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

Optimal. Leaf size=737 \[ -\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{4 b f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt{1-c^2 x^2}}-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{8 b^2 f g \sqrt{d-c^2 d x^2}}{9 c^2}+\frac{4 b^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{b^2 g^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{b^2 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}} \]

[Out]

(8*b^2*f*g*Sqrt[d - c^2*d*x^2])/(9*c^2) - (b^2*f^2*x*Sqrt[d - c^2*d*x^2])/4 + (b^2*g^2*x*Sqrt[d - c^2*d*x^2])/
(64*c^2) - (b^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/32 + (4*b^2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*c^2) + (b^
2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64
*c^3*Sqrt[1 - c^2*x^2]) + (4*b*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f
^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (4*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^
2*x^2]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (f^2*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (g^2*x^3*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^2)
+ (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (g^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x])^3)/(24*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.02713, antiderivative size = 737, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {4777, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707} \[ -\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{4 b f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt{1-c^2 x^2}}-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{8 b^2 f g \sqrt{d-c^2 d x^2}}{9 c^2}+\frac{4 b^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{b^2 g^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{b^2 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b^2*f*g*Sqrt[d - c^2*d*x^2])/(9*c^2) - (b^2*f^2*x*Sqrt[d - c^2*d*x^2])/4 + (b^2*g^2*x*Sqrt[d - c^2*d*x^2])/
(64*c^2) - (b^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/32 + (4*b^2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*c^2) + (b^
2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64
*c^3*Sqrt[1 - c^2*x^2]) + (4*b*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f
^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (4*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^
2*x^2]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (f^2*x*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x])^2)/2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (g^2*x^3*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^2)
+ (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (g^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcSin[c*x])^3)/(24*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]

Rubi steps

\begin{align*} \int (f+g x)^2 \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)^2 \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^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+2 f g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+g^2 x^2 \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^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}{\sqrt{1-c^2 x^2}}+\frac{\left (2 f 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 (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{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{\left (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}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b c f^2 \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 (4 b f 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}{3 c \sqrt{1-c^2 x^2}}+\frac{\left (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 (b c 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{4 b f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{b c f^2 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{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c 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{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{f^2 \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^2 \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 (4 b^2 f 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}{3 \sqrt{1-c^2 x^2}}+\frac{\left (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 (b 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 (b^2 c^2 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{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{4 b f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{b c f^2 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{b 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{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c 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{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{f^2 \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{g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 f^2 \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 (2 b^2 f 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 )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 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 (b^2 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{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 g^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{b^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{4 b f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{b c f^2 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{b 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{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c 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{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{f^2 \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{g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 f 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 )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 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 (b^2 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{8 b^2 f g \sqrt{d-c^2 d x^2}}{9 c^2}-\frac{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 g^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{4 b^2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{b^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{b^2 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 f g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{b c f^2 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{b 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{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c 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{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{f^2 \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{g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.955986, size = 441, normalized size = 0.6 \[ \frac{\sqrt{d-c^2 d x^2} \left (-\frac{g^2 \left (-3 b^2 \left (c x \left (2 a c x+b \sqrt{1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )+6 b c x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 \left (a+b \sin ^{-1}(c x)\right )^3\right )}{48 b c^3}+\frac{1}{2} f^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b f^2 \left (c x \left (2 a c x+b \sqrt{1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{4 c}-\frac{2 f g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}-\frac{4 b f g \left (3 a c x \left (c^2 x^2-3\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2-7\right )+3 b c x \left (c^2 x^2-3\right ) \sin ^{-1}(c x)\right )}{27 c^2}+\frac{1}{4} g^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b g^2 \left (8 a c^4 x^4+b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4-3\right ) \sin ^{-1}(c x)\right )}{64 c^3}+\frac{f^2 \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c}\right )}{\sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*((f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*c^2) + (f^2*(a + b*ArcSin[c*x])^3)/(6*b
*c) - (4*b*f*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]))/(27*c^2) - (b*f^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*c) - (b*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) - (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])))/(48*b*c^3)))/Sqrt[1 - c^2*x^2]

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Maple [B]  time = 0.646, size = 2051, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*c^2/(c^2*x^2-1)*arcsin(c*x)*x^4+1/8*a^2*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))+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/(c^2*x^2-1)*arcsin(c*x)+a*b*(-d*(c^2*
x^2-1))^(1/2)*f^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^3-1/2*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^2-1/8*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*g^2+1/4*
a*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*c^2/(c^2*x^2-1
)*arcsin(c*x)^2*x^4+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4-1/8*b^
2*(-d*(c^2*x^2-1))^(1/2)*g^2/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f
^2*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*g^2*c^2/(c^2*x^2-1)*arcsin(
c*x)*x^5+1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*g^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-1/8*a*b*(-d*(c^2*x^2-1))^(1/2
)*g^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c^2*x^2-1)*arcsin(c*x)*x^2+1/2*
a*b*(-d*(c^2*x^2-1))^(1/2)*f^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c/(c^2*
x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x+4/9*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2
+1)^(1/2)*x^3+4/9*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3-4/3*a*b*(-d*(c^2*x^2-1))
^(1/2)*f*g/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-2/3*a^2*f*g/c^2/d*(-c^2*d*x^2+d)^(3/2)-1/4*a^2*g^2*x*(-c^2*d*x^2
+d)^(3/2)/c^2/d-28/27*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/(c^2*x^2-1)-3/8*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2/(c^2*x
^2-1)*arcsin(c*x)^2*x^3+32/27*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/(c^2*x^2-1)*x^2-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f^
2/(c^2*x^2-1)*arcsin(c*x)^2*x-1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*c^2/(c^2*x^2-1)*x^5-1/64*b^2*(-d*(c^2*x^2-1)
)^(1/2)*g^2/c^2/(c^2*x^2-1)*x-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f^2*c^2/(c^2*x^2-1)*x^3+1/64*a*b*(-d*(c^2*x^2-1))
^(1/2)*g^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-a*b*(-d*(c^2*x^2-1))^(1/2)*f^2/(c^2*x^2-1)*arcsin(c*x)*x-1/4*a*b
*(-d*(c^2*x^2-1))^(1/2)*f^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-3/4*a*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c^2*x^2-1)*ar
csin(c*x)*x^3-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/(c^2*x^2-1)*arcsin(c*x)^2*x^2-1/6*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^2-1/24*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*g^2+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^5+1/8*b^2*(-d*(c^
2*x^2-1))^(1/2)*g^2/c^2/(c^2*x^2-1)*arcsin(c*x)^2*x+1/64*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2/c^3/(c^2*x^2-1)*arcsin
(c*x)*(-c^2*x^2+1)^(1/2)-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f^2/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4/27*
b^2*(-d*(c^2*x^2-1))^(1/2)*f*g*c^2/(c^2*x^2-1)*x^4+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/(c^2*x^2-1)*arcsin(c
*x)^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3+1/8*a^2*g^2/c^2*x*(-c^2*d*x^2+d)^(1
/2)+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f^2/(c^2*x^2-1)*x+3/64*b^2*(-d*(c^2*x^2-1))^(1/2)*g^2/(c^2*x^2-1)*x^3+1/2*a
^2*f^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/2*a^2*f^2*x*(-c^2*d*x^2+d)^(1/2)

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

[Out]

integral((a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*g
^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), 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)**2*(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 )}^{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*(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)^2*(b*arcsin(c*x) + a)^2, x)