\(\int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx\) [59]

Optimal result

Integrand size = 33, antiderivative size = 737 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {8 b^2 f g \sqrt {d-c^2 d x^2}}{9 c^2}-\frac {1}{4} b^2 f^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {b^2 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {4 b c f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {4861, 4847, 4741, 4737, 4723, 327, 222, 4767, 4739, 455, 45, 4783, 4795} \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {4 b f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}-\frac {4 b c f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {b^2 f^2 \arcsin (c x) \sqrt {d-c^2 d x^2}}{4 c \sqrt {1-c^2 x^2}}-\frac {b^2 g^2 \arcsin (c x) \sqrt {d-c^2 d x^2}}{64 c^3 \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 f^2 x \sqrt {d-c^2 d x^2}+\frac {8 b^2 f g \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {b^2 g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 g^2 x^3 \sqrt {d-c^2 d x^2} \]

[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 45

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 222

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 327

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 455

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 4723

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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4739

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 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4783

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[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[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/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^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] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4795

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

Rule 4847

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 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 {d-c^2 d x^2} \int (f+g x)^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2+2 f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {\left (f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, 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 ) (a+b \arcsin (c x)) \, 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 (a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, dx}{2 \sqrt {1-c^2 x^2}} \\ & = \frac {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}}-\frac {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}}-\frac {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}}-\frac {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}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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 {(a+b \arcsin (c x))^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 (a+b \arcsin (c x)) \, 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} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {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}}+\frac {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}}-\frac {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}}-\frac {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}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {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}}+\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 ) \text {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} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {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}}-\frac {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}}+\frac {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}}-\frac {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}}-\frac {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}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {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}}-\frac {\left (2 b^2 f g \sqrt {d-c^2 d x^2}\right ) \text {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} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {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}}-\frac {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}}+\frac {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}}-\frac {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}}-\frac {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}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.60 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {1}{4} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 (a+b \arcsin (c x))^3}{6 b c}-\frac {4 b f g \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^2}-\frac {b f^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )}{4 c}-\frac {b g^2 \left (8 a c^4 x^4+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b \left (-3+8 c^4 x^4\right ) \arcsin (c x)\right )}{64 c^3}-\frac {g^2 \left (6 b c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 (a+b \arcsin (c x))^3-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )\right )}{48 b c^3}\right )}{\sqrt {1-c^2 x^2}} \]

[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]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 1852, normalized size of antiderivative = 2.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1852\)
parts	\(\text {Expression too large to display}\)	\(1852\)

[In]

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

[Out]

a^2*(f^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+g^2*(-1
/4*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/c^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x
/(-c^2*d*x^2+d)^(1/2))))-2/3*f*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b^2*(-1/24*(-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*(4*c^2*f^2+g^2)+1/512*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^
4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(4*I*arcsin(c*x)+8*arcsi
n(c*x)^2-1)/c^3/(c^2*x^2-1)+1/108*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3
*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/
2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f^2*(2*I*arcsin(c*x)+2*arcsin(c*x)^2
-1)/c/(c^2*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arcsin(c*x)^2-2+2*I*arc
sin(c*x))/c^2/(c^2*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(c*x)^2-2
-2*I*arcsin(c*x))/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^
2*x^2+1)^(1/2)-2*c*x)*f^2*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/c/(c^2*x^2-1)+1/108*(-d*(c^2*x^2-1))^(1/2)*(4*I
*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*f*g*(-6*I*arcsin(c*x)+9*arcsin(c
*x)^2-2)/c^2/(c^2*x^2-1)+1/512*(-d*(c^2*x^2-1))^(1/2)*(8*I*(-c^2*x^2+1)^(1/2)*c^4*x^4+8*c^5*x^5-8*I*(-c^2*x^2+
1)^(1/2)*x^2*c^2-12*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(-4*I*arcsin(c*x)+8*arcsin(c*x)^2-1)/c^3/(c^2*x^2-
1))+2*a*b*(-1/16*(-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*(4*c^2*f^2+g^2)+1/256
*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I
*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(4*arcsin(c*x)+I)/c^3/(c^2*x^2-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*
x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(I+3*arcsin(c*x))/c^2/(c^2*x^2-1)+1/16*(-
d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f^2*(I+2*arcsin(c*
x))/c/(c^2*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arcsin(c*x)+I)/c^2/(c^2
*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(c*x)-I)/c^2/(c^2*x^2-1)+1/
16*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f^2*(-I+2*arcs
in(c*x))/c/(c^2*x^2-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^
(1/2)*x*c-5*c^2*x^2+1)*f*g*(-I+3*arcsin(c*x))/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(8*I*(-c^2*x^2+1)^(
1/2)*c^4*x^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+4*c*x)*g^2*(-I+4*arcsin(
c*x))/c^3/(c^2*x^2-1))

Fricas [F]

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

[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)

Sympy [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}\, dx \]

[In]

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

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2*(f + g*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a^2*f^2 + 1/8*a^2*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c
^2*d*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d)*arcsin(c*x)/c^3) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a^2*f*g/(c^2*d) + sqrt(d
)*integrate(((b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^2*x
^2 + 2*a*b*f*g*x + a*b*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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