3.170 \(\int (c e+d e x)^2 (a+b \cosh ^{-1}(c+d x))^{7/2} \, dx\)

Optimal. Leaf size=509 \[ -\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}+\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}-\frac{35 b^3 e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}-\frac{175 b^3 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac{7 b e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}-\frac{7 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d} \]

[Out]

(-175*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(54*d) - (35*b^3*e^2*Sqrt[-1
+ c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(216*d) + (35*b^2*e^2*(c + d*x)*(a + b*
ArcCosh[c + d*x])^(3/2))/(18*d) + (35*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(3/2))/(108*d) - (7*b*e^2*S
qrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(9*d) - (7*b*e^2*Sqrt[-1 + c + d*x]*(c + d
*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(7/
2))/(3*d) - (105*b^(7/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d) - (35*b^(7/2)
*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d) + (105*b^(7/2)*e^2*S
qrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (35*b^(7/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]
*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d*E^((3*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 2.14238, antiderivative size = 509, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {5866, 12, 5664, 5759, 5718, 5654, 5658, 3308, 2180, 2205, 2204, 5670, 5448} \[ -\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}+\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}-\frac{35 b^3 e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}-\frac{175 b^3 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac{7 b e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}-\frac{7 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(-175*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(54*d) - (35*b^3*e^2*Sqrt[-1
+ c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(216*d) + (35*b^2*e^2*(c + d*x)*(a + b*
ArcCosh[c + d*x])^(3/2))/(18*d) + (35*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(3/2))/(108*d) - (7*b*e^2*S
qrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(9*d) - (7*b*e^2*Sqrt[-1 + c + d*x]*(c + d
*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(7/
2))/(3*d) - (105*b^(7/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d) - (35*b^(7/2)
*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d) + (105*b^(7/2)*e^2*S
qrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (35*b^(7/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]
*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d*E^((3*a)/b))

Rule 5866

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
]

Rule 12

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

Rule 5664

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 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5654

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

Rule 5658

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

Rule 3308

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 2180

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] &&  !$UseGamma === True

Rule 2205

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 2204

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 5670

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

Rule 5448

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]

Rubi steps

\begin{align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (7 b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{6 d}\\ &=-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (7 b e^2\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{9 d}+\frac{\left (35 b^2 e^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{36 d}\\ &=\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac{\left (35 b^2 e^2\right ) \operatorname{Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{18 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{108 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{12 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{432 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{432 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{216 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{24 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{24 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{\sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{432 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{432 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{432 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{48 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{48 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1728 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1728 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{216 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{24 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{24 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{175 b^{7/2} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{216 d}+\frac{175 b^{7/2} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}\\ &=-\frac{175 b^3 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{7 b e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{105 b^{7/2} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{35 b^{7/2} e^2 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}+\frac{105 b^{7/2} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 b^{7/2} e^2 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}\\ \end{align*}

Mathematica [B]  time = 13.5723, size = 1523, normalized size = 2.99 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

e^2*((a^3*Sqrt[a + b*ArcCosh[c + d*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[3/2, a/b + Arc
Cosh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 9*E^((2*a)
/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b
*ArcCosh[c + d*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c + d*x]))/b]))/(72*d*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c
+ d*x])^2/b^2)]) + (a^2*Sqrt[b]*(9*(-12*Sqrt[b]*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*Ar
cCosh[c + d*x]] + 8*Sqrt[b]*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + (2*a + 3*b)*Sqrt[Pi]*Erf
i[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[
c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])
/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sq
rt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcCosh[c + d*x]]*(2*ArcCosh[c + d*x]*Cosh[3*Ar
cCosh[c + d*x]] - Sinh[3*ArcCosh[c + d*x]])))/(96*d) + (a*(-27*(-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]*(12*a^2 + 12*a*b + 5*b^2)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[
(3*a)/b] - Sinh[(3*a)/b]) - Sqrt[b]*(12*a^2 - 12*a*b + 5*b^2)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d
*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*b*Sqrt[a + b*ArcCosh[c + d*x]]*(b*(5 + 12*ArcCosh[c + d*x]
^2)*Cosh[3*ArcCosh[c + d*x]] + 2*(a - 5*b*ArcCosh[c + d*x])*Sinh[3*ArcCosh[c + d*x]])))/(576*d) + (-81*(4*b*Sq
rt[a + b*ArcCosh[c + d*x]]*(Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(4*a^2 - 4*a*b*ArcCosh[c + d*x] +
7*b^2*(15 + 4*ArcCosh[c + d*x]^2)) - 2*b*(c + d*x)*(-10*a + b*ArcCosh[c + d*x]*(35 + 4*ArcCosh[c + d*x]^2)))
+ Sqrt[b]*(8*a^3 + 36*a^2*b + 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(-Cosh[a
/b] + Sinh[a/b]) + Sqrt[b]*(-8*a^3 + 36*a^2*b - 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/
Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(72*a^3 + 108*a^2*b + 90*a*b^2 + 35*b^3)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*
Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) - Sqrt[b]*(-72*a^3 + 108*a^2*b - 90*a*b
^2 + 35*b^3)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) -
12*b*Sqrt[a + b*ArcCosh[c + d*x]]*(-2*b*(-10*a + b*ArcCosh[c + d*x]*(35 + 36*ArcCosh[c + d*x]^2))*Cosh[3*ArcCo
sh[c + d*x]] + (12*a^2 - 12*a*b*ArcCosh[c + d*x] + 7*b^2*(5 + 12*ArcCosh[c + d*x]^2))*Sinh[3*ArcCosh[c + d*x]]
))/(10368*d))

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Maple [F]  time = 0.244, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^(7/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{2}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^2*(b*arccosh(d*x + c) + a)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2*(a+b*acosh(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

sage0*x