)*x^3*f^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)*g/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2-3/8*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*f*g^2+3/2*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2
*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5+3/4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x+2*a*b*(-d*
(c^2*x^2-1))^(1/2)*g*c^2/(c^2*x^2-1)*arcsin(c*x)*x^4*f^2+3/8*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2*c/(c^2*x^2-1)*(-
c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4

________________________________________________________________________________________

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*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2),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} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} +{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\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*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2
*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*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*(a+b*asin(c*x))**2*(-c**2*d*x**2+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-c^{2} d x^{2} + d}{\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*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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