3.2.67 \(\int (c e+d e x) (a+b \cosh ^{-1}(c+d x))^{5/2} \, dx\) [167]

Optimal. Leaf size=269 \[ -\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {15 b^{5/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 b^{5/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d} \]

[Out]

-1/4*e*(a+b*arccosh(d*x+c))^(5/2)/d+1/2*e*(d*x+c)^2*(a+b*arccosh(d*x+c))^(5/2)/d-15/512*b^(5/2)*e*exp(2*a/b)*e
rf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d-15/512*b^(5/2)*e*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)-5/8*b*e*(d*x+c)*(a+b*arccosh(d*x+c))^(3/2)*(d*x+c-1)^(1/2)
*(d*x+c+1)^(1/2)/d-15/64*b^2*e*(a+b*arccosh(d*x+c))^(1/2)/d+15/32*b^2*e*(d*x+c)^2*(a+b*arccosh(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.71, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5996, 12, 5884, 5939, 5893, 5953, 3393, 3388, 2211, 2236, 2235} \begin {gather*} -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {5 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*b^2*e*Sqrt[a + b*ArcCosh[c + d*x]])/(64*d) + (15*b^2*e*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(32*d) -
(5*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(8*d) - (e*(a + b*ArcCo
sh[c + d*x])^(5/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(5/2))/(2*d) - (15*b^(5/2)*e*E^((2*a)/b)*S
qrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*d) - (15*b^(5/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2
]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*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*} \int (c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int x \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{16 d}\\ &=\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}\\ &=\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{128 d}\\ &=-\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}-\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{128 d}-\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{128 d}\\ &=-\frac {15 b^2 e \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {15 b^{5/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 b^{5/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1846\) vs. \(2(269)=538\).
time = 8.14, size = 1846, normalized size = 6.86 \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

e*((a^2*c*Sqrt[-1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c + d*x]])/Sq
rt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c + d*x])/b)]))/(
2*d*E^(a/b)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (a*b*c*Sqrt[-1 + c + d*x]*(-12*Sqrt[(-1 +
c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*A
rcCosh[c + d*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/S
qrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(4
*d*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) - (c*Sqrt[-1 + c + d*x]*(-4*b*Sqrt[a + b*ArcCosh[c +
d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*ArcCosh[c + d*x]) + b*(c + d*x)*(15 + 4*Arc
Cosh[c + d*x]^2)) + Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cos
h[a/b] - Sinh[a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Co
sh[a/b] + Sinh[a/b])))/(16*d*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (a^2*Sqrt[-1 + c + d*x]*(
-32*c*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*Sqrt[a + b*ArcCosh[c + d*x]]*Cosh[2*ArcCosh[c + d*x]] + 8*Sqr
t[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(
Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] - 8*Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b
]]*Sinh[a/b] + 8*Sqrt[b]*c*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + Sqrt[b
]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt
[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/(32*d*Sqrt[(-1 + c + d*x)/(1 + c
+ d*x)]*Sqrt[1 + c + d*x]) + (a*Sqrt[-1 + c + d*x]*(-16*c*(-12*b*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d
*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*b*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + Sqrt[b]*(2*a
+ 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[b]*Sqrt[
Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(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[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*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]])))/(64*d*Sqrt[(-1 + c + d
*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) - (Sqrt[-1 + c + d*x]*(-32*c*(-4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(2*Sqrt[
(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*ArcCosh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d*x]^
2)) + Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[
a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh
[a/b])) + 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]])))/(512*d*Sqrt[(-1
+ c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e \left (\int a^{2} c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int a^{2} d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int b^{2} c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int b^{2} d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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