\(\int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx\) [11]

Optimal result

Integrand size = 21, antiderivative size = 135 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{75 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d} \] Output:

-8/75*b*e^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-4/75*b*e^4*(d*x+c-1)^(1/2)*( 
d*x+c)^2*(d*x+c+1)^(1/2)/d-1/25*b*e^4*(d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^ 
(1/2)/d+1/5*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^4 \left (-\frac {1}{75} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (8+4 (c+d x)^2+3 (c+d x)^4\right )+\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))\right )}{d} \] Input:

Integrate[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x]),x]
 

Output:

(e^4*(-1/75*(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(8 + 4*(c + d*x)^2 + 3 
*(c + d*x)^4)) + ((c + d*x)^5*(a + b*ArcCosh[c + d*x]))/5))/d
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6411, 27, 6298, 111, 27, 111, 27, 83}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e^4 (c+d x)^4 (a+b \text {arccosh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \int (c+d x)^4 (a+b \text {arccosh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \int \frac {(c+d x)^5}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \left (\frac {1}{5} \int \frac {4 (c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \left (\frac {4}{5} \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \left (\frac {4}{5} \left (\frac {1}{3} \int \frac {2 (c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4\right )\right )}{d}\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))-\frac {1}{5} b \left (\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4+\frac {4}{5} \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+\frac {2}{3} \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )\right )}{d}\)

Input:

Int[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x]),x]
 

Output:

(e^4*(-1/5*(b*((Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/5 + (4*( 
(2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/3 + (Sqrt[-1 + c + d*x]*(c + d*x) 
^2*Sqrt[1 + c + d*x])/3))/5)) + ((c + d*x)^5*(a + b*ArcCosh[c + d*x]))/5)) 
/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 83

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> 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 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(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 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58

method	result	size
derivativedivides	\(\frac {\frac {e^{4} a \left (d x +c \right )^{5}}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)
default	\(\frac {\frac {e^{4} a \left (d x +c \right )^{5}}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)
parts	\(\frac {e^{4} a \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(80\)
orering	\(\frac {\left (27 d^{6} x^{6}+162 c \,d^{5} x^{5}+405 c^{2} d^{4} x^{4}+540 c^{3} d^{3} x^{3}+405 c^{4} d^{2} x^{2}+4 d^{4} x^{4}+162 c^{5} d x +16 c \,d^{3} x^{3}+27 c^{6}+24 c^{2} d^{2} x^{2}+16 c^{3} d x +4 c^{4}+16 d^{2} x^{2}+32 c d x +16 c^{2}-32\right ) \left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}{75 \left (d x +c \right )^{5} d}-\frac {\left (3 d^{4} x^{4}+12 c \,d^{3} x^{3}+18 c^{2} d^{2} x^{2}+12 c^{3} d x +3 c^{4}+4 d^{2} x^{2}+8 c d x +4 c^{2}+8\right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (4 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d e +\frac {\left (d e x +c e \right )^{4} b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{75 d^{2} \left (d x +c \right )^{4}}\)	\(294\)

Input:

int((d*e*x+c*e)^4*(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/5*e^4*a*(d*x+c)^5+e^4*b*(1/5*(d*x+c)^5*arccosh(d*x+c)-1/75*(d*x+c-1 
)^(1/2)*(d*x+c+1)^(1/2)*(3*(d*x+c)^4+4*(d*x+c)^2+8)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (115) = 230\).

Time = 0.10 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.07 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {15 \, a d^{5} e^{4} x^{5} + 75 \, a c d^{4} e^{4} x^{4} + 150 \, a c^{2} d^{3} e^{4} x^{3} + 150 \, a c^{3} d^{2} e^{4} x^{2} + 75 \, a c^{4} d e^{4} x + 15 \, {\left (b d^{5} e^{4} x^{5} + 5 \, b c d^{4} e^{4} x^{4} + 10 \, b c^{2} d^{3} e^{4} x^{3} + 10 \, b c^{3} d^{2} e^{4} x^{2} + 5 \, b c^{4} d e^{4} x + b c^{5} e^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b d^{4} e^{4} x^{4} + 12 \, b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d e^{4} x + {\left (3 \, b c^{4} + 4 \, b c^{2} + 8 \, b\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{75 \, d} \] Input:

