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

Optimal result

Integrand size = 31, antiderivative size = 670 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=-\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {1-c^2 x^2}}-\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}+\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}+\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2}-\frac {g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4}+\frac {g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

1/2*f^3*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-3/8*f*g^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/4*f*
g^2*x^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-f^2*g*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-1
/3*g^3*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+1/5*g^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*
d*x^2+d)^(1/2)/c^4-b*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/15*b*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(
-c^2*x^2+1)^(1/2)+1/4*b*c*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/16*b*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2
)/c/(-c^2*x^2+1)^(1/2)+1/3*b*c*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/45*b*g^3*x^3*(-c^2*d*x^2+d)
^(1/2)/c/(-c^2*x^2+1)^(1/2)+3/16*b*c*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/25*b*c*g^3*x^5*(-c^2*
d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/4*f^3*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-3/16
*f*g^2*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {4862, 4848, 4742, 4738, 30, 4768, 4784, 4796, 272, 45, 4780, 12} \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2}-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}-\frac {g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {1-c^2 x^2}}+\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}+\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}-\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

-((b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[1 - c^2*x^2])) - (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^
2*x^2]) + (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) - (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*
Sqrt[1 - c^2*x^2]) + (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) - (b*g^3*x^3*Sqrt[d - c^2*d*x^2
])/(45*c*Sqrt[1 - c^2*x^2]) + (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (b*c*g^3*x^5*Sqrt
[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/2 - (3*f*g^2*x*Sqrt[
d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/4 - (f^2*g
*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/c^2 - (g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cCos[c*x]))/(3*c^4) + (g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^4) - (f^3*Sqrt[d - c^
2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) - (3*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2
)/(16*b*c^3*Sqrt[1 - c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4738

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

Rule 4742

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

Rule 4768

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcCos[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*ArcCos[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 4780

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 4784

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 4796

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 4848

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[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 4862

Int[((a_.) + ArcCos[(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*ArcCos[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)^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\sqrt {d-c^2 d x^2} \int \left (f^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+3 f^2 g x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+3 f g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+g^3 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2}-\frac {g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4}+\frac {g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}+\frac {\left (f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b c f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{c \sqrt {1-c^2 x^2}}+\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b c g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {1-c^2 x^2}}+\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}+\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2}-\frac {g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4}+\frac {g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt {1-c^2 x^2}} \\ & = -\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {1-c^2 x^2}}-\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}+\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}+\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2}-\frac {g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4}+\frac {g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.66 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {240 a \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (-16 g^3-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-3600 a c \sqrt {d} f \left (4 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-2400 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)-\cos (3 \arccos (c x))\right )+3600 b c^3 f^3 \sqrt {d-c^2 d x^2} (\cos (2 \arccos (c x))+2 \arccos (c x) (-\arccos (c x)+\sin (2 \arccos (c x))))+675 b c f g^2 \sqrt {d-c^2 d x^2} \left (-8 \arccos (c x)^2+\cos (4 \arccos (c x))+4 \arccos (c x) \sin (4 \arccos (c x))\right )-8 b g^3 \sqrt {d-c^2 d x^2} \left (16 c x \left (30+5 c^2 x^2-9 c^4 x^4\right )+15 \arccos (c x) \left (30 \sqrt {1-c^2 x^2}-5 \sin (3 \arccos (c x))-3 \sin (5 \arccos (c x))\right )\right )}{28800 c^4 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

(240*a*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f
^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)) - 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*ArcTan
[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 2400*b*c^2*f^2*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*(1 - c
^2*x^2)^(3/2)*ArcCos[c*x] - Cos[3*ArcCos[c*x]]) + 3600*b*c^3*f^3*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCos[c*x]] + 2*A
rcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])) + 675*b*c*f*g^2*Sqrt[d - c^2*d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*
ArcCos[c*x]] + 4*ArcCos[c*x]*Sin[4*ArcCos[c*x]]) - 8*b*g^3*Sqrt[d - c^2*d*x^2]*(16*c*x*(30 + 5*c^2*x^2 - 9*c^4
*x^4) + 15*ArcCos[c*x]*(30*Sqrt[1 - c^2*x^2] - 5*Sin[3*ArcCos[c*x]] - 3*Sin[5*ArcCos[c*x]])))/(28800*c^4*Sqrt[
1 - c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.73 (sec) , antiderivative size = 1396, normalized size of antiderivative = 2.08

method	result	size
default	\(\text {Expression too large to display}\)	\(1396\)
parts	\(\text {Expression too large to display}\)	\(1396\)

[In]

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

[Out]

a*(f^3*(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^3*(-1/5
*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(-c^2*d*x^2+d)^(3/2))+3*f*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))))-f^2*g*(-c
^2*d*x^2+d)^(3/2)/c^2/d)+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)*arccos(c*x)^2*f*(4*
c^2*f^2+3*g^2)+1/800*(-d*(c^2*x^2-1))^(1/2)*(16*I*c^5*x^5*(-c^2*x^2+1)^(1/2)+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2
)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*g^3*(I+5*arccos(c*x))/c^4/(c^2*x^2-1)+3/256*(-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)*f*g^2*(4*arccos(c*x)+I)/c^3/(c^2*x^2-1)+1/288*(-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)*g*(12*I*f^2*c^2+36*arccos(c*x)*c^2*f^2+I*g^2+3*a
rccos(c*x)*g^2)/c^4/(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^3*(I+2*arccos(c*x))/c/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+
c^2*x^2-1)*g*(6*I*f^2*c^2+6*arccos(c*x)*c^2*f^2+I*g^2+arccos(c*x)*g^2)/c^4/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(
1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(-6*I*f^2*c^2+6*arccos(c*x)*c^2*f^2-I*g^2+arccos(c*x)*g^2)/c^4/(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^
3*(-I+2*arccos(c*x))/c/(c^2*x^2-1)+1/288*(-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)*g*(-12*I*f^2*c^2+36*arccos(c*x)*c^2*f^2-I*g^2+3*arccos(c*x)*g^2)/c^4/(c^2*
x^2-1)+3/256*(-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)*f*g^2*(-I+4*arccos(c*x))/c^3/(c^2*x^2-1)+1/800*(-d*(c^2*x^2-1))^(1/2)*(
16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2
+1)^(1/2)*x*c-1)*g^3*(-I+5*arccos(c*x))/c^4/(c^2*x^2-1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]

[In]

integrate((g*x+f)**3*(a+b*acos(c*x))*(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))*(f + g*x)**3, x)

Maxima [F]

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

[In]

integrate((g*x+f)^3*(a+b*arccos(c*x))*(-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*f^3 - 1/15*a*g^3*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d)
+ 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 3/8*a*f*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) - (-c^2*d*x^2 + d)^(3/2)*a*f^2*g/(c^2*d) + sqrt(d)*integrate((b*g^3*x^3 + 3*b*
f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)

Giac [F(-2)]

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

[In]

integrate((g*x+f)^3*(a+b*arccos(c*x))*(-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. \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]

[In]

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

[Out]

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