\(\int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx\) [189]

Optimal result

Integrand size = 23, antiderivative size = 216 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \]

[Out]

-2/3*e*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/d+2/3*e*erfi(2^(1/2
)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/d/exp(2*a/b)-2/3*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x
+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^(3/2)+4/3*e/b^2/d/(a+b*arccosh(d*x+c))^(1/2)-8/3*e*(d*x+c)^2/b^2/d/(a+b*a
rccosh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5996, 12, 5886, 5951, 5887, 5556, 3389, 2211, 2236, 2235, 5893} \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {2 \pi } e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 \sqrt {2 \pi } e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}} \]

[In]

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

[Out]

(-2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (4*e)/(3*b^2*d*
Sqrt[a + b*ArcCosh[c + d*x]]) - (8*e*(c + d*x)^2)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*e*E^((2*a)/b)*Sq
rt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sq
rt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((2*a)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

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

Rule 5556

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 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 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] && LtQ[n, -2]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(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*Arc
Cosh[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 5951

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

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e x}{(a+b \text {arccosh}(x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{(a+b \text {arccosh}(x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {(2 e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac {(4 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arccosh}(x)}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {(16 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {(16 e) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {(8 e) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {(4 e) \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^3 d}+\frac {(4 e) \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {(8 e) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{3 b^3 d}+\frac {(8 e) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{3 b^3 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(687\) vs. \(2(216)=432\).

Time = 3.69 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.18 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\frac {e \left (4 a \sqrt {b} c (c+d x)+4 b^{3/2} c (c+d x) \text {arccosh}(c+d x)-2 \sqrt {b} c e^{-\text {arccosh}(c+d x)} \left (1+e^{2 \text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-4 a \sqrt {b} \cosh (2 \text {arccosh}(c+d x))-4 b^{3/2} \text {arccosh}(c+d x) \cosh (2 \text {arccosh}(c+d x))+2 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{3/2} \cosh \left (\frac {a}{b}\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-2 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-2 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{3/2} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+2 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+2 \sqrt {b} c e^{a/b} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-2 b^{3/2} c e^{-\frac {a}{b}} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+2 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{3/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+2 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{3/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-2 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{3/2} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-2 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{3/2} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-b^{3/2} \sinh (2 \text {arccosh}(c+d x))\right )}{3 b^{5/2} d (a+b \text {arccosh}(c+d x))^{3/2}} \]

[In]

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

[Out]

(e*(4*a*Sqrt[b]*c*(c + d*x) + 4*b^(3/2)*c*(c + d*x)*ArcCosh[c + d*x] - (2*Sqrt[b]*c*(1 + E^(2*ArcCosh[c + d*x]
))*(a + b*ArcCosh[c + d*x]))/E^ArcCosh[c + d*x] - 4*a*Sqrt[b]*Cosh[2*ArcCosh[c + d*x]] - 4*b^(3/2)*ArcCosh[c +
d*x]*Cosh[2*ArcCosh[c + d*x]] + 2*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Cosh[a/b]*Erf[Sqrt[a + b*ArcCosh[
c + d*x]]/Sqrt[b]] - 2*Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Cosh[(2*a)/b]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh
[c + d*x]])/Sqrt[b]] - 2*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Cosh[a/b]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]
/Sqrt[b]] + 2*Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x
]])/Sqrt[b]] + 2*Sqrt[b]*c*E^(a/b)*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, a/b + ArcC
osh[c + d*x]] - (2*b^(3/2)*c*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)])/
E^(a/b) + 2*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 2*
c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(3/2)*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 2*Sqrt[2*Pi]*
(a + b*ArcCosh[c + d*x])^(3/2)*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - 2*Sqrt[2*Pi
]*(a + b*ArcCosh[c + d*x])^(3/2)*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - b^(3/2)*
Sinh[2*ArcCosh[c + d*x]]))/(3*b^(5/2)*d*(a + b*ArcCosh[c + d*x])^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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