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c)),x, algorithm="fricas")
 

Output:

1/75*(15*a*d^5*e^4*x^5 + 75*a*c*d^4*e^4*x^4 + 150*a*c^2*d^3*e^4*x^3 + 150* 
a*c^3*d^2*e^4*x^2 + 75*a*c^4*d*e^4*x + 15*(b*d^5*e^4*x^5 + 5*b*c*d^4*e^4*x 
^4 + 10*b*c^2*d^3*e^4*x^3 + 10*b*c^3*d^2*e^4*x^2 + 5*b*c^4*d*e^4*x + b*c^5 
*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (3*b*d^4*e^4*x^4 
+ 12*b*c*d^3*e^4*x^3 + 2*(9*b*c^2 + 2*b)*d^2*e^4*x^2 + 4*(3*b*c^3 + 2*b*c) 
*d*e^4*x + (3*b*c^4 + 4*b*c^2 + 8*b)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 
))/d
 

Sympy [F]

\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=e^{4} \left (\int a c^{4}\, dx + \int a d^{4} x^{4}\, dx + \int b c^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 a c d^{3} x^{3}\, dx + \int 6 a c^{2} d^{2} x^{2}\, dx + \int 4 a c^{3} d x\, dx + \int b d^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b c d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 6 b c^{2} d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b c^{3} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate((d*e*x+c*e)**4*(a+b*acosh(d*x+c)),x)
 

Output:

e**4*(Integral(a*c**4, x) + Integral(a*d**4*x**4, x) + Integral(b*c**4*aco 
sh(c + d*x), x) + Integral(4*a*c*d**3*x**3, x) + Integral(6*a*c**2*d**2*x* 
*2, x) + Integral(4*a*c**3*d*x, x) + Integral(b*d**4*x**4*acosh(c + d*x), 
x) + Integral(4*b*c*d**3*x**3*acosh(c + d*x), x) + Integral(6*b*c**2*d**2* 
x**2*acosh(c + d*x), x) + Integral(4*b*c**3*d*x*acosh(c + d*x), x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (115) = 230\).

Time = 0.06 (sec) , antiderivative size = 1241, normalized size of antiderivative = 9.19 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\text {Too large to display} \] Input:

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c)),x, algorithm="maxima")
 

Output:

1/5*a*d^4*e^4*x^5 + a*c*d^3*e^4*x^4 + 2*a*c^2*d^2*e^4*x^3 + 2*a*c^3*d*e^4* 
x^2 + (2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2* 
x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 
- (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d 
^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*b*c^3*d*e^4 + 1/3*(6*x^3* 
arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3 
*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt 
(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d 
+ 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + 
 c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*b* 
c^2*d^2*e^4 + 1/24*(24*x^4*arccosh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + 
c^2 - 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^2/d^3 + 105*c^ 
4*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 35*sq 
rt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*log(2*d^2*x + 
 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 - 105*sqrt(d^2*x^2 + 2 
*c*d*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)* 
x/d^4 + 9*(c^2 - 1)^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 
 - 1)*d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c/d^5)*d)*b* 
c*d^3*e^4 + 1/600*(120*x^5*arccosh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + 
 c^2 - 1)*x^4/d^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^3/d^3 + 12...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (115) = 230\).

Time = 0.76 (sec) , antiderivative size = 846, normalized size of antiderivative = 6.27 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx =\text {Too large to display} \] Input:

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c)),x, algorithm="giac")
 

Output:

