\(\int x (a+b \text {arccosh}(c x))^{3/2} \, dx\) [146]

Optimal result

Integrand size = 14, antiderivative size = 184 \[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=-\frac {3 b x \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{8 c}-\frac {(a+b \text {arccosh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3 b^{3/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^2} \] Output:

-3/8*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2)/c-1/4*(a+b*a 
rccosh(c*x))^(3/2)/c^2+1/2*x^2*(a+b*arccosh(c*x))^(3/2)-3/128*b^(3/2)*exp( 
2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/c^2+ 
3/128*b^(3/2)*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/ 
2))/c^2/exp(2*a/b)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\frac {3 b^{3/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-3 b^{3/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c x)} (4 a \cosh (2 \text {arccosh}(c x))+4 b \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 b \sinh (2 \text {arccosh}(c x)))}{128 c^2} \] Input:

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

Output:

(3*b^(3/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Co 
sh[(2*a)/b] - Sinh[(2*a)/b]) - 3*b^(3/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + 
b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[a + b*A 
rcCosh[c*x]]*(4*a*Cosh[2*ArcCosh[c*x]] + 4*b*ArcCosh[c*x]*Cosh[2*ArcCosh[c 
*x]] - 3*b*Sinh[2*ArcCosh[c*x]]))/(128*c^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.70 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6299, 6354, 6302, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}

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.

\(\displaystyle \int x (a+b \text {arccosh}(c x))^{3/2} \, dx\)

\(\Big \downarrow \) 6299

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \int \frac {x^2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx\)

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}}d\sqrt {a+b \text {arccosh}(c x)}-i \int e^{\frac {2 (a+b \text {arccosh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arccosh}(c x)}\right )}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}}d\sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}+\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int \frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}+\frac {(a+b \text {arccosh}(c x))^{3/2}}{3 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c^2}\right )\)

Input:

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

Output:

(x^2*(a + b*ArcCosh[c*x])^(3/2))/2 - (3*b*c*((x*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c^2) + (a + b*ArcCosh[c*x])^(3/2)/(3*b*c^3 
) - ((I/8)*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A 
rcCosh[c*x]])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + 
b*ArcCosh[c*x]])/Sqrt[b]])/E^((2*a)/b)))/c^3))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6299

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

rule 6302

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

rule 6308

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

rule 6354

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

Maple [F]

\[\int x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}d x\]

Input:

int(x*(a+b*arccosh(c*x))^(3/2),x)
 

Output:

int(x*(a+b*arccosh(c*x))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate(x*(a+b*acosh(c*x))**(3/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(x*(a + b*acosh(c*x))**(3/2), x)
 

Maxima [F]

\[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x \,d x } \] Input:

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

Output:

integrate((b*arccosh(c*x) + a)^(3/2)*x, x)
 

Giac [F]

\[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x \,d x } \] Input:

integrate(x*(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)^(3/2)*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2} \,d x \] Input:

int(x*(a + b*acosh(c*x))^(3/2),x)
 

Output:

int(x*(a + b*acosh(c*x))^(3/2), x)
 

Reduce [F]

\[ \int x (a+b \text {arccosh}(c x))^{3/2} \, dx=\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, \mathit {acosh} \left (c x \right ) x d x \right ) b +\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x d x \right ) a \] Input:

int(x*(a+b*acosh(c*x))^(3/2),x)
 

Output:

int(sqrt(acosh(c*x)*b + a)*acosh(c*x)*x,x)*b + int(sqrt(acosh(c*x)*b + a)* 
x,x)*a