3.470 \(\int x^4 (d+e x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=319 \[ \frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b \left (1-c^2 x^2\right )^2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b e \left (1-c^2 x^2\right )^4 \left (9 c^2 d+14 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*(63*c^4*d^2
+ 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(21*c^4*d^2 + 90*c^2*d*e
+ 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*e*(9*c^2*d + 14*e)*(1 - c^2*x^2)^4)/
(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*
x^5*(a + b*ArcCosh[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCosh[c*x]))/7 + (e^2*x^9*(a + b*ArcCosh[c*x]))/9

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Rubi [A]  time = 0.411012, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {270, 5790, 12, 520, 1251, 897, 1153} \[ \frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b \left (1-c^2 x^2\right )^2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b e \left (1-c^2 x^2\right )^4 \left (9 c^2 d+14 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]

[Out]

(b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*(63*c^4*d^2
+ 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(21*c^4*d^2 + 90*c^2*d*e
+ 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*e*(9*c^2*d + 14*e)*(1 - c^2*x^2)^4)/
(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*
x^5*(a + b*ArcCosh[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCosh[c*x]))/7 + (e^2*x^9*(a + b*ArcCosh[c*x]))/9

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5790

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] &
& (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

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

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^4 \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{x^2}{c^2}\right )^2 \left (\frac{63 c^4 d^2+90 c^2 d e+35 e^2}{c^4}-\frac{\left (-90 c^2 d e-70 e^2\right ) x^2}{c^4}+\frac{35 e^2 x^4}{c^4}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{315 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{63 c^4 d^2+90 c^2 d e+35 e^2}{c^8}+\frac{2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) x^2}{c^8}+\frac{3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) x^4}{c^8}+\frac{10 e \left (9 c^2 d+14 e\right ) x^6}{c^8}+\frac{35 e^2 x^8}{c^8}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{315 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{7} d e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.263065, size = 192, normalized size = 0.6 \[ \frac{315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+160 c^2 e \left (81 d+14 e x^2\right )+4480 e^2\right )}{c^9}+315 b x^5 \cosh ^{-1}(c x) \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{99225} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]

[Out]

(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^2 + 160*c^2*e*(81*d +
14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) +
c^8*(3969*d^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcCosh[
c*x])/99225

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Maple [A]  time = 0.014, size = 227, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{9}{x}^{9}}{9}}+{\frac{2\,{c}^{9}de{x}^{7}}{7}}+{\frac{{c}^{9}{x}^{5}{d}^{2}}{5}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{2}{c}^{9}{x}^{9}}{9}}+{\frac{2\,{\rm arccosh} \left (cx\right ){c}^{9}de{x}^{7}}{7}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{9}{x}^{5}{d}^{2}}{5}}-{\frac{1225\,{c}^{8}{e}^{2}{x}^{8}+4050\,{c}^{8}de{x}^{6}+3969\,{c}^{8}{d}^{2}{x}^{4}+1400\,{c}^{6}{e}^{2}{x}^{6}+4860\,{c}^{6}de{x}^{4}+5292\,{c}^{6}{d}^{2}{x}^{2}+1680\,{c}^{4}{e}^{2}{x}^{4}+6480\,{c}^{4}de{x}^{2}+10584\,{d}^{2}{c}^{4}+2240\,{c}^{2}{e}^{2}{x}^{2}+12960\,{c}^{2}de+4480\,{e}^{2}}{99225}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x)

[Out]

1/c^5*(a/c^4*(1/9*e^2*c^9*x^9+2/7*c^9*d*e*x^7+1/5*c^9*x^5*d^2)+b/c^4*(1/9*arccosh(c*x)*e^2*c^9*x^9+2/7*arccosh
(c*x)*c^9*d*e*x^7+1/5*arccosh(c*x)*c^9*x^5*d^2-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^2*x^8+4050*c^8*
d*e*x^6+3969*c^8*d^2*x^4+1400*c^6*e^2*x^6+4860*c^6*d*e*x^4+5292*c^6*d^2*x^2+1680*c^4*e^2*x^4+6480*c^4*d*e*x^2+
10584*c^4*d^2+2240*c^2*e^2*x^2+12960*c^2*d*e+4480*e^2)))

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Maxima [A]  time = 1.02647, size = 412, normalized size = 1.29 \begin{align*} \frac{1}{9} \, a e^{2} x^{9} + \frac{2}{7} \, a d e x^{7} + \frac{1}{5} \, a d^{2} x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac{2}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e + \frac{1}{2835} \,{\left (315 \, x^{9} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

1/9*a*e^2*x^9 + 2/7*a*d*e*x^7 + 1/5*a*d^2*x^5 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*s
qrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d^2 + 2/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 -
1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*d*e +
1/2835*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2
- 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^10)*c)*b*e^2

