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

Optimal. Leaf size=692 \[ -\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}+\frac{4 b g^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b^2 f g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt{d-c^2 d x^2}}+\frac{14 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt{d-c^2 d x^2}} \]

[Out]

(6*b^2*f^2*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (14*b^2*g^3*(1 - c^2*x^2))/(9*c^4*Sqrt[d - c^2*d*x^2])
+ (3*b^2*f*g^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2)^2)/(27*c^4*Sqrt[d - c^
2*d*x^2]) - (3*b^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (6*b*f^2*g*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2]) + (4*b*g^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3
*Sqrt[d - c^2*d*x^2]) + (3*b*f*g^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) + (2*b
*g^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c^2*x^2)*(a + b*ArcS
in[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c^4*Sqrt[d - c^2*d*x^2]
) - (3*f*g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c^2*x^2)*(a +
b*ArcSin[c*x])^2)/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d -
c^2*d*x^2]) + (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c^3*Sqrt[d - c^2*d*x^2])

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Rubi [A]  time = 0.702042, antiderivative size = 692, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {4777, 4773, 3317, 3296, 2638, 3311, 32, 2635, 8, 2633} \[ -\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}+\frac{4 b g^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b^2 f g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt{d-c^2 d x^2}}+\frac{14 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*f^2*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (14*b^2*g^3*(1 - c^2*x^2))/(9*c^4*Sqrt[d - c^2*d*x^2])
+ (3*b^2*f*g^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2)^2)/(27*c^4*Sqrt[d - c^
2*d*x^2]) - (3*b^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (6*b*f^2*g*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2]) + (4*b*g^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3
*Sqrt[d - c^2*d*x^2]) + (3*b*f*g^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) + (2*b
*g^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c^2*x^2)*(a + b*ArcS
in[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c^4*Sqrt[d - c^2*d*x^2]
) - (3*f*g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c^2*x^2)*(a +
b*ArcSin[c*x])^2)/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d -
c^2*d*x^2]) + (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c^3*Sqrt[d - c^2*d*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 4773

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

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 (c f+g \sin (x))^3 \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (c^3 f^3 (a+b x)^2+3 c^2 f^2 g (a+b x)^2 \sin (x)+3 c f g^2 (a+b x)^2 \sin ^2(x)+g^3 (a+b x)^2 \sin ^3(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{\left (3 f^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 f g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{\left (6 b f^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 f g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 f g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (2 g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{9 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c^3 \sqrt{d-c^2 d x^2}}-\frac{\left (6 b^2 f^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 f g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{9 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 b^2 f g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{4 b g^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c^3 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{14 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 b^2 f g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{4 b g^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c^3 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.47757, size = 582, normalized size = 0.84 \[ \frac{-36 a^2 d \left (1-c^2 x^2\right )^{3/2} \left (c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+4 g^3\right )-108 a^2 c \sqrt{d} f \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (2 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-1296 a b c^2 d f^2 g \left (c^2 x^2-1\right ) \left (c x-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)\right )-216 a b c^3 d f^3 \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^2+162 a b c d f g^2 \left (c^2 x^2-1\right ) \left (-2 \sin ^{-1}(c x)^2+2 \sin \left (2 \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+\cos \left (2 \sin ^{-1}(c x)\right )\right )-48 a b d g^3 \left (c^2 x^2-1\right ) \left (c^3 x^3-3 \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)+6 c x\right )+648 b^2 c^2 d f^2 g \left (1-c^2 x^2\right ) \left (2 c x \sin ^{-1}(c x)-\sqrt{1-c^2 x^2} \left (\sin ^{-1}(c x)^2-2\right )\right )-72 b^2 c^3 d f^3 \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^3+27 b^2 c d f g^2 \left (1-c^2 x^2\right ) \left (4 \sin ^{-1}(c x)^3+\left (3-6 \sin ^{-1}(c x)^2\right ) \sin \left (2 \sin ^{-1}(c x)\right )-6 \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )\right )-2 b^2 d g^3 \left (1-c^2 x^2\right ) \left (81 \sqrt{1-c^2 x^2} \left (\sin ^{-1}(c x)^2-2\right )+6 \sin ^{-1}(c x) \left (\sin \left (3 \sin ^{-1}(c x)\right )-27 c x\right )-\left (9 \sin ^{-1}(c x)^2-2\right ) \cos \left (3 \sin ^{-1}(c x)\right )\right )}{216 c^4 d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*a^2*d*(1 - c^2*x^2)^(3/2)*(4*g^3 + c^2*g*(18*f^2 + 9*f*g*x + 2*g^2*x^2)) - 216*a*b*c^3*d*f^3*(-1 + c^2*x^
2)*ArcSin[c*x]^2 - 72*b^2*c^3*d*f^3*(-1 + c^2*x^2)*ArcSin[c*x]^3 - 1296*a*b*c^2*d*f^2*g*(-1 + c^2*x^2)*(c*x -
Sqrt[1 - c^2*x^2]*ArcSin[c*x]) - 48*a*b*d*g^3*(-1 + c^2*x^2)*(6*c*x + c^3*x^3 - 3*Sqrt[1 - c^2*x^2]*(2 + c^2*x
^2)*ArcSin[c*x]) + 648*b^2*c^2*d*f^2*g*(1 - c^2*x^2)*(2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^
2)) - 108*a^2*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d
*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 162*a*b*c*d*f*g^2*(-1 + c^2*x^2)*(-2*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 2
*ArcSin[c*x]*Sin[2*ArcSin[c*x]]) + 27*b^2*c*d*f*g^2*(1 - c^2*x^2)*(4*ArcSin[c*x]^3 - 6*ArcSin[c*x]*Cos[2*ArcSi
n[c*x]] + (3 - 6*ArcSin[c*x]^2)*Sin[2*ArcSin[c*x]]) - 2*b^2*d*g^3*(1 - c^2*x^2)*(81*Sqrt[1 - c^2*x^2]*(-2 + Ar
cSin[c*x]^2) - (-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] + 6*ArcSin[c*x]*(-27*c*x + Sin[3*ArcSin[c*x]])))/(216
*c^4*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.696, size = 1876, normalized size = 2.7 \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)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)

