\(\int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [70]

Optimal result

Integrand size = 33, antiderivative size = 692 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 b^2 f g^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {6 b f^2 g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}+\frac {4 b g^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {3 b f g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}+\frac {2 b g^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{2 b c^3 \sqrt {d-c^2 d x^2}} \]

[Out]

6*b^2*f^2*g*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)+14/9*b^2*g^3*(-c^2*x^2+1)/c^4/(-c^2*d*x^2+d)^(1/2)+3/4*b^2*f
*g^2*x*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)-2/27*b^2*g^3*(-c^2*x^2+1)^2/c^4/(-c^2*d*x^2+d)^(1/2)-3*f^2*g*(-c^
2*x^2+1)*(a+b*arcsin(c*x))^2/c^2/(-c^2*d*x^2+d)^(1/2)-2/3*g^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c^4/(-c^2*d*x^2
+d)^(1/2)-3/2*f*g^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c^2/(-c^2*d*x^2+d)^(1/2)-1/3*g^3*x^2*(-c^2*x^2+1)*(a+b*
arcsin(c*x))^2/c^2/(-c^2*d*x^2+d)^(1/2)-3/4*b^2*f*g^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+
6*b*f^2*g*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+4/3*b*g^3*x*(a+b*arcsin(c*x))*(-c^2*x^
2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+3/2*b*f*g^2*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2
)+2/9*b*g^3*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/3*f^3*(a+b*arcsin(c*x))^3*(-c^2*
x^2+1)^(1/2)/b/c/(-c^2*d*x^2+d)^(1/2)+1/2*f*g^2*(a+b*arcsin(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c^3/(-c^2*d*x^2+d)^(1
/2)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {4861, 4857, 3398, 3377, 2718, 3392, 32, 2715, 8, 2713} \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {6 b f^2 g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b f g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b g^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 g^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c^4 \sqrt {d-c^2 d x^2}}+\frac {f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{2 b c^3 \sqrt {d-c^2 d x^2}}+\frac {4 b g^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b^2 f g^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {6 b^2 f^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 f g^2 x \left (1-c^2 x^2\right )}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 g^3 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}} \]

[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 8

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

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 2713

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]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3398

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 4857

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 4861

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

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

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 582, normalized size of antiderivative = 0.84 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-36 a^2 d \left (1-c^2 x^2\right )^{3/2} \left (4 g^3+c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )-216 a b c^3 d f^3 \left (-1+c^2 x^2\right ) \arcsin (c x)^2-72 b^2 c^3 d f^3 \left (-1+c^2 x^2\right ) \arcsin (c x)^3-1296 a b c^2 d f^2 g \left (-1+c^2 x^2\right ) \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)\right )-48 a b d g^3 \left (-1+c^2 x^2\right ) \left (6 c x+c^3 x^3-3 \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )+648 b^2 c^2 d f^2 g \left (1-c^2 x^2\right ) \left (2 c x \arcsin (c x)-\sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )\right )-108 a^2 c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+162 a b c d f g^2 \left (-1+c^2 x^2\right ) \left (-2 \arcsin (c x)^2+\cos (2 \arcsin (c x))+2 \arcsin (c x) \sin (2 \arcsin (c x))\right )+27 b^2 c d f g^2 \left (1-c^2 x^2\right ) \left (4 \arcsin (c x)^3-6 \arcsin (c x) \cos (2 \arcsin (c x))+\left (3-6 \arcsin (c x)^2\right ) \sin (2 \arcsin (c x))\right )-2 b^2 d g^3 \left (1-c^2 x^2\right ) \left (81 \sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )-\left (-2+9 \arcsin (c x)^2\right ) \cos (3 \arcsin (c x))+6 \arcsin (c x) (-27 c x+\sin (3 \arcsin (c x)))\right )}{216 c^4 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]

[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])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 1636, normalized size of antiderivative = 2.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1636\)
parts	\(\text {Expression too large to display}\)	\(1636\)

[In]

int((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^3*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2
/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+3*f*g^2*(-1/2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)
^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))-3*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+b^2*(-1/6*(-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*(2*c^2*f^2+3*g^2)+1/432*(-d*(c^2*x^2-1))^(1/2)*(2*c^2*x^2-2*I*c*x*
(-c^2*x^2+1)^(1/2)-1)*g^3*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(c^
2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(8*I*arcsin(c*x)*c^2*f^2+4*arcsin(c*x)^2*c^2*f^2+2*I*arcsin(c*x)*g^2+arcsi
n(c*x)^2*g^2-8*c^2*f^2-2*g^2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1
)*g*(4*arcsin(c*x)^2*c^2*f^2-8*I*arcsin(c*x)*c^2*f^2+arcsin(c*x)^2*g^2-2*I*arcsin(c*x)*g^2-8*c^2*f^2-2*g^2)/c^
4/d/(c^2*x^2-1)+1/432*(-d*(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*g^3*(-6*I*arcsin(c*x)+9*
arcsin(c*x)^2-2)/c^4/d/(c^2*x^2-1)+3/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*g^2*f*arcsi
n(c*x)+3/16*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*g^2*f*(2*arcsin(c*x)^2-1)*x-1/216*(-d*(c^2*x^2-1))^(1/2)/
c^4/d/(c^2*x^2-1)*g^3*(9*arcsin(c*x)^2-2)*cos(4*arcsin(c*x))+1/36*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*arc
sin(c*x)*g^3*sin(4*arcsin(c*x))+3/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*f*arcsin(c*x)*cos(3*arcsin(c*
x))+3/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*f*(2*arcsin(c*x)^2-1)*sin(3*arcsin(c*x)))+2*a*b*(-1/4*(-
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*(2*c^2*f^2+3*g^2)+1/144*(-d*(c^2*x^2
-1))^(1/2)*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)*g^3*(I+3*arcsin(c*x))/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1
))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(4*arcsin(c*x)*c^2*f^2+4*I*c^2*f^2+arcsin(c*x)*g^2+I*g^2)/c^4/
d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(4*arcsin(c*x)*c^2*f^2-4*I*c^2
*f^2+arcsin(c*x)*g^2-I*g^2)/c^4/d/(c^2*x^2-1)+1/144*(-d*(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x
^2-1)*g^3*(-I+3*arcsin(c*x))/c^4/d/(c^2*x^2-1)+3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1
)*f*g^2+3/8*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*f*g^2*arcsin(c*x)*x-1/24*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^
2*x^2-1)*arcsin(c*x)*g^3*cos(4*arcsin(c*x))+1/72*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*g^3*sin(4*arcsin(c*x
))+3/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*cos(3*arcsin(c*x))+3/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^
2*x^2-1)*f*g^2*arcsin(c*x)*sin(3*arcsin(c*x)))

Fricas [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]

[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 >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/3*a^2*g^3*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4*d)) - 3/2*a^2*f*g^2*(sqrt(-c^2*d*
x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + a*b*f^3*arcsin(c*x)^2/(c*sqrt(d)) + 6*a*b*f^2*g*x/(c*sqrt(d)
) + a^2*f^3*arcsin(c*x)/(c*sqrt(d)) - 6*sqrt(-c^2*d*x^2 + d)*a*b*f^2*g*arcsin(c*x)/(c^2*d) - 3*sqrt(-c^2*d*x^2
+ d)*a^2*f^2*g/(c^2*d) - sqrt(d)*integrate(((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)*arctan2
(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^2*d*x^2 - d), x)

Giac [F]

\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]

[In]

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

[Out]

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