\(\int \frac {a+b \text {arccosh}(c x)}{(d+e x^2)^{5/2}} \, dx\) [146]

Optimal result

Integrand size = 20, antiderivative size = 180 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/3*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(1/2)+1/3*x*(a+ 
b*arccosh(c*x))/d/(e*x^2+d)^(3/2)+2/3*x*(a+b*arccosh(c*x))/d^2/(e*x^2+d)^( 
1/2)-2/3*b*(c^2*x^2-1)^(1/2)*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d) 
^(1/2))/d^2/e^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.79 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.52 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}+\frac {a x \left (3 d+2 e x^2\right )}{d^2}+\frac {b x \left (3 d+2 e x^2\right ) \text {arccosh}(c x)}{d^2}+\frac {4 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (d+e x^2\right ) \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \operatorname {EllipticPi}\left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c d^2 \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{3 \left (d+e x^2\right )^{3/2}} \] Input:

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

Output:

(-((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + e))) + (a*x* 
(3*d + 2*e*x^2))/d^2 + (b*x*(3*d + 2*e*x^2)*ArcCosh[c*x])/d^2 + (4*b*(-1 + 
 c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[ 
e])*(-1 + c*x))]*(d + e*x^2)*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*Sqrt[d] + S 
qrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/( 
1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt 
[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + 
I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d]) + I*Sqrt[e])*Sqrt[((c^ 
2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[ 
d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticPi[(2*c*Sqrt[d])/(c* 
Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sq 
rt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] 
+ I*Sqrt[e])^2]))/(c*d^2*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + (I*Sqrt[e] 
*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]))/(3*(d + e*x^2)^(3 
/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6323, 27, 1076, 435, 87, 66, 221}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6323

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1076

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

\(\Big \downarrow \) 435

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {2 e x^2+3 d}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx^2}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (2 \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+\frac {2 d \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (4 \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}+\frac {2 d \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {4 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}+\frac {2 d \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(x*(a + b*ArcCosh[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x 
]))/(3*d^2*Sqrt[d + e*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*((2*d*Sqrt[-1 + c^2* 
x^2])/((c^2*d + e)*Sqrt[d + e*x^2]) + (4*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^ 
2])/(c*Sqrt[d + e*x^2])])/(c*Sqrt[e])))/(6*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
])
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 87

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

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

rule 1076

Int[((g_.)*(x_))^(m_.)*((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*( 
x_)^(n2_.))^(r_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^( 
q_.), x_Symbol] :> Simp[(e1 + f1*x^(n/2))^FracPart[r]*((e2 + f2*x^(n/2))^Fr 
acPart[r]/(e1*e2 + f1*f2*x^n)^FracPart[r])   Int[(g*x)^m*(a + b*x^n)^p*(c + 
 d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e1, f1, e2, f 
2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]
 

rule 6323

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] 
|| ILtQ[p + 1/2, 0])
 

Maple [F]

\[\int \frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (148) = 296\).

Time = 0.15 (sec) , antiderivative size = 724, normalized size of antiderivative = 4.02 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right ) + 2 \, {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/6*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + 
b*d*e^2)*x^2)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e 
 - c^2*e^2)*x^2 - 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x 
^2 + d)*sqrt(e) + e^2) + 2*(2*(b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^2*d^2*e + 
 b*d*e^2)*x)*sqrt(e*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 2*(2*(a*c^2*d* 
e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2*e + a*d*e^2)*x - (b*c*d*e^2*x^2 + b*c*d^2* 
e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 
 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3*e^3)*x^2), 1/3*((b*c^2*d^3 + (b*c^2 
*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(-e)*ar 
ctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt( 
-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + (2*(b*c^2*d*e^2 + b*e 
^3)*x^3 + 3*(b*c^2*d^2*e + b*d*e^2)*x)*sqrt(e*x^2 + d)*log(c*x + sqrt(c^2* 
x^2 - 1)) + (2*(a*c^2*d*e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2*e + a*d*e^2)*x - ( 
b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^2*d^5*e 
+ d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3*e^3)*x^2)]
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*acosh(c*x))/(e*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(d + e*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( 
log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e*x^2 + d)^(5/2), x)
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\mathit {acosh} \left (c x \right ) b +a}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x \] Input:

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

Output:

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