3.113 \(\int (c e+d e x)^4 (a+b \cosh ^{-1}(c+d x))^3 \, dx\)

Optimal. Leaf size=382 \[ \frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{3 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{8 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{6 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{625 d}-\frac{272 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5625 d}-\frac{4144 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1}}{5625 d}+\frac{16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d} \]

[Out]

(16*a*b^2*e^4*x)/25 - (4144*b^3*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(5625*d) - (272*b^3*e^4*Sqrt[-1 + c
+ d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5625*d) - (6*b^3*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/
(625*d) + (16*b^3*e^4*(c + d*x)*ArcCosh[c + d*x])/(25*d) + (8*b^2*e^4*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(7
5*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x]))/(125*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x
]*(a + b*ArcCosh[c + d*x])^2)/(25*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCos
h[c + d*x])^2)/(25*d) - (3*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/
(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^3)/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.723173, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5866, 12, 5662, 5759, 5718, 5654, 74, 100} \[ \frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{3 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{8 b e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{6 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{625 d}-\frac{272 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5625 d}-\frac{4144 b^3 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1}}{5625 d}+\frac{16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(16*a*b^2*e^4*x)/25 - (4144*b^3*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(5625*d) - (272*b^3*e^4*Sqrt[-1 + c
+ d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5625*d) - (6*b^3*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/
(625*d) + (16*b^3*e^4*(c + d*x)*ArcCosh[c + d*x])/(25*d) + (8*b^2*e^4*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(7
5*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x]))/(125*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x
]*(a + b*ArcCosh[c + d*x])^2)/(25*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCos
h[c + d*x])^2)/(25*d) - (3*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/
(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^3)/(5*d)

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 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 74

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

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int (c e+d e x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (3 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{5 d}\\ &=-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (12 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{25 d}+\frac{\left (6 b^2 e^4\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}\\ &=\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (8 b e^4\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{25 d}+\frac{\left (8 b^2 e^4\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac{\left (6 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{125 d}\\ &=-\frac{6 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{625 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{8 b e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}+\frac{\left (16 b^2 e^4\right ) \operatorname{Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac{\left (6 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{4 x^3}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{625 d}-\frac{\left (8 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{75 d}\\ &=\frac{16}{25} a b^2 e^4 x-\frac{8 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{225 d}-\frac{6 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{625 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{8 b e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (8 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{2 x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{225 d}-\frac{\left (24 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{625 d}+\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{25 d}\\ &=\frac{16}{25} a b^2 e^4 x-\frac{272 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5625 d}-\frac{6 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{625 d}+\frac{16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{8 b e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (8 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{2 x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{625 d}-\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{225 d}-\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=\frac{16}{25} a b^2 e^4 x-\frac{32 b^3 e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{45 d}-\frac{272 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5625 d}-\frac{6 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{625 d}+\frac{16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{8 b e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac{\left (16 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{625 d}\\ &=\frac{16}{25} a b^2 e^4 x-\frac{4144 b^3 e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{5625 d}-\frac{272 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5625 d}-\frac{6 b^3 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{625 d}+\frac{16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac{8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac{6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac{8 b e^4 \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{4 b e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac{3 b e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac{e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.586373, size = 404, normalized size = 1.06 \[ \frac{e^4 \left (3 a \left (25 a^2+6 b^2\right ) (c+d x)^5+\frac{1}{15} b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (-27 \left (25 a^2+2 b^2\right ) (c+d x)^4-4 \left (225 a^2+68 b^2\right ) (c+d x)^2-8 \left (225 a^2+518 b^2\right )\right )-b \cosh ^{-1}(c+d x) \left (-225 a^2 (c+d x)^5+90 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4+120 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2+240 a b \sqrt{c+d x-1} \sqrt{c+d x+1}-18 b^2 (c+d x)^5-40 b^2 (c+d x)^3-240 b^2 (c+d x)\right )+40 a b^2 (c+d x)^3+240 a b^2 (c+d x)-15 b^2 \cosh ^{-1}(c+d x)^2 \left (-15 a (c+d x)^5+3 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4+4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2+8 b \sqrt{c+d x-1} \sqrt{c+d x+1}\right )+75 b^3 (c+d x)^5 \cosh ^{-1}(c+d x)^3\right )}{375 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(e^4*(240*a*b^2*(c + d*x) + 40*a*b^2*(c + d*x)^3 + 3*a*(25*a^2 + 6*b^2)*(c + d*x)^5 + (b*Sqrt[-1 + c + d*x]*Sq
rt[1 + c + d*x]*(-8*(225*a^2 + 518*b^2) - 4*(225*a^2 + 68*b^2)*(c + d*x)^2 - 27*(25*a^2 + 2*b^2)*(c + d*x)^4))
/15 - b*(-240*b^2*(c + d*x) - 40*b^2*(c + d*x)^3 - 225*a^2*(c + d*x)^5 - 18*b^2*(c + d*x)^5 + 240*a*b*Sqrt[-1
+ c + d*x]*Sqrt[1 + c + d*x] + 120*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] + 90*a*b*Sqrt[-1 + c +
d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] - 15*b^2*(-15*a*(c + d*x)^5 + 8*b*Sqrt[-1 + c + d*x]*Sqr
t[1 + c + d*x] + 4*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] + 3*b*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqr
t[1 + c + d*x])*ArcCosh[c + d*x]^2 + 75*b^3*(c + d*x)^5*ArcCosh[c + d*x]^3))/(375*d)

