3.11 \(\int (f+g x)^2 (d-c^2 d x^2)^{5/2} (a+b \cos ^{-1}(c x)) \, dx\)

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.95887, antiderivative size = 940, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 261, 4678, 194, 4700, 4698, 4708, 266, 43} \[ \frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}+\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{1-c^2 x^2}}-\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}-\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c} \]

Antiderivative was successfully verified.

[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 4778

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

Rule 4764

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 4650

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4648

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcCos[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] + Dist[(b*c*n*Sqrt[d + e*x^2])/(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 4642

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

Rule 30

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

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 261

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 4678

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 194

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 4700

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4698

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[Sqrt[d + e*x^2]/((m + 2)*Sqrt[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*Sqrt[d + e*x^2])/(f*(m
+ 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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4708

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

Rule 266

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 43

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])

Rubi steps

\begin{align*} \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right )+2 f g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )\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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{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} \left (a+b \cos ^{-1}(c x)\right ) \, 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} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{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} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(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 \left (a+b \cos ^{-1}(c x)\right )}{\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 ) \operatorname{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} \left (a+b \cos ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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 \cos ^{-1}(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} \left (a+b \cos ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 4.516, size = 794, normalized size = 0.84 \[ \frac{d^2 \left (\sqrt{d-c^2 d x^2} \left (3010560 a c^7 f^2 x^5 \sqrt{1-c^2 x^2}-9784320 a c^5 f^2 x^3 \sqrt{1-c^2 x^2}[phone] a c^3 f^2 x \sqrt{1-c^2 x^2}+5160960 a c^7 f g x^6 \sqrt{1-c^2 x^2}-15482880 a c^5 f g x^4 \sqrt{1-c^2 x^2}[phone] a c^3 f g x^2 \sqrt{1-c^2 x^2}-5160960 a c f g \sqrt{1-c^2 x^2}+2257920 a c^7 g^2 x^7 \sqrt{1-c^2 x^2}-6397440 a c^5 g^2 x^5 \sqrt{1-c^2 x^2}+5550720 a c^3 g^2 x^3 \sqrt{1-c^2 x^2}-705600 a c g^2 x \sqrt{1-c^2 x^2}+141120 b \left (15 c^2 f^2+g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right )-211680 b c^2 f^2 \cos \left (4 \cos ^{-1}(c x)\right )+15680 b c^2 f^2 \cos \left (6 \cos ^{-1}(c x)\right )-2822400 b c^2 f g x+564480 b c f g \cos \left (3 \cos ^{-1}(c x)\right )-112896 b c f g \cos \left (5 \cos ^{-1}(c x)\right )+11520 b c f g \cos \left (7 \cos ^{-1}(c x)\right )+35280 b g^2 \cos \left (4 \cos ^{-1}(c x)\right )-15680 b g^2 \cos \left (6 \cos ^{-1}(c x)\right )+2205 b g^2 \cos \left (8 \cos ^{-1}(c x)\right )\right )-705600 a \sqrt{d} \sqrt{1-c^2 x^2} \left (8 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-352800 b \sqrt{d-c^2 d x^2} \left (8 c^2 f^2+g^2\right ) \cos ^{-1}(c x)^2+168 b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left (25200 c^2 f^2 \sin \left (2 \cos ^{-1}(c x)\right )-5040 c^2 f^2 \sin \left (4 \cos ^{-1}(c x)\right )+560 c^2 f^2 \sin \left (6 \cos ^{-1}(c x)\right )+111872 c^3 f g x^2 \sqrt{1-c^2 x^2}-58112 c f g \sqrt{1-c^2 x^2}-27648 c f g \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )-3840 c f g \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )-8960 c f g \sin \left (3 \cos ^{-1}(c x)\right )-5376 c f g \sin \left (5 \cos ^{-1}(c x)\right )+1680 g^2 \sin \left (2 \cos ^{-1}(c x)\right )+840 g^2 \sin \left (4 \cos ^{-1}(c x)\right )-560 g^2 \sin \left (6 \cos ^{-1}(c x)\right )+105 g^2 \sin \left (8 \cos ^{-1}(c x)\right )\right )\right )}{18063360 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[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 [A]  time = 0.702, size = 1633, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*a*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d-1/8*a*g^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+5/192*a*g^2/c^2*d*x*(-c^2*d*x^2+d
)^(3/2)+5/128*a*g^2/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/128*a*g^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-
c^2*d*x^2+d)^(1/2))+13/96*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4*f^2-59/768*b*(-d
*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4*g^2-11/32*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2
-1)*(-c^2*x^2+1)^(1/2)*x^2*f^2+5/256*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2*g^2-1/6
4*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8+17/288*b*(-d*(c^2*x^2-1))^(1/2)*g^2*
d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-1/36*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2
)*x^6*f^2+1/48*a*g^2/c^2*x*(-c^2*d*x^2+d)^(5/2)-133/384*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c^2*x^2-1)*arccos(c*
x)*x^3-11/16*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arccos(c*x)*x*f^2+1/6*a*f^2*x*(-c^2*d*x^2+d)^(5/2)-2/49*
b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^7+6/35*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*
c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5-2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x
^3+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^
2*c^6/(c^2*x^2-1)*arccos(c*x)*x^8-8/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x^6+12/7*b*
(-d*(c^2*x^2-1))^(1/2)*f*g*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^4+5/128*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/c^2/(c^2
*x^2-1)*arccos(c*x)*x+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/c^2/(c^2*x^2-1)*arccos(c*x)+1/6*b*(-d*(c^2*x^2-1))^
(1/2)*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^7*f^2-17/24*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x
^5*f^2+59/48*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^3*f^2-8/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g
*d^2/(c^2*x^2-1)*arccos(c*x)*x^2+5/32*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccos(c*x)^2*
d^2*f^2+5/256*b*(-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*d^2*g^2+1/8*b*(-d*(c^2
*x^2-1))^(1/2)*g^2*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^9-23/48*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^4/(c^2*x^2-1)*
arccos(c*x)*x^7+127/192*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^5+5/16*a*f^2*d^2*x*(-c^
2*d*x^2+d)^(1/2)+5/16*a*f^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+5/24*a*f^2*d*x*(-c^
2*d*x^2+d)^(3/2)+299/2304*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*f^2+359/73728*b*(-d*(c
^2*x^2-1))^(1/2)*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*g^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (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} +{\left (a c^{4} d^{2} f^{2} - 2 \, a c^{2} d^{2} g^{2}\right )} x^{4} -{\left (2 \, a c^{2} d^{2} f^{2} - a d^{2} g^{2}\right )} x^{2} +{\left (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} +{\left (b c^{4} d^{2} f^{2} - 2 \, b c^{2} d^{2} g^{2}\right )} x^{4} -{\left (2 \, b c^{2} d^{2} f^{2} - b d^{2} g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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