3.481 \(\int x^2 (d+e x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=365 \[ \frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b e \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left (1-c^2 x^2\right )^2 \left (378 c^4 d^2 e+105 c^6 d^3+405 c^2 d e^2+140 e^3\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left (1-c^2 x^2\right )^4 \left (27 c^2 d+28 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
) - (b*(105*c^6*d^3 + 378*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
(b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(1 - c^2*x^2)^5)/(81
*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^3*(a + b*ArcCosh[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcCosh[c*x]))/5 +
(3*d*e^2*x^7*(a + b*ArcCosh[c*x]))/7 + (e^3*x^9*(a + b*ArcCosh[c*x]))/9

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Rubi [A]  time = 0.541253, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {270, 5790, 12, 1610, 1799, 1620} \[ \frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b e \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left (1-c^2 x^2\right )^2 \left (378 c^4 d^2 e+105 c^6 d^3+405 c^2 d e^2+140 e^3\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left (1-c^2 x^2\right )^4 \left (27 c^2 d+28 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
) - (b*(105*c^6*d^3 + 378*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
(b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(1 - c^2*x^2)^5)/(81
*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^3*(a + b*ArcCosh[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcCosh[c*x]))/5 +
(3*d*e^2*x^7*(a + b*ArcCosh[c*x]))/7 + (e^3*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 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 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 \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 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 \left (\frac{105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3}{c^8 \sqrt{-1+c^2 x}}+\frac{\left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) \sqrt{-1+c^2 x}}{c^8}+\frac{3 e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (-1+c^2 x\right )^{3/2}}{c^8}+\frac{5 e^2 \left (27 c^2 d+28 e\right ) \left (-1+c^2 x\right )^{5/2}}{c^8}+\frac{35 e^3 \left (-1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (63 c^4 d^2+135 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{b e^2 \left (27 c^2 d+28 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.311631, size = 236, normalized size = 0.65 \[ \frac{315 a x^3 \left (189 d^2 e x^2+105 d^3+135 d e^2 x^4+35 e^3 x^6\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^8 \left (11907 d^2 e x^4+11025 d^3 x^2+6075 d e^2 x^6+1225 e^3 x^8\right )+2 c^6 \left (7938 d^2 e x^2+11025 d^3+3645 d e^2 x^4+700 e^3 x^6\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+80 c^2 e^2 \left (243 d+28 e x^2\right )+4480 e^3\right )}{c^9}+315 b x^3 \cosh ^{-1}(c x) \left (189 d^2 e x^2+105 d^3+135 d e^2 x^4+35 e^3 x^6\right )}{99225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*a*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^3
+ 80*c^2*e^2*(243*d + 28*e*x^2) + 24*c^4*e*(1323*d^2 + 405*d*e*x^2 + 70*e^2*x^4) + 2*c^6*(11025*d^3 + 7938*d^2
*e*x^2 + 3645*d*e^2*x^4 + 700*e^3*x^6) + c^8*(11025*d^3*x^2 + 11907*d^2*e*x^4 + 6075*d*e^2*x^6 + 1225*e^3*x^8)
))/c^9 + 315*b*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6)*ArcCosh[c*x])/99225