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Maple [A]  time = 0.05, size = 602, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^3,x)

[Out]

1/d*(1/5*(d*x+c)^5*e^4*a^3+e^4*b^3*(1/5*(d*x+c)^3*arccosh(d*x+c)^3*(d*x+c-1)*(d*x+c+1)+1/5*arccosh(d*x+c)^3*(d
*x+c-1)*(d*x+c+1)*(d*x+c)+1/5*arccosh(d*x+c)^3*(d*x+c)-3/25*(d*x+c)^4*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+
1)^(1/2)-8/25*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-4/25*arccosh(d*x+c)^2*(d*x+c)^2*(d*x+c-1)^(1/2)
*(d*x+c+1)^(1/2)+6/125*arccosh(d*x+c)*(d*x+c-1)*(d*x+c+1)*(d*x+c)^3+58/375*arccosh(d*x+c)*(d*x+c-1)*(d*x+c+1)*
(d*x+c)+298/375*(d*x+c)*arccosh(d*x+c)-6/625*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-4144/5625*(d*x+c-1)^(1/
2)*(d*x+c+1)^(1/2)-272/5625*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))+3*e^4*a*b^2*(1/5*(d*x+c)^3*arccosh(d*x+
c)^2*(d*x+c-1)*(d*x+c+1)+1/5*arccosh(d*x+c)^2*(d*x+c-1)*(d*x+c+1)*(d*x+c)+1/5*arccosh(d*x+c)^2*(d*x+c)-2/25*(d
*x+c)^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-16/75*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-8/
75*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2+2/125*(d*x+c-1)*(d*x+c+1)*(d*x+c)^3+58/1125*(d*x+c
-1)*(d*x+c+1)*(d*x+c)+298/1125*d*x+298/1125*c)+3*e^4*a^2*b*(1/5*(d*x+c)^5*arccosh(d*x+c)-1/75*(d*x+c-1)^(1/2)*
(d*x+c+1)^(1/2)*(3*(d*x+c)^4+4*(d*x+c)^2+8)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.77614, size = 2325, normalized size = 6.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/5625*(45*(25*a^3 + 6*a*b^2)*d^5*e^4*x^5 + 225*(25*a^3 + 6*a*b^2)*c*d^4*e^4*x^4 + 150*(4*a*b^2 + 3*(25*a^3 +
6*a*b^2)*c^2)*d^3*e^4*x^3 + 450*(4*a*b^2*c + (25*a^3 + 6*a*b^2)*c^3)*d^2*e^4*x^2 + 225*(8*a*b^2*c^2 + (25*a^3
+ 6*a*b^2)*c^4 + 16*a*b^2)*d*e^4*x + 1125*(b^3*d^5*e^4*x^5 + 5*b^3*c*d^4*e^4*x^4 + 10*b^3*c^2*d^3*e^4*x^3 + 10
*b^3*c^3*d^2*e^4*x^2 + 5*b^3*c^4*d*e^4*x + b^3*c^5*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + 2
25*(15*a*b^2*d^5*e^4*x^5 + 75*a*b^2*c*d^4*e^4*x^4 + 150*a*b^2*c^2*d^3*e^4*x^3 + 150*a*b^2*c^3*d^2*e^4*x^2 + 75
*a*b^2*c^4*d*e^4*x + 15*a*b^2*c^5*e^4 - (3*b^3*d^4*e^4*x^4 + 12*b^3*c*d^3*e^4*x^3 + 2*(9*b^3*c^2 + 2*b^3)*d^2*
e^4*x^2 + 4*(3*b^3*c^3 + 2*b^3*c)*d*e^4*x + (3*b^3*c^4 + 4*b^3*c^2 + 8*b^3)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2
- 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 15*(9*(25*a^2*b + 2*b^3)*d^5*e^4*x^5 + 45*(25*a^2*b
+ 2*b^3)*c*d^4*e^4*x^4 + 10*(4*b^3 + 9*(25*a^2*b + 2*b^3)*c^2)*d^3*e^4*x^3 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^
3)*c^3)*d^2*e^4*x^2 + 15*(8*b^3*c^2 + 3*(25*a^2*b + 2*b^3)*c^4 + 16*b^3)*d*e^4*x + (40*b^3*c^3 + 9*(25*a^2*b +
2*b^3)*c^5 + 240*b^3*c)*e^4 - 30*(3*a*b^2*d^4*e^4*x^4 + 12*a*b^2*c*d^3*e^4*x^3 + 2*(9*a*b^2*c^2 + 2*a*b^2)*d^
2*e^4*x^2 + 4*(3*a*b^2*c^3 + 2*a*b^2*c)*d*e^4*x + (3*a*b^2*c^4 + 4*a*b^2*c^2 + 8*a*b^2)*e^4)*sqrt(d^2*x^2 + 2*
c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (27*(25*a^2*b + 2*b^3)*d^4*e^4*x^4 + 108*
(25*a^2*b + 2*b^3)*c*d^3*e^4*x^3 + 2*(450*a^2*b + 136*b^3 + 81*(25*a^2*b + 2*b^3)*c^2)*d^2*e^4*x^2 + 4*(27*(25
*a^2*b + 2*b^3)*c^3 + 2*(225*a^2*b + 68*b^3)*c)*d*e^4*x + (27*(25*a^2*b + 2*b^3)*c^4 + 1800*a^2*b + 4144*b^3 +
4*(225*a^2*b + 68*b^3)*c^2)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

