\(\int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) [165]

Optimal result

Integrand size = 25, antiderivative size = 469 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d} \]

[Out]

-3/32*e^3*(a+b*arccosh(d*x+c))^(5/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^(5/2)/d-15/1024*b^(5/2)*e^3*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)/d-15/1024*b^(5/2)*e^3*erfi(2^(1/2)*(a+
b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-15/16384*b^(5/2)*e^3*exp(4*a/b)*erf(2*(a+b*arcc
osh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d-15/16384*b^(5/2)*e^3*erfi(2*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)
/d/exp(4*a/b)-15/64*b*e^3*(d*x+c)*(a+b*arccosh(d*x+c))^(3/2)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-5/32*b*e^3*(d*x
+c)^3*(a+b*arccosh(d*x+c))^(3/2)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-225/2048*b^2*e^3*(a+b*arccosh(d*x+c))^(1/2)
/d+45/256*b^2*e^3*(d*x+c)^2*(a+b*arccosh(d*x+c))^(1/2)/d+15/256*b^2*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5996, 12, 5884, 5939, 5893, 5953, 3393, 3388, 2211, 2236, 2235} \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {5 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {15 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d} \]

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(5/2),x]

[Out]

(-225*b^2*e^3*Sqrt[a + b*ArcCosh[c + d*x]])/(2048*d) + (45*b^2*e^3*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(
256*d) + (15*b^2*e^3*(c + d*x)^4*Sqrt[a + b*ArcCosh[c + d*x]])/(256*d) - (15*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x
)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(64*d) - (5*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 +
c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(32*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^(5/2))/(32*d) + (e^3*(c + d*
x)^4*(a + b*ArcCosh[c + d*x])^(5/2))/(4*d) - (15*b^(5/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c
+ d*x]])/Sqrt[b]])/(16384*d) - (15*b^(5/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]
])/Sqrt[b]])/(512*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16384*d*E^((4
*a)/b)) - (15*b^(5/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(512*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 3388

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

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[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 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 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(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] + (Dist[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] - Dist[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] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

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 e^3 x^3 (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arccosh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (5 b e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int x^3 \sqrt {a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{64 d} \\ & = \frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{64 d}+\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int x \sqrt {a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{128 d}-\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \text {arccosh}(x)}} \, dx,x,c+d x\right )}{512 d} \\ & = \frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d}-\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \text {arccosh}(x)}} \, dx,x,c+d x\right )}{512 d} \\ & = \frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d} \\ & = -\frac {45 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{512 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8192 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8192 d}-\frac {\left (15 b^2 e^3\right ) \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 )}{2048 d}-\frac {\left (15 b^2 e^3\right ) \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 )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{1024 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \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 )}{1024 d}-\frac {\left (15 b^2 e^3\right ) \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 )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \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 )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \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 )}{2048 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {\left (45 b^2 e^3\right ) \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 )}{1024 d}-\frac {\left (45 b^2 e^3\right ) \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 )}{1024 d} \\ & = -\frac {225 b^2 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{2048 d}+\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}+\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{256 d}-\frac {15 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{64 d}-\frac {5 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}-\frac {15 b^{5/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {15 b^{5/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16384 d}-\frac {15 b^{5/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{512 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(968\) vs. \(2(469)=938\).

