\(\int (f+g x)^2 (d-c^2 d x^2)^{5/2} (a+b \arccos (c x)) \, dx\) [11]

Optimal result

Integrand size = 31, antiderivative size = 940 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

-1/36*b*d^2*f^2*(-c^2*x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2
)-5/128*d^2*g^2*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*g^2*x^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)
^(1/2)+5/24*d^2*f^2*x*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+5/48*d^2*g^2*x^3*(-c^2*x^2+1)*(a+b*a
rccos(c*x))*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f^2*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+1/8*d^2*g
^2*x^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-2/7*d^2*f*g*(-c^2*x^2+1)^3*(a+b*arccos(c*x))*(-c^
2*d*x^2+d)^(1/2)/c^2-2/7*b*d^2*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+25/96*b*c*d^2*f^2*x^2*(-c^2*d*x
^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-5/256*b*d^2*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2/7*b*c*d^2*f*g*x
^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-5/96*b*c^3*d^2*f^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+59/7
68*b*c*d^2*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-6/35*b*c^3*d^2*f*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x
^2+1)^(1/2)-17/288*b*c^3*d^2*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2/49*b*c^5*d^2*f*g*x^7*(-c^2*d*x^
2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/64*b*c^5*d^2*g^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-5/32*d^2*f^2*(a+b
*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-5/256*d^2*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 = 0.61 (sec) , antiderivative size = 940, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {4862, 4848, 4744, 4742, 4738, 30, 14, 267, 4768, 200, 4788, 4784, 4796, 272, 45} \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {b c^5 d^2 g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 \sqrt {d-c^2 d x^2} x^6}{288 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 \sqrt {d-c^2 d x^2} x^4}{768 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3+\frac {1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3+\frac {5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3+\frac {2 b c d^2 f g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x}{128 c^2}+\frac {1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x+\frac {5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x-\frac {2 b d^2 f g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]

[In]

Int[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

(-2*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) + (25*b*c*d^2*f^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sq
rt[1 - c^2*x^2]) - (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d^2*f*g*x^3*Sqrt[d
- c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (59*
b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1 - c^2*x^2]) - (6*b*c^3*d^2*f*g*x^5*Sqrt[d - c^2*d*x^2])/(35*S
qrt[1 - c^2*x^2]) - (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) + (2*b*c^5*d^2*f*g*x^7*
Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) -
(b*d^2*f^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c
*x]))/16 - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(128*c^2) + (5*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2
]*(a + b*ArcCos[c*x]))/64 + (5*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/24 + (5*d^2*g^
2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/48 + (d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^
2]*(a + b*ArcCos[c*x]))/6 + (d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 - (2*d^2*f
*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (5*d^2*f^2*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2]) - (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(256*b*c^3*Sq
rt[1 - c^2*x^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 267

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

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 4744

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((
a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n,
x], x] + Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcC
os[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 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 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 4788

