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

Optimal. Leaf size=568 \[ \frac {1}{6} d^2 f x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 g (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

5/16*d^2*f*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+5/24*d^2*f*x*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*
x^2+d)^(1/2)+1/6*d^2*f*x*(-c*x+1)^2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-1/7*d^2*g*(-c*x+1)^3*(c*
x+1)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/7*b*d^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^
(1/2)-25/96*b*c*d^2*f*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*b*c*d^2*g*x^3*(-c^2*d*x^2+d)^(1
/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b*c^3*d^2*f*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/35*b*c
^3*d^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/49*b*c^5*d^2*g*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1
)^(1/2)/(c*x+1)^(1/2)+1/36*b*d^2*f*(-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*d^2*
f*(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)

________________________________________________________________________________________

Rubi [A]  time = 0.90, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 261, 5718, 194} \[ \frac {1}{6} d^2 f x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 g (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^2*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*x^2*Sqrt[d - c^2*d*x^2])/(9
6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*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*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])
/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
(b*d^2*f*(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*x*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCosh[c*x]))/16 + (5*d^2*f*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/24 +
(d^2*f*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*(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*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*
b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*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 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 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 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 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) \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) \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 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+g x (-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 \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 (d^2 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 {1}{6} d^2 f 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 {d^2 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 \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 \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 (b d^2 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 {b d^2 f \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 x (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 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 {d^2 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 \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 \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 (b d^2 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 {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f \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 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (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 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 {d^2 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 \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 \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 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 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 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 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 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f \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 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (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 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 {d^2 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 \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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.21, size = 644, normalized size = 1.13 \[ \frac {d^2 \left (-882000 a c \sqrt {d} f \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 )+8400 a \sqrt {\frac {c x-1}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} \left (48 g \left (c^2 x^2-1\right )^3+7 c^2 f x \left (8 c^4 x^4-26 c^2 x^2+33\right )\right )-352800 b c f \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 )+44100 b c f \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+1225 b c f \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+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 )+12 \cosh ^{-1}(c x) \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 )\right )+78400 b g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-1568 b 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 )+4 b g \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 )\right )}{2822400 c^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(8400*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 -
26*c^2*x^2 + 8*c^4*x^4)) - 882000*a*c*Sqrt[d]*f*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))] + 78400*b*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*f*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*Ar
cCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 44100*b*c*f*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]]) - 1568*b*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]*S
inh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]) + 1225*b*c*f*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]
^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*Ar
cCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])) + 4*b*g*Sqrt[d - c^2*d*x^2]*(55125*c*x - 55125*S
qrt[(-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*ArcCosh[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^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

________________________________________________________________________________________

fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g x^{5} + a c^{4} d^{2} f x^{4} - 2 \, a c^{2} d^{2} g x^{3} - 2 \, a c^{2} d^{2} f x^{2} + a d^{2} g x + a d^{2} f + {\left (b c^{4} d^{2} g x^{5} + b c^{4} d^{2} f x^{4} - 2 \, b c^{2} d^{2} g x^{3} - 2 \, b c^{2} d^{2} f x^{2} + b d^{2} g x + b d^{2} f\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)*(-c^2*d*x^2+d)^(5/2)*(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

________________________________________________________________________________________

maple [A]  time = 0.90, size = 877, normalized size = 1.54 \[ -\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a f d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a f \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{16}+\frac {5 a f \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 \sqrt {c^{2} d}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c^{6} \mathrm {arccosh}\left (c x \right ) x^{8}}{7 \left (c x +1\right ) \left (c x -1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c^{4} \mathrm {arccosh}\left (c x \right ) x^{6}}{7 \left (c x +1\right ) \left (c x -1\right )}+\frac {6 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{4}}{7 \left (c x +1\right ) \left (c x -1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} \mathrm {arccosh}\left (c x \right ) x^{2}}{7 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c^{6} \mathrm {arccosh}\left (c x \right ) x^{7}}{6 \left (c x +1\right ) \left (c x -1\right )}-\frac {17 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c^{4} \mathrm {arccosh}\left (c x \right ) x^{5}}{24 \left (c x +1\right ) \left (c x -1\right )}+\frac {59 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{48 \left (c x +1\right ) \left (c x -1\right )}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} \mathrm {arccosh}\left (c x \right ) x}{16 \left (c x +1\right ) \left (c x -1\right )}+\frac {299 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2}}{2304 \sqrt {c x +1}\, \sqrt {c x -1}\, c}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c^{5} x^{7}}{49 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c^{3} x^{5}}{35 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} c \,x^{3}}{7 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} x}{7 \sqrt {c x +1}\, c \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c^{5} x^{6}}{36 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {13 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c^{3} x^{4}}{96 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,d^{2} c \,x^{2}}{32 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,d^{2} \mathrm {arccosh}\left (c x \right )}{7 \left (c x +1\right ) c^{2} \left (c x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a*g/c^2/d*(-c^2*d*x^2+d)^(7/2)+1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*f*
d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-5/32*b*(-d*
(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*f*arccosh(c*x)^2*d^2+1/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+
1)*c^6/(c*x-1)*arccosh(c*x)*x^8-4/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^6+6/7*b*
(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^4-4/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)/(
c*x-1)*arccosh(c*x)*x^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c*x+1)/(c*x-1)*c^6*arccosh(c*x)*x^7-17/24*b*(-d*(c
^2*x^2-1))^(1/2)*f*d^2/(c*x+1)/(c*x-1)*c^4*arccosh(c*x)*x^5+59/48*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c*x+1)/(c*x-
1)*c^2*arccosh(c*x)*x^3-11/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x+299/2304*b*(-d*(c^
2*x^2-1))^(1/2)*f*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)/c-1/49*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)^(1/2)*c^5/(c*x
-1)^(1/2)*x^7+3/35*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^5-1/7*b*(-d*(c^2*x^2-1))^(
1/2)*g*d^2/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3+1/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*
x-1/36*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5*x^6+13/96*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2
/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*x^4-11/32*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*x^2+1/
7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, {\left (8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x + 10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c}\right )} a f - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a g}{7 \, c^{2} d} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b f \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a*g/(c^2*d) + integrate((-c^2*d*x^2 + d)^(5/2)*b*g*x*log(c*x +
sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(5/2)*b*f*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________