3.58 \(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=1029 \[ \frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d g^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{7 c^2}-\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d g^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {3 d f^2 g (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2} \]

[Out]

3/8*d*f^3*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-3/16*d*f*g^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2
+3/8*d*f*g^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f^3*x*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2
*d*x^2+d)^(1/2)+1/2*d*f*g^2*x^3*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-3/5*d*f^2*g*(-c*x+1)^
2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-2/35*d*g^3*(-c*x+1)^2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c
^2*d*x^2+d)^(1/2)/c^4-1/7*d*g^3*x^2*(-c*x+1)^2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/5*b*d*f
^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/35*b*d*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2
)/(c*x+1)^(1/2)-5/16*b*c*d*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/32*b*d*f*g^2*x^2*(-c^2*d
*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/5*b*c*d*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/
2)+1/105*b*d*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*c^3*d*f^3*x^4*(-c^2*d*x^2+d)^(1
/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/32*b*c*d*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/25*b*c
^3*d*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-
1)^(1/2)/(c*x+1)^(1/2)+1/12*b*c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/49*b*c^3*d*g^
3*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*d*f^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/
c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/32*d*f*g^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+
1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.06, antiderivative size = 1029, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 5718, 194, 5745, 5743, 5759, 100, 12, 74, 5733, 373} \[ \frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d g^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{7 c^2}-\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d g^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {3 d f^2 g (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35
*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
(3*b*d*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d
*x^2])/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*g^3*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]) + (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2
*d*x^2])/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]) - (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f*g^2*x^6*Sqr
t[d - c^2*d*x^2])/(12*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (3*d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x]))/(16*c^2) + (3*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (d*f^3*x*(1 - c*x)
*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 + (d*f*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*
(a + b*ArcCosh[c*x]))/2 - (3*d*f^2*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2)
- (2*d*g^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(35*c^4) - (d*g^3*x^2*(1 - c*x)^
2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^2) - (3*d*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[
c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*
c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

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 74

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

Rule 100

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

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 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 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 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 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 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 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

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 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 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 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 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]

Rubi steps

\begin {align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} (f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+3 f^2 g x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+3 f g^2 x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+g^3 x^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d f^3 \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}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d f 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}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} d f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d f 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 {3 d f^2 g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {2 d g^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {d g^3 x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac {\left (3 d f^3 \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}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d f 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}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (2+5 c^2 x^2\right )}{35 c^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d f 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 {3 d f^2 g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {2 d g^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {d g^3 x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (3 d f^3 \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}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d f 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}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (2+5 c^2 x^2\right ) \, dx}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d f 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 {3 d f^2 g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {2 d g^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {d g^3 x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d f 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}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (2+c^2 x^2-8 c^4 x^4+5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d f 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 {3 d f^2 g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {2 d g^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {d g^3 x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 4.45, size = 901, normalized size = 0.88 \[ \frac {-352800 b c^3 d \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right ) f^3+22050 b c^3 d \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2-4 \sinh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+\cosh \left (4 \cosh ^{-1}(c x)\right )\right ) f^3+235200 b c^2 d g \sqrt {d-c^2 d x^2} \left (12 \left (\frac {c x-1}{c x+1}\right )^{3/2} \cosh ^{-1}(c x) (c x+1)^3+9 c x-\cosh \left (3 \cosh ^{-1}(c x)\right )\right ) f^2-2352 b c^2 d g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right ) f^2-529200 a c d^{3/2} \left (2 c^2 f^2+g^2\right ) \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right ) f-66150 b c d g^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2-4 \sinh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+\cosh \left (4 \cosh ^{-1}(c x)\right )\right ) f-3675 b c d g^2 \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+12 \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )\right ) f-5040 a d \sqrt {\frac {c x-1}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} \left (4 x^3 \left (35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right ) c^6-2 x \left (175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right ) c^4+g \left (336 f^2+105 g x f+16 g^2 x^2\right ) c^2+32 g^3\right )+784 b d g^3 \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )-4 b d g^3 \sqrt {d-c^2 d x^2} \left (55125 c x-55125 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-1225 \cosh \left (3 \cosh ^{-1}(c x)\right )-1323 \cosh \left (5 \cosh ^{-1}(c x)\right )-225 \cosh \left (7 \cosh ^{-1}(c x)\right )+3675 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+6615 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )+1575 \cosh ^{-1}(c x) \sinh \left (7 \cosh ^{-1}(c x)\right )\right )}{2822400 c^4 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5040*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*
g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*
f*g^2*x^2 + 64*g^3*x^3)) - 529200*a*c*d^(3/2)*f*(2*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[
(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 235200*b*c^2*d*f^2*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-
1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]) - 352800*b*c^3*d*f^3*Sqrt[d - c^2*d
*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 22050*b*c^3*d*f^3*Sqrt[d
- c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 66150*b*c*d*f*
g^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 2352
*b*c^2*d*f^2*g*Sqrt[d - c^2*d*x^2]*(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*ArcCo
sh[c*x]]) + 784*b*d*g^3*Sqrt[d - c^2*d*x^2]*(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]*Sin
h[5*ArcCosh[c*x]]) - 3675*b*c*d*f*g^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Co
sh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]
] + Sinh[6*ArcCosh[c*x]])) - 4*b*d*g^3*Sqrt[d - c^2*d*x^2]*(55125*c*x - 55125*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 +
c*x)*ArcCosh[c*x] - 1225*Cosh[3*ArcCosh[c*x]] - 1323*Cosh[5*ArcCosh[c*x]] - 225*Cosh[7*ArcCosh[c*x]] + 3675*Ar
cCosh[c*x]*Sinh[3*ArcCosh[c*x]] + 6615*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]] + 1575*ArcCosh[c*x]*Sinh[7*ArcCosh[c*
x]]))/(2822400*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d g^{3} x^{5} + 3 \, a c^{2} d f g^{2} x^{4} - 3 \, a d f^{2} g x - a d f^{3} + {\left (3 \, a c^{2} d f^{2} g - a d g^{3}\right )} x^{3} + {\left (a c^{2} d f^{3} - 3 \, a d f g^{2}\right )} x^{2} + {\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} + {\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} + {\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(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