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 2*p + 1)))*Simp[(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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 {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+2 f g x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 d^2 f g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \, dx}{6 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (2 b d^2 f g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}} \\ & = -\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt {1-c^2 x^2}}-\frac {\left (2 b d^2 f g \sqrt {d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 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}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{48 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 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}{64 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{48 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (5 b d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.60 (sec) , antiderivative size = 794, normalized size of antiderivative = 0.84 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 \left (-352800 b \left (8 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \arccos (c x)^2-705600 a \sqrt {d} \left (8 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-2822400 b c^2 f g x-5160960 a c f g \sqrt {1-c^2 x^2}[phone] a c^3 f^2 x \sqrt {1-c^2 x^2}-705600 a c g^2 x \sqrt {1-c^2 x^2}[phone] a c^3 f g x^2 \sqrt {1-c^2 x^2}-9784320 a c^5 f^2 x^3 \sqrt {1-c^2 x^2}+5550720 a c^3 g^2 x^3 \sqrt {1-c^2 x^2}-15482880 a c^5 f g x^4 \sqrt {1-c^2 x^2}+3010560 a c^7 f^2 x^5 \sqrt {1-c^2 x^2}-6397440 a c^5 g^2 x^5 \sqrt {1-c^2 x^2}+5160960 a c^7 f g x^6 \sqrt {1-c^2 x^2}+2257920 a c^7 g^2 x^7 \sqrt {1-c^2 x^2}+141120 b \left (15 c^2 f^2+g^2\right ) \cos (2 \arccos (c x))+564480 b c f g \cos (3 \arccos (c x))-211680 b c^2 f^2 \cos (4 \arccos (c x))+35280 b g^2 \cos (4 \arccos (c x))-112896 b c f g \cos (5 \arccos (c x))+15680 b c^2 f^2 \cos (6 \arccos (c x))-15680 b g^2 \cos (6 \arccos (c x))+11520 b c f g \cos (7 \arccos (c x))+2205 b g^2 \cos (8 \arccos (c x))\right )+168 b \sqrt {d-c^2 d x^2} \arccos (c x) \left (-58112 c f g \sqrt {1-c^2 x^2}+111872 c^3 f g x^2 \sqrt {1-c^2 x^2}-27648 c f g \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))-3840 c f g \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+25200 c^2 f^2 \sin (2 \arccos (c x))+1680 g^2 \sin (2 \arccos (c x))-8960 c f g \sin (3 \arccos (c x))-5040 c^2 f^2 \sin (4 \arccos (c x))+840 g^2 \sin (4 \arccos (c x))-5376 c f g \sin (5 \arccos (c x))+560 c^2 f^2 \sin (6 \arccos (c x))-560 g^2 \sin (6 \arccos (c x))+105 g^2 \sin (8 \arccos (c x))\right )\right )}{18063360 c^3 \sqrt {1-c^2 x^2}} \]

[In]

Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

(d^2*(-352800*b*(8*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 705600*a*Sqrt[d]*(8*c^2*f^2 + 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))] + Sqrt[d - c^2*d*x^2]*(-2822400*b*c^2*
f*g*x - 5160960*a*c*f*g*Sqrt[1 - c^2*x^2] + 12418560*a*c^3*f^2*x*Sqrt[1 - c^2*x^2] - 705600*a*c*g^2*x*Sqrt[1 -
c^2*x^2] + 15482880*a*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] - 9784320*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] + 5550720*a*c^3
*g^2*x^3*Sqrt[1 - c^2*x^2] - 15482880*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 3010560*a*c^7*f^2*x^5*Sqrt[1 - c^2*x^2
] - 6397440*a*c^5*g^2*x^5*Sqrt[1 - c^2*x^2] + 5160960*a*c^7*f*g*x^6*Sqrt[1 - c^2*x^2] + 2257920*a*c^7*g^2*x^7*
Sqrt[1 - c^2*x^2] + 141120*b*(15*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] + 564480*b*c*f*g*Cos[3*ArcCos[c*x]] - 21168
0*b*c^2*f^2*Cos[4*ArcCos[c*x]] + 35280*b*g^2*Cos[4*ArcCos[c*x]] - 112896*b*c*f*g*Cos[5*ArcCos[c*x]] + 15680*b*
c^2*f^2*Cos[6*ArcCos[c*x]] - 15680*b*g^2*Cos[6*ArcCos[c*x]] + 11520*b*c*f*g*Cos[7*ArcCos[c*x]] + 2205*b*g^2*Co
s[8*ArcCos[c*x]]) + 168*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-58112*c*f*g*Sqrt[1 - c^2*x^2] + 111872*c^3*f*g*x^2
*Sqrt[1 - c^2*x^2] - 27648*c*f*g*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 3840*c*f*g*(1 - c^2*x^2)^(3/2)*Cos[4
*ArcCos[c*x]] + 25200*c^2*f^2*Sin[2*ArcCos[c*x]] + 1680*g^2*Sin[2*ArcCos[c*x]] - 8960*c*f*g*Sin[3*ArcCos[c*x]]
- 5040*c^2*f^2*Sin[4*ArcCos[c*x]] + 840*g^2*Sin[4*ArcCos[c*x]] - 5376*c*f*g*Sin[5*ArcCos[c*x]] + 560*c^2*f^2*
Sin[6*ArcCos[c*x]] - 560*g^2*Sin[6*ArcCos[c*x]] + 105*g^2*Sin[8*ArcCos[c*x]])))/(18063360*c^3*Sqrt[1 - c^2*x^2
])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 2204, normalized size of antiderivative = 2.34

method	result	size
default	\(\text {Expression too large to display}\)	\(2204\)
parts	\(\text {Expression too large to display}\)	\(2204\)

