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

Optimal. Leaf size=1015 \[ -\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{c x-1} \sqrt{c x+1}}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3-\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d^2 f g (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.10583, antiderivative size = 1015, 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 = {5836, 5822, 5685, 5683, 5676, 30, 14, 261, 5718, 194, 5745, 5743, 5759, 266, 43} \[ -\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{c x-1} \sqrt{c x+1}}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3-\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d^2 f g (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5836

Int[((a_.) + ArcCosh[(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*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x
)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[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]

Rule 5822

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5685

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)
*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)]
&& GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

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 5745

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*
x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[
{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 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 \cosh ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} (f+g x)^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (f^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+2 f g x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+g^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 d^2 f g \sqrt{d-c^2 d x^2}\right ) \int x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac{\left (5 d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}-\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}\\ &=\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{24} d^2 f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-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 x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\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 x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}+\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}+\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 x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-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 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\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 x} \sqrt{1+c x}}-\frac{\left (5 b c d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 \sqrt{-1+c x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}-\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 x} \sqrt{1+c x}}+\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 x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{25 b c d^2 f^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d^2 f^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{59 b c d^2 g^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-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 \cosh ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{128 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{25 b c d^2 f^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b d^2 g^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d^2 f^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{59 b c d^2 g^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-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 \cosh ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 7.34846, size = 1282, normalized size = 1.26 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((-2*a*d^2*f*g)/(7*c^2) + (a*d^2*(88*c^2*f^2 - 5*g^2)*x)/(128*c^2) + (6*a*d^2*f*g*x^
2)/7 + (a*d^2*(-104*c^2*f^2 + 59*g^2)*x^3)/192 - (6*a*c^2*d^2*f*g*x^4)/7 + (a*c^2*d^2*(8*c^2*f^2 - 17*g^2)*x^5
)/48 + (2*a*c^4*d^2*f*g*x^6)/7 + (a*c^4*d^2*g^2*x^7)/8) - (5*a*d^(5/2)*(8*c^2*f^2 + g^2)*ArcTan[(c*x*Sqrt[-(d*
(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(128*c^3) - (b*d^2*f*g*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x -
12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[c*x]]))/(18*c^2*Sqrt[(-1 + c*x)/(1 +
c*x)]*(1 + c*x)) - (b*d^2*f^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh
[c*x] - Sinh[2*ArcCosh[c*x]])))/(8*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f^2*Sqrt[-(d*(-1 + c*x)*(1
+ c*x))]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(64*c*Sqrt[(-1 + c*
x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*g^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]]
- 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(128*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f*g*Sqrt[-(d*
(-1 + c*x)*(1 + c*x))]*(-450*c*x + 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 25*Cosh[3*ArcCosh[c
*x]] + 9*Cosh[5*ArcCosh[c*x]] - 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]))/
(900*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(18*Cosh[2*ArcCosh
[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*(36*ArcCosh[c*x]^2 + Cosh[6*ArcCosh[c*x]] + 18*ArcCosh[c*x]*Sinh[2*ArcCosh
[c*x]] - 18*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]])))/(2304*c*Sqrt[(-1 + c*x)
/(1 + c*x)]*(1 + c*x)) - (b*d^2*g^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCos
h[c*x]] - 2*(36*ArcCosh[c*x]^2 + Cosh[6*ArcCosh[c*x]] + 18*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 18*ArcCosh[c*x]
*Sinh[4*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]])))/(1152*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
- (b*d^2*f*g*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-55125*c*x + 1225*Cosh[3*ArcCosh[c*x]] + 3*(18375*Sqrt[(-1 + c*
x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 441*Cosh[5*ArcCosh[c*x]] + 75*Cosh[7*ArcCosh[c*x]] - 1225*ArcCosh[c*x]*
Sinh[3*ArcCosh[c*x]] - 2205*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]] - 525*ArcCosh[c*x]*Sinh[7*ArcCosh[c*x]])))/(3528
00*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*g^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(1440*ArcCosh[c*x]^2
- 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] + 11
52*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 576*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 384*ArcCosh[c*x]*Sinh[6*ArcCosh
[c*x]] - 72*ArcCosh[c*x]*Sinh[8*ArcCosh[c*x]]))/(73728*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [A]  time = 0.526, size = 1540, normalized size = 1.5 \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*arccosh(c*x)),x)

[Out]

-133/384*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3-11/16*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(c*x+1)/(c*x-1)*arccosh(c*x)*x*f^2-5/32*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*
d^2*f^2-5/256*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*d^2*g^2+13/96*b*(-d*(c^2
*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^4*f^2-11/32*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c/
(c*x-1)^(1/2)*x^2*f^2-1/64*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)^(1/2)*c^5/(c*x-1)^(1/2)*x^8+17/288*b*(-d*(
c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^6-59/768*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)^
(1/2)*c/(c*x-1)^(1/2)*x^4+5/256*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2-1/36*b*(-d*
(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^5/(c*x-1)^(1/2)*x^6*f^2-1/8*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))-2/7*a*f*g/c^2/d*(-c^2*d*x^2+d)^(7/2)+1/48*a*g^2/c^2*x*(-c^2*d*x^2
+d)^(5/2)+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)+5/16*a*f^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-2/49*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)^(1/2)*c^5/(c*x-1
)^(1/2)*x^7+6/35*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^5-2/7*b*(-d*(c^2*x^2-1))^(
1/2)*f*g*d^2/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1
/2)*x-8/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^2+1/8*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d
^2/(c*x+1)*c^6/(c*x-1)*arccosh(c*x)*x^9-23/48*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x
)*x^7+127/192*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5+5/128*b*(-d*(c^2*x^2-1))^(
1/2)*g^2*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)/c^2/(c*x-1)*arcco
sh(c*x)+1/6*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6/(c*x-1)*arccosh(c*x)*x^7*f^2-17/24*b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^5*f^2+59/48*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^2/(c*x-1)*arccosh
(c*x)*x^3*f^2+1/6*a*f^2*x*(-c^2*d*x^2+d)^(5/2)+2/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)*c^6/(c*x-1)*arccos
h(c*x)*x^8-8/7*b*(-d*(c^2*x^2-1))^(1/2)*f*g*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^6+12/7*b*(-d*(c^2*x^2-1))^(
1/2)*f*g*d^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^4+359/73728*b*(-d*(c^2*x^2-1))^(1/2)*g^2*d^2/(c*x+1)^(1/2)/c^3
/(c*x-1)^(1/2)+299/2304*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*f^2

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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*arccosh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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 )} \operatorname{arcosh}\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*arccosh(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)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)

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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*acosh(c*x)),x)

[Out]

Timed out

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Giac [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*arccosh(c*x)),x, algorithm="giac")

[Out]

Timed out