[Out]

-3/2*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c/d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-6*a*b*(-d*(c^2*x^2-1))^(1/2)*g/c/
d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2-3/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c/d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*
x^2+1)^(1/2)*x^2-6*b^2*(-d*(c^2*x^2-1))^(1/2)*g/c/d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x*f^2-3/2*b^2*(
-d*(c^2*x^2-1))^(1/2)*f*g^2/d/(c^2*x^2-1)*arcsin(c*x)^2*x^3-3*b^2*(-d*(c^2*x^2-1))^(1/2)*g/d/(c^2*x^2-1)*arcsi
n(c*x)^2*x^2*f^2-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/(c^2*x^2-1)*arcsin(c*x)^3*f^3-1/3*b^2*(
-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d/(c^2*x^2-1)*arcsin(c*x)^2*x^2-3/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/d/(c^2*
x^2-1)*x+3*b^2*(-d*(c^2*x^2-1))^(1/2)*g/c^2/d/(c^2*x^2-1)*arcsin(c*x)^2*f^2-2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3
/d/(c^2*x^2-1)*arcsin(c*x)*x^4+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d/(c^2*x^2-1)*arcsin(c*x)+6*b^2*(-d*(c^2
*x^2-1))^(1/2)*g/d/(c^2*x^2-1)*x^2*f^2-2/3*a^2*g^3/d/c^4*(-c^2*d*x^2+d)^(1/2)+3/2*a^2*f*g^2/c^2/(c^2*d)^(1/2)*
arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-3*a^2*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2)-1/3*a^2*g^3*x^2/c^2/d*(-c^
2*d*x^2+d)^(1/2)+2/27*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/d/(c^2*x^2-1)*x^4-40/27*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^
4/d/(c^2*x^2-1)+a^2*f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/3*a*b*(-d*(c^2*x^2-1))^(1
/2)*g^3/c^2/d/(c^2*x^2-1)*arcsin(c*x)*x^2+6*a*b*(-d*(c^2*x^2-1))^(1/2)*g/c^2/d/(c^2*x^2-1)*arcsin(c*x)*f^2-2/9
*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c/d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^3-4/3*b^2*(-d*(c^2*x^2-1))^(1
/2)*g^3/c^3/d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x+3/4*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^3/d/(c^2*x^2
-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(
c*x)^3*f*g^2+3/2*b^2*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/d/(c^2*x^2-1)*arcsin(c*x)^2*x-2/9*a*b*(-d*(c^2*x^2-1))^(
1/2)*g^3/c/d/(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)*g^3/c^3/d/(c^2*x^2-1)*(-c^2*x^2
+1)^(1/2)*x-3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/d/(c^2*x^2-1)*arcsin(c*x)*x^3+3/4*a*b*(-d*(c^2*x^2-1))^(1/2)*f*
g^2/c^3/d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-6*a*b*(-d*(c^2*x^2-1))^(1/2)*g/d/(c^2*x^2-1)*arcsin(c*x)*x^2*f^2-a*b*
(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/(c^2*x^2-1)*arcsin(c*x)^2*f^3-3/2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c
^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^2*f*g^2+3*a*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2/c^2/d/(c^2*x^2-1)*arc
sin(c*x)*x-3/2*a^2*f*g^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+38/27*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d/(c^2*x^2-1)*x
^2+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d/(c^2*x^2-1)*arcsin(c*x)^2-6*b^2*(-d*(c^2*x^2-1))^(1/2)*g/c^2/d/(c^
2*x^2-1)*f^2-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/d/(c^2*x^2-1)*arcsin(c*x)^2*x^4+3/4*b^2*(-d*(c^2*x^2-1))^(1/2)
*f*g^2/d/(c^2*x^2-1)*x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(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 (-\frac{{\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}}{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)/(c^2*d*x^2 - d), x)

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

Exception raised: TypeError

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