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Maple [A]  time = 0.012, size = 289, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{9}{x}^{9}}{9}}+{\frac{3\,d{e}^{2}{c}^{9}{x}^{7}}{7}}+{\frac{3\,{c}^{9}{d}^{2}e{x}^{5}}{5}}+{\frac{{x}^{3}{c}^{9}{d}^{3}}{3}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{3}{c}^{9}{x}^{9}}{9}}+{\frac{3\,{\rm arccosh} \left (cx\right )d{e}^{2}{c}^{9}{x}^{7}}{7}}+{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{9}{d}^{2}e{x}^{5}}{5}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{9}{x}^{3}{d}^{3}}{3}}-{\frac{1225\,{c}^{8}{e}^{3}{x}^{8}+6075\,{c}^{8}d{e}^{2}{x}^{6}+11907\,{c}^{8}{d}^{2}e{x}^{4}+1400\,{c}^{6}{e}^{3}{x}^{6}+11025\,{c}^{8}{d}^{3}{x}^{2}+7290\,{c}^{6}d{e}^{2}{x}^{4}+15876\,{c}^{6}{d}^{2}e{x}^{2}+1680\,{c}^{4}{e}^{3}{x}^{4}+22050\,{d}^{3}{c}^{6}+9720\,{c}^{4}d{e}^{2}{x}^{2}+31752\,{c}^{4}{d}^{2}e+2240\,{c}^{2}{e}^{3}{x}^{2}+19440\,d{e}^{2}{c}^{2}+4480\,{e}^{3}}{99225}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^6*(1/9*e^3*c^9*x^9+3/7*d*e^2*c^9*x^7+3/5*c^9*d^2*e*x^5+1/3*x^3*c^9*d^3)+b/c^6*(1/9*arccosh(c*x)*e^3
*c^9*x^9+3/7*arccosh(c*x)*d*e^2*c^9*x^7+3/5*arccosh(c*x)*c^9*d^2*e*x^5+1/3*arccosh(c*x)*c^9*x^3*d^3-1/99225*(c
*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^3*x^8+6075*c^8*d*e^2*x^6+11907*c^8*d^2*e*x^4+1400*c^6*e^3*x^6+11025*c^8*
d^3*x^2+7290*c^6*d*e^2*x^4+15876*c^6*d^2*e*x^2+1680*c^4*e^3*x^4+22050*c^6*d^3+9720*c^4*d*e^2*x^2+31752*c^4*d^2
*e+2240*c^2*e^3*x^2+19440*c^2*d*e^2+4480*e^3)))

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Maxima [A]  time = 1.16467, size = 505, normalized size = 1.38 \begin{align*} \frac{1}{9} \, a e^{3} x^{9} + \frac{3}{7} \, a d e^{2} x^{7} + \frac{3}{5} \, a d^{2} e x^{5} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} + \frac{1}{25} \,{\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} e + \frac{3}{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^{2} + \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^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*e^3*x^9 + 3/7*a*d*e^2*x^7 + 3/5*a*d^2*e*x^5 + 1/3*a*d^3*x^3 + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2
- 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^3 + 1/25*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*
sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d^2*e + 3/245*(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
^2 + 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^3

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Fricas [A]  time = 2.40888, size = 701, normalized size = 1.92 \begin{align*} \frac{11025 \, a c^{9} e^{3} x^{9} + 42525 \, a c^{9} d e^{2} x^{7} + 59535 \, a c^{9} d^{2} e x^{5} + 33075 \, a c^{9} d^{3} x^{3} + 315 \,{\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{8} e^{3} x^{8} + 22050 \, b c^{6} d^{3} + 31752 \, b c^{4} d^{2} e + 25 \,{\left (243 \, b c^{8} d e^{2} + 56 \, b c^{6} e^{3}\right )} x^{6} + 19440 \, b c^{2} d e^{2} + 3 \,{\left (3969 \, b c^{8} d^{2} e + 2430 \, b c^{6} d e^{2} + 560 \, b c^{4} e^{3}\right )} x^{4} + 4480 \, b e^{3} +{\left (11025 \, b c^{8} d^{3} + 15876 \, b c^{6} d^{2} e + 9720 \, b c^{4} d e^{2} + 2240 \, b c^{2} e^{3}\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^2*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/99225*(11025*a*c^9*e^3*x^9 + 42525*a*c^9*d*e^2*x^7 + 59535*a*c^9*d^2*e*x^5 + 33075*a*c^9*d^3*x^3 + 315*(35*b
*c^9*e^3*x^9 + 135*b*c^9*d*e^2*x^7 + 189*b*c^9*d^2*e*x^5 + 105*b*c^9*d^3*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (
1225*b*c^8*e^3*x^8 + 22050*b*c^6*d^3 + 31752*b*c^4*d^2*e + 25*(243*b*c^8*d*e^2 + 56*b*c^6*e^3)*x^6 + 19440*b*c
^2*d*e^2 + 3*(3969*b*c^8*d^2*e + 2430*b*c^6*d*e^2 + 560*b*c^4*e^3)*x^4 + 4480*b*e^3 + (11025*b*c^8*d^3 + 15876
*b*c^6*d^2*e + 9720*b*c^4*d*e^2 + 2240*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^9