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Fricas [A]  time = 2.38452, size = 562, normalized size = 1.76 \begin{align*} \frac{11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \,{\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \,{\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \,{\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \,{\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{99225 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/99225*(11025*a*c^9*e^2*x^9 + 28350*a*c^9*d*e*x^7 + 19845*a*c^9*d^2*x^5 + 315*(35*b*c^9*e^2*x^9 + 90*b*c^9*d*
e*x^7 + 63*b*c^9*d^2*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^8*e^2*x^8 + 10584*b*c^4*d^2 + 50*(81*b*c^8*
d*e + 28*b*c^6*e^2)*x^6 + 12960*b*c^2*d*e + 3*(1323*b*c^8*d^2 + 1620*b*c^6*d*e + 560*b*c^4*e^2)*x^4 + 4480*b*e
^2 + 4*(1323*b*c^6*d^2 + 1620*b*c^4*d*e + 560*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^9

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Sympy [A]  time = 31.7405, size = 422, normalized size = 1.32 \begin{align*} \begin{cases} \frac{a d^{2} x^{5}}{5} + \frac{2 a d e x^{7}}{7} + \frac{a e^{2} x^{9}}{9} + \frac{b d^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{2 b d e x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b e^{2} x^{9} \operatorname{acosh}{\left (c x \right )}}{9} - \frac{b d^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{2 b d e x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{b e^{2} x^{8} \sqrt{c^{2} x^{2} - 1}}{81 c} - \frac{4 b d^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{12 b d e x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b e^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{567 c^{3}} - \frac{8 b d^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} - \frac{16 b d e x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{945 c^{5}} - \frac{32 b d e \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} - \frac{64 b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{2835 c^{7}} - \frac{128 b e^{2} \sqrt{c^{2} x^{2} - 1}}{2835 c^{9}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{2} x^{5}}{5} + \frac{2 d e x^{7}}{7} + \frac{e^{2} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)**2*(a+b*acosh(c*x)),x)

[Out]

Piecewise((a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 + b*d**2*x**5*acosh(c*x)/5 + 2*b*d*e*x**7*acosh(c*x)
/7 + b*e**2*x**9*acosh(c*x)/9 - b*d**2*x**4*sqrt(c**2*x**2 - 1)/(25*c) - 2*b*d*e*x**6*sqrt(c**2*x**2 - 1)/(49*
c) - b*e**2*x**8*sqrt(c**2*x**2 - 1)/(81*c) - 4*b*d**2*x**2*sqrt(c**2*x**2 - 1)/(75*c**3) - 12*b*d*e*x**4*sqrt
(c**2*x**2 - 1)/(245*c**3) - 8*b*e**2*x**6*sqrt(c**2*x**2 - 1)/(567*c**3) - 8*b*d**2*sqrt(c**2*x**2 - 1)/(75*c
**5) - 16*b*d*e*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) - 16*b*e**2*x**4*sqrt(c**2*x**2 - 1)/(945*c**5) - 32*b*d*e
*sqrt(c**2*x**2 - 1)/(245*c**7) - 64*b*e**2*x**2*sqrt(c**2*x**2 - 1)/(2835*c**7) - 128*b*e**2*sqrt(c**2*x**2 -
1)/(2835*c**9), Ne(c, 0)), ((a + I*pi*b/2)*(d**2*x**5/5 + 2*d*e*x**7/7 + e**2*x**9/9), True))

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Giac [A]  time = 1.36795, size = 383, normalized size = 1.2 \begin{align*} \frac{1}{5} \, a d^{2} x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b d^{2} + \frac{1}{2835} \,{\left (315 \, a x^{9} +{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 180 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 420 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} - 1}}{c^{9}}\right )} b\right )} e^{2} + \frac{2}{245} \,{\left (35 \, a d x^{7} +{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b d\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

1/5*a*d^2*x^5 + 1/75*(15*x^5*log(c*x + sqrt(c^2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^2*x^2 - 1)^(3/2) +
15*sqrt(c^2*x^2 - 1))/c^5)*b*d^2 + 1/2835*(315*a*x^9 + (315*x^9*log(c*x + sqrt(c^2*x^2 - 1)) - (35*(c^2*x^2 -
1)^(9/2) + 180*(c^2*x^2 - 1)^(7/2) + 378*(c^2*x^2 - 1)^(5/2) + 420*(c^2*x^2 - 1)^(3/2) + 315*sqrt(c^2*x^2 - 1)
)/c^9)*b)*e^2 + 2/245*(35*a*d*x^7 + (35*x^7*log(c*x + sqrt(c^2*x^2 - 1)) - (5*(c^2*x^2 - 1)^(7/2) + 21*(c^2*x^
2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2 - 1))/c^7)*b*d)*e