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maple [A]  time = 1.15, size = 1638, normalized size = 1.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^7+11/8*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2
*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5+3/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)
*x-3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^6*f^2+9/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/
(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^4*f^2+3/8*a*f^3*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/
2))-2/35*a*g^3/d/c^4*(-c^2*d*x^2+d)^(5/2)-3/5*a*f^2*g/c^2/d*(-c^2*d*x^2+d)^(5/2)-1/7*a*g^3*x^2*(-c^2*d*x^2+d)^
(5/2)/c^2/d+1/8*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(3/2)+13/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)*c^2/(c*x-1)*ar
ccosh(c*x)*x^6-1/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x^2+3/5*b*(-d*(c^2*x^2-1))
^(1/2)*g*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*f^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c*x+1)*c^4/(c*x-1)*arccosh
(c*x)*x^5+7/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^3+1/12*b*(-d*(c^2*x^2-1))^(1/2
)*f*g^2*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^6-7/32*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)^(1/2)*c/(c*x-1)^(1
/2)*x^4+3/32*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+3/25*b*(-d*(c^2*x^2-1))^(1/2)*
g*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^5*f^2-2/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3
*f^2+3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x*f^2-9/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*
x+1)/(c*x-1)*arccosh(c*x)*x^2*f^2-17/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3-3/32
*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*f*arccosh(c*x)^2*d*g^2+3/8*a*f^3*d*x*(-c^2*d*x^2+d)^
(1/2)+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^4-5/16*b*(-d*(c^2*x^2-1))^(1/2)*f^
3*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^2+1/49*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^7-
8/175*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^5-3/16*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(
1/2)/(c*x+1)^(1/2)/c*f^3*arccosh(c*x)^2*d+2/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)/c^4/(c*x-1)*arccosh(c*x)
-5/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x-9/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)
/(c*x-1)*arccosh(c*x)*x^4+1/105*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^3+2/35*b*(-d*(c
^2*x^2-1))^(1/2)*g^3*d/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)*x+7/768*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/(c*x+1)^(1/2)/
c^3/(c*x-1)^(1/2)+1/4*a*f^3*x*(-c^2*d*x^2+d)^(3/2)-1/7*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c*x+1)*c^4/(c*x-1)*arcc
osh(c*x)*x^8-1/2*a*f*g^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+3/16*a*f*g^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/16*a*f*g^2/c
^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+17/128*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c*x+1
)^(1/2)/c/(c*x-1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{3} - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a g^{3} + \frac {1}{16} \, a f g^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a f^{2} g}{5 \, c^{2} d} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b g^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b f g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b f^{2} g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b f^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a*f^3 - 1/35*(5*(-c^2*
d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a*g^3 + 1/16*a*f*g^2*(2*(-c^2*d*x^2 + d)^(3/2
)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*arcsin(c*x)/c^3) - 3
/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d) + integrate((-c^2*d*x^2 + d)^(3/2)*b*g^3*x^3*log(c*x + sqrt(c*x + 1)
*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(3/2)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*(-c^2*d*x^2
+ d)^(3/2)*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(3/2)*b*f^3*log(c*x + sqrt(c*x
+ 1)*sqrt(c*x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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