\(\int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx\) [14]

Optimal result

Integrand size = 16, antiderivative size = 191 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}-\frac {b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \text {arccosh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e} \]

[Out]

-1/32*b*(8*c^4*d^4+24*c^2*d^2*e^2+3*e^4)*arccosh(c*x)/c^4/e+1/4*(e*x+d)^4*(a+b*arccosh(c*x))/e-7/48*b*d*(e*x+d
)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/16*b*(e*x+d)^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/96*b*(4*d*(19*c^2*d^2+16*e^
2)+e*(26*c^2*d^2+9*e^2)*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5963, 102, 158, 152, 54} \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b \text {arccosh}(c x) \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )}{32 c^4 e}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac {b \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{16 c}-\frac {7 b d \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{48 c} \]

[In]

Int[(d + e*x)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(-7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(48*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(16*c)
- (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x))/(96*c^3) - (b*(8*c^
4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcCosh[c*x]))/(4*e)

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 102

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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 158

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

Rule 5963

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

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {(b c) \int \frac {(d+e x)^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 e} \\ & = -\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b \int \frac {(d+e x)^2 \left (4 c^2 d^2+3 e^2+7 c^2 d e x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c e} \\ & = -\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b \int \frac {(d+e x) \left (c^2 d \left (12 c^2 d^2+23 e^2\right )+c^2 e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{48 c^3 e} \\ & = -\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {\left (b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3 e} \\ & = -\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}-\frac {b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \text {arccosh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.01 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {24 a c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+24 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {arccosh}(c x)-9 b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{96 c^4} \]

[In]

Integrate[(d + e*x)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d + 9*e*x)
+ c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 24*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x
^3)*ArcCosh[c*x] - 9*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(96*c^4)

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.64

method	result	size
parts	\(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}\right )}{c}\)	\(314\)
derivativedivides	\(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\)	\(331\)
default	\(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\)	\(331\)

[In]

int((e*x+d)^3*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/4*a*(e*x+d)^4/e+b/c*(1/4*c/e*arccosh(c*x)*d^4+arccosh(c*x)*c*x*d^3+3/2*c*arccosh(c*x)*d^2*e*x^2+c*e^2*arccos
h(c*x)*d*x^3+1/4*c*e^3*arccosh(c*x)*x^4-1/96/c^3/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(24*c^4*d^4*ln(c*x+(c^2*x^2-1)^
(1/2))+96*c^3*d^3*e*(c^2*x^2-1)^(1/2)+72*c^3*d^2*e^2*x*(c^2*x^2-1)^(1/2)+32*c^3*d*e^3*(c^2*x^2-1)^(1/2)*x^2+6*
e^4*c^3*x^3*(c^2*x^2-1)^(1/2)+72*c^2*d^2*e^2*ln(c*x+(c^2*x^2-1)^(1/2))+64*c*d*e^3*(c^2*x^2-1)^(1/2)+9*e^4*c*x*
(c^2*x^2-1)^(1/2)+9*e^4*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \, {\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x - 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} + 64 \, b c d e^{2} + 9 \, {\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{96 \, c^{4}} \]

[In]

integrate((e*x+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/96*(24*a*c^4*e^3*x^4 + 96*a*c^4*d*e^2*x^3 + 144*a*c^4*d^2*e*x^2 + 96*a*c^4*d^3*x + 3*(8*b*c^4*e^3*x^4 + 32*b
*c^4*d*e^2*x^3 + 48*b*c^4*d^2*e*x^2 + 32*b*c^4*d^3*x - 24*b*c^2*d^2*e - 3*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1))
- (6*b*c^3*e^3*x^3 + 32*b*c^3*d*e^2*x^2 + 96*b*c^3*d^3 + 64*b*c*d*e^2 + 9*(8*b*c^3*d^2*e + b*c*e^3)*x)*sqrt(c^
2*x^2 - 1))/c^4

Sympy [F]

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

[In]

integrate((e*x+d)**3*(a+b*acosh(c*x)),x)

[Out]

Integral((a + b*acosh(c*x))*(d + e*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.39 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]

[In]

integrate((e*x+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c
^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*b*d^2*e + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(
c^2*x^2 - 1)/c^4))*b*d*e^2 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c
^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*e^3 + a*d^3*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b
*d^3/c

Giac [F(-2)]

Exception generated. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((e*x+d)^3*(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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

int((a + b*acosh(c*x))*(d + e*x)^3,x)

[Out]

int((a + b*acosh(c*x))*(d + e*x)^3, x)