[In]

int((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(f^2*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(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*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2*(1
/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(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))))))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+b*(-3/1024*(-d*(c^2*x
^2-1))^(1/2)*(c*x*(-c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*(22*I*c^2*f^2+32*arccos(c*x)*c^2*f^2+I*g^2+4*arccos(c*x)*g^2
)*sin(3*arccos(c*x))*d^2/c^3/(c^2*x^2-1)+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*
c^9*x^9-256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5-32*I*(-c^2*x
^2+1)^(1/2)*x^2*c^2-88*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+8*c*x)*g^2*(8*arccos(c*x)+I)*d^2/c^3/(c^2*x^2-1)+1/3136*(-
d*(c^2*x^2-1))^(1/2)*(64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+
56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*f*g*(I+7*arccos(c*x))*d^2
/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7-48*I*(-c^2*x^2+1)^(
1/2)*x^4*c^4-64*c^5*x^5+18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-6*c*x)*(6*arccos(c*x)*
c^2*f^2+I*c^2*f^2-6*arccos(c*x)*g^2-I*g^2)*d^2/c^3/(c^2*x^2-1)-1/1024*(-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)*(24*arccos(c*x)
*c^2*f^2+6*I*f^2*c^2-4*arccos(c*x)*g^2-I*g^2)*d^2/c^3/(c^2*x^2-1)-5/64*(-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*(arccos(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/64*(-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*(arccos(c*x)-I)*d^2/c^2/(c^2*x^2-1)+1/256*(-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)*(30*arccos(c*x)*c^2*f^2+2*arccos(c*x)*g^2-15*I*c^2*f^2-I*g^2)*d^
2/c^3/(c^2*x^2-1)+1/64*(-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*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)
^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^
3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*(6*arccos(c*x)*c^2*f^2-I*c^2*f^2-6*arccos(c*x)*g^2+I*g^2)*d^2/c^3/(c^2*x^2-1)+1/
16384*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-3
20*c^7*x^7-160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*c^2*x^2-88*c^3*x^3-I*(-c^2*x^2
+1)^(1/2)+8*c*x)*g^2*(-I+8*arccos(c*x))*d^2/c^3/(c^2*x^2-1)+1/3920*(-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*(11*I+70*arccos(c*x))*cos(6*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+3/7840*(-d*(c^2*x^2-1))^(1/2
)*(c*x*(-c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*f*g*(9*I+35*arccos(c*x))*sin(6*arccos(c*x))*d^2/c^2/(c^2*x^2-1)-1/80*(-
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*(I+5*arccos(c*x))*cos(4*arccos(c*x))*d^2/c^2/(c^
2*x^2-1)+5/256*(-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*(8*c^2*f^2+g^2)*d^2-1/1
024*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(54*I*c^2*f^2+144*arccos(c*x)*c^2*f^2+5*I*g^2+
4*arccos(c*x)*g^2)*cos(3*arccos(c*x))*d^2/c^3/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(c*x*(-c^2*x^2+1)^(1/2)
+I*c^2*x^2-I)*f*g*(3*I+5*arccos(c*x))*sin(4*arccos(c*x))*d^2/c^2/(c^2*x^2-1))

Fricas [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g^2*x^6 + 2*a*c^4*d^2*f*g*x^5 - 4*a*c^2*d^2*f*g*x^3 + 2*a*d^2*f*g*x + a*d^2*f^2 + (a*c^4*d
^2*f^2 - 2*a*c^2*d^2*g^2)*x^4 - (2*a*c^2*d^2*f^2 - a*d^2*g^2)*x^2 + (b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 -
4*b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 - 2*b*c^2*d^2*g^2)*x^4 - (2*b*c^2*d^2*f^2 -
b*d^2*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**2*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x)),x)

[Out]

Timed out

Maxima [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="maxima")

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a*f^2 + 1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*
d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a*g^2 - 2/7*(-c^2*d
*x^2 + d)^(7/2)*a*f*g/(c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 - 4*b*c^2*d^2*f*g*x
^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 - 2*b*c^2*d^2*g^2)*x^4 - (2*b*c^2*d^2*f^2 - b*d^2*g^2)*x^2)*sq
rt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),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. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[In]

int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2), x)