1/5*a*d^4*e^4*x^5 + a*c*d^3*e^4*x^4 + 2*a*c^2*d^2*e^4*x^3 + 2*a*c^3*d*e^4* 
x^2 - (d*(c*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))* 
abs(d)))/(d*abs(d)) + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x + 
 c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c^4*e^4 + (2*x^2*log(d*x + c + 
sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x 
/d^2 - 3*c/d^3) - (2*c^2 + 1)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2* 
c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d)))*d)*b*c^3*d*e^4 + 1/3*(6*x^3*log(d 
*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^ 
2 - 1)*(x*(2*x/d^2 - 5*c/d^3) + (11*c^2*d + 4*d)/d^5) + 3*(2*c^3 + 3*c)*lo 
g(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^3* 
abs(d)))*d)*b*c^2*d^2*e^4 + 1/24*(24*x^4*log(d*x + c + sqrt(d^2*x^2 + 2*c* 
d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*((2*x*(3*x/d^2 - 7*c/ 
d^3) + (26*c^2*d^3 + 9*d^3)/d^7)*x - 5*(10*c^3*d^2 + 11*c*d^2)/d^7) - 3*(8 
*c^4 + 24*c^2 + 3)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 
 - 1))*abs(d)))/(d^4*abs(d)))*d)*b*c*d^3*e^4 + 1/600*(120*x^5*log(d*x + c 
+ sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)* 
((2*(3*x*(4*x/d^2 - 9*c/d^3) + (47*c^2*d^5 + 16*d^5)/d^9)*x - 7*(22*c^3*d^ 
4 + 23*c*d^4)/d^9)*x + (274*c^4*d^3 + 607*c^2*d^3 + 64*d^3)/d^9) + 15*(8*c 
^5 + 40*c^3 + 15*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^ 
2 - 1))*abs(d)))/(d^5*abs(d)))*d)*b*d^4*e^4 + a*c^4*e^4*x
 

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \] Input:

int((c*e + d*e*x)^4*(a + b*acosh(c + d*x)),x)
 

Output:

int((c*e + d*e*x)^4*(a + b*acosh(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.21 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^{4} \left (75 \mathit {acosh} \left (d x +c \right ) b \,c^{5}+75 \mathit {acosh} \left (d x +c \right ) b \,c^{4} d x +150 \mathit {acosh} \left (d x +c \right ) b \,c^{3} d^{2} x^{2}+150 \mathit {acosh} \left (d x +c \right ) b \,c^{2} d^{3} x^{3}+75 \mathit {acosh} \left (d x +c \right ) b c \,d^{4} x^{4}+15 \mathit {acosh} \left (d x +c \right ) b \,d^{5} x^{5}+72 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,c^{4}-12 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,c^{3} d x -18 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,c^{2} d^{2} x^{2}-4 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,c^{2}-12 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b c \,d^{3} x^{3}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b c d x -3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,d^{4} x^{4}-4 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b \,d^{2} x^{2}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, b -75 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, b \,c^{4}-60 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) b \,c^{5}+75 a \,c^{4} d x +150 a \,c^{3} d^{2} x^{2}+150 a \,c^{2} d^{3} x^{3}+75 a c \,d^{4} x^{4}+15 a \,d^{5} x^{5}\right )}{75 d} \] Input:

int((d*e*x+c*e)^4*(a+b*acosh(d*x+c)),x)
 

Output:

(e**4*(75*acosh(c + d*x)*b*c**5 + 75*acosh(c + d*x)*b*c**4*d*x + 150*acosh 
(c + d*x)*b*c**3*d**2*x**2 + 150*acosh(c + d*x)*b*c**2*d**3*x**3 + 75*acos 
h(c + d*x)*b*c*d**4*x**4 + 15*acosh(c + d*x)*b*d**5*x**5 + 72*sqrt(c**2 + 
2*c*d*x + d**2*x**2 - 1)*b*c**4 - 12*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)* 
b*c**3*d*x - 18*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*c**2*d**2*x**2 - 4* 
sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*c**2 - 12*sqrt(c**2 + 2*c*d*x + d** 
2*x**2 - 1)*b*c*d**3*x**3 - 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*c*d*x 
 - 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*b*d**4*x**4 - 4*sqrt(c**2 + 2*c* 
d*x + d**2*x**2 - 1)*b*d**2*x**2 - 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)* 
b - 75*sqrt(c + d*x + 1)*sqrt(c + d*x - 1)*b*c**4 - 60*log(sqrt(c**2 + 2*c 
*d*x + d**2*x**2 - 1) + c + d*x)*b*c**5 + 75*a*c**4*d*x + 150*a*c**3*d**2* 
x**2 + 150*a*c**2*d**3*x**3 + 75*a*c*d**4*x**4 + 15*a*d**5*x**5))/(75*d)