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Sympy [A]  time = 36.9676, size = 532, normalized size = 1.46 \begin{align*} \begin{cases} \frac{a d^{3} x^{3}}{3} + \frac{3 a d^{2} e x^{5}}{5} + \frac{3 a d e^{2} x^{7}}{7} + \frac{a e^{3} x^{9}}{9} + \frac{b d^{3} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{3 b d^{2} e x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{3 b d e^{2} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b e^{3} x^{9} \operatorname{acosh}{\left (c x \right )}}{9} - \frac{b d^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{3 b d^{2} e x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{3 b d e^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{b e^{3} x^{8} \sqrt{c^{2} x^{2} - 1}}{81 c} - \frac{2 b d^{3} \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} - \frac{4 b d^{2} e x^{2} \sqrt{c^{2} x^{2} - 1}}{25 c^{3}} - \frac{18 b d e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b e^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{567 c^{3}} - \frac{8 b d^{2} e \sqrt{c^{2} x^{2} - 1}}{25 c^{5}} - \frac{24 b d e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{945 c^{5}} - \frac{48 b d e^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} - \frac{64 b e^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{2835 c^{7}} - \frac{128 b e^{3} \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^{3} x^{3}}{3} + \frac{3 d^{2} e x^{5}}{5} + \frac{3 d e^{2} x^{7}}{7} + \frac{e^{3} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**3/3 + 3*a*d**2*e*x**5/5 + 3*a*d*e**2*x**7/7 + a*e**3*x**9/9 + b*d**3*x**3*acosh(c*x)/3 +
3*b*d**2*e*x**5*acosh(c*x)/5 + 3*b*d*e**2*x**7*acosh(c*x)/7 + b*e**3*x**9*acosh(c*x)/9 - b*d**3*x**2*sqrt(c**2
*x**2 - 1)/(9*c) - 3*b*d**2*e*x**4*sqrt(c**2*x**2 - 1)/(25*c) - 3*b*d*e**2*x**6*sqrt(c**2*x**2 - 1)/(49*c) - b
*e**3*x**8*sqrt(c**2*x**2 - 1)/(81*c) - 2*b*d**3*sqrt(c**2*x**2 - 1)/(9*c**3) - 4*b*d**2*e*x**2*sqrt(c**2*x**2
- 1)/(25*c**3) - 18*b*d*e**2*x**4*sqrt(c**2*x**2 - 1)/(245*c**3) - 8*b*e**3*x**6*sqrt(c**2*x**2 - 1)/(567*c**
3) - 8*b*d**2*e*sqrt(c**2*x**2 - 1)/(25*c**5) - 24*b*d*e**2*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) - 16*b*e**3*x*
*4*sqrt(c**2*x**2 - 1)/(945*c**5) - 48*b*d*e**2*sqrt(c**2*x**2 - 1)/(245*c**7) - 64*b*e**3*x**2*sqrt(c**2*x**2
- 1)/(2835*c**7) - 128*b*e**3*sqrt(c**2*x**2 - 1)/(2835*c**9), Ne(c, 0)), ((a + I*pi*b/2)*(d**3*x**3/3 + 3*d*
*2*e*x**5/5 + 3*d*e**2*x**7/7 + e**3*x**9/9), True))

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Giac [A]  time = 1.41622, size = 479, normalized size = 1.31 \begin{align*} \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b d^{3} + \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^{3} + \frac{3}{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^{2} + \frac{1}{25} \,{\left (15 \, a d^{2} x^{5} +{\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}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*d^3*x^3 + 1/9*(3*x^3*log(c*x + sqrt(c^2*x^2 - 1)) - ((c^2*x^2 - 1)^(3/2) + 3*sqrt(c^2*x^2 - 1))/c^3)*b*d
^3 + 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^3 + 3/245*(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^2 + 1/25*(15*a*d^2*x^5 + (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)*e