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Sympy [A]  time = 23.0477, size = 2518, normalized size = 6.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4*(a+b*acosh(d*x+c))**3,x)

[Out]

Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e**4*x**3 + a**3*c*d**3*e**4*x**4 + a
**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e**4*acosh(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*acosh(c + d*x) - 3*a**2*
b*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(25*d) + 6*a**2*b*c**3*d*e**4*x**2*acosh(c + d*x) - 12*a**2*b
*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*acosh(c + d*x) - 18*a**2*b
*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 4*a**2*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2
- 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*acosh(c + d*x) - 12*a**2*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)/25 - 8*a**2*b*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 + 3*a**2*b*d**4*e**4*x**5*acosh(c +
d*x)/5 - 3*a**2*b*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 4*a**2*b*d*e**4*x**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1)/25 - 8*a**2*b*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(25*d) + 3*a*b**2*c**5*e**4*ac
osh(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x*acosh(c + d*x)**2 + 6*a*b**2*c**4*e**4*x/25 - 6*a*b**2*c**4*e**4*
sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x**2*acosh(c + d*x)**2 + 12*
a*b**2*c**3*d*e**4*x**2/25 - 24*a*b**2*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 6*
a*b**2*c**2*d**2*e**4*x**3*acosh(c + d*x)**2 + 12*a*b**2*c**2*d**2*e**4*x**3/25 - 36*a*b**2*c**2*d*e**4*x**2*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**2*c**2*e**4*x/25 - 8*a*b**2*c**2*e**4*sqrt(c**2
+ 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(25*d) + 3*a*b**2*c*d**3*e**4*x**4*acosh(c + d*x)**2 + 6*a*b**2*c*d
**3*e**4*x**4/25 - 24*a*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**
2*c*d*e**4*x**2/25 - 16*a*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 3*a*b**2*d**4
*e**4*x**5*acosh(c + d*x)**2/5 + 6*a*b**2*d**4*e**4*x**5/125 - 6*a*b**2*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**2*d**2*e**4*x**3/75 - 8*a*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2
*x**2 - 1)*acosh(c + d*x)/25 + 16*a*b**2*e**4*x/25 - 16*a*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh
(c + d*x)/(25*d) + b**3*c**5*e**4*acosh(c + d*x)**3/(5*d) + 6*b**3*c**5*e**4*acosh(c + d*x)/(125*d) + b**3*c**
4*e**4*x*acosh(c + d*x)**3 + 6*b**3*c**4*e**4*x*acosh(c + d*x)/25 - 3*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)*acosh(c + d*x)**2/(25*d) - 6*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(625*d) + 2*b**3*
c**3*d*e**4*x**2*acosh(c + d*x)**3 + 12*b**3*c**3*d*e**4*x**2*acosh(c + d*x)/25 - 12*b**3*c**3*e**4*x*sqrt(c**
2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 24*b**3*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/6
25 + 8*b**3*c**3*e**4*acosh(c + d*x)/(75*d) + 2*b**3*c**2*d**2*e**4*x**3*acosh(c + d*x)**3 + 12*b**3*c**2*d**2
*e**4*x**3*acosh(c + d*x)/25 - 18*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2
/25 - 36*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*c**2*e**4*x*acosh(c + d*x)/25
- 4*b**3*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(25*d) - 272*b**3*c**2*e**4*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 - 1)/(5625*d) + b**3*c*d**3*e**4*x**4*acosh(c + d*x)**3 + 6*b**3*c*d**3*e**4*x**4*aco
sh(c + d*x)/25 - 12*b**3*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 24*b**3*
c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*c*d*e**4*x**2*acosh(c + d*x)/25 - 8*b**3*c*
e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 544*b**3*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)/5625 + 16*b**3*c*e**4*acosh(c + d*x)/(25*d) + b**3*d**4*e**4*x**5*acosh(c + d*x)**3/5 + 6*b**3*d**
4*e**4*x**5*acosh(c + d*x)/125 - 3*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/
25 - 6*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*d**2*e**4*x**3*acosh(c + d*x)/75
- 4*b**3*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 272*b**3*d*e**4*x**2*sqrt(c**
2 + 2*c*d*x + d**2*x**2 - 1)/5625 + 16*b**3*e**4*x*acosh(c + d*x)/25 - 8*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)*acosh(c + d*x)**2/(25*d) - 4144*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(5625*d), Ne(d, 0)),
(c**4*e**4*x*(a + b*acosh(c))**3, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{4}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arccosh(d*x + c) + a)^3, x)