Time = 8.60 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.06 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e^3 \left (\frac {a^2 e^{-\frac {4 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+4 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{128 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}}+\frac {a \sqrt {b} \left ((8 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )-\sinh \left (\frac {4 a}{b}\right )\right )+(8 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )+\sinh \left (\frac {4 a}{b}\right )\right )+8 \left ((4 a+3 b) \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+(4 a-3 b) \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (4 \text {arccosh}(c+d x) \cosh (2 \text {arccosh}(c+d x))-3 \sinh (2 \text {arccosh}(c+d x)))\right )+8 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (8 \text {arccosh}(c+d x) \cosh (4 \text {arccosh}(c+d x))-3 \sinh (4 \text {arccosh}(c+d x)))\right )}{1024 d}+\frac {-\sqrt {b} \left (64 a^2+48 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )-\sinh \left (\frac {4 a}{b}\right )\right )-\sqrt {b} \left (64 a^2-48 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {4 a}{b}\right )+\sinh \left (\frac {4 a}{b}\right )\right )-16 \left (\sqrt {b} \left (16 a^2+24 a b+15 b^2\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \left (16 a^2-24 a b+15 b^2\right ) \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-8 b \sqrt {a+b \text {arccosh}(c+d x)} \left (b \left (15+16 \text {arccosh}(c+d x)^2\right ) \cosh (2 \text {arccosh}(c+d x))+4 (a-5 b \text {arccosh}(c+d x)) \sinh (2 \text {arccosh}(c+d x))\right )\right )+8 b \sqrt {a+b \text {arccosh}(c+d x)} \left (b \left (15+64 \text {arccosh}(c+d x)^2\right ) \cosh (4 \text {arccosh}(c+d x))+8 (a-5 b \text {arccosh}(c+d x)) \sinh (4 \text {arccosh}(c+d x))\right )}{16384 d}\right ) \]

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(5/2),x]

[Out]

e^3*((a^2*Sqrt[a + b*ArcCosh[c + d*x]]*(Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-4*(a + b*ArcCosh[c + d*x]))/
b] + 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcCosh[c + d*x]))/b] + E^((6*a)
/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*(4*Sqrt[2]*Gamma[3/2, (2*(a + b*ArcCosh[c + d*x]))/b] + E^((2*a)/b)*Ga
mma[3/2, (4*(a + b*ArcCosh[c + d*x]))/b])))/(128*d*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])^2/b^2)]) + (a*S
qrt[b]*((8*a + 3*b)*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] - Sinh[(4*a)/b]) +
(8*a - 3*b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] + Sinh[(4*a)/b]) + 8*((4*a +
3*b)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + (4*a -
3*b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[
b]*Sqrt[a + b*ArcCosh[c + d*x]]*(4*ArcCosh[c + d*x]*Cosh[2*ArcCosh[c + d*x]] - 3*Sinh[2*ArcCosh[c + d*x]])) +
8*Sqrt[b]*Sqrt[a + b*ArcCosh[c + d*x]]*(8*ArcCosh[c + d*x]*Cosh[4*ArcCosh[c + d*x]] - 3*Sinh[4*ArcCosh[c + d*x
]])))/(1024*d) + (-(Sqrt[b]*(64*a^2 + 48*a*b + 15*b^2)*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]
*(Cosh[(4*a)/b] - Sinh[(4*a)/b])) - Sqrt[b]*(64*a^2 - 48*a*b + 15*b^2)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c +
d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] + Sinh[(4*a)/b]) - 16*(Sqrt[b]*(16*a^2 + 24*a*b + 15*b^2)*Sqrt[2*Pi]*Erfi[(Sqrt
[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + Sqrt[b]*(16*a^2 - 24*a*b + 15*b^2
)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - 8*b*Sqrt[a
+ b*ArcCosh[c + d*x]]*(b*(15 + 16*ArcCosh[c + d*x]^2)*Cosh[2*ArcCosh[c + d*x]] + 4*(a - 5*b*ArcCosh[c + d*x])*
Sinh[2*ArcCosh[c + d*x]])) + 8*b*Sqrt[a + b*ArcCosh[c + d*x]]*(b*(15 + 64*ArcCosh[c + d*x]^2)*Cosh[4*ArcCosh[c
+ d*x]] + 8*(a - 5*b*ArcCosh[c + d*x])*Sinh[4*ArcCosh[c + d*x]]))/(16384*d))

Maple [F]

\[\int \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

[In]

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x)

[Out]

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^3*(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(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^(5/2), x)

Giac [F]

\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(5/2),x)

[Out]

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(5/2), x)