\(\int x^2 (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [481]

Optimal result

Integrand size = 21, antiderivative size = 365 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (27 c^2 d+28 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 5958, 12, 1624, 1813, 1634} \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\frac {b e^2 \left (1-c^2 x^2\right )^4 \left (27 c^2 d+28 e\right )}{441 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{525 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \left (1-c^2 x^2\right )^2 \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right )}{945 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{315 c^9 \sqrt {c x-1} \sqrt {c x+1}} \]

[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 12

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

Rule 276

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 1624

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 1634

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]

Rule 1813

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 5958

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]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-(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 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\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 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\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 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {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 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {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 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.65 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^3+80 c^2 e^2 \left (243 d+28 e x^2\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+2 c^6 \left (11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right )+c^8 \left (11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right )\right )}{c^9}+315 b x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right ) \text {arccosh}(c x)}{99225} \]

[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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.75

method	result	size
parts	\(a \left (\frac {1}{9} e^{3} x^{9}+\frac {3}{7} d \,e^{2} x^{7}+\frac {3}{5} d^{2} e \,x^{5}+\frac {1}{3} d^{3} x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arccosh}\left (c x \right ) e^{3} x^{9}}{9}+\frac {3 c^{3} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{7}}{7}+\frac {3 c^{3} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) c^{3} x^{3} d^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} x^{4} e^{3}+22050 d^{3} c^{6}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} x^{2} e^{3}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225 c^{6}}\right )}{c^{3}}\)	\(273\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} c^{9} d^{3} x^{3}+\frac {3}{5} c^{9} d^{2} e \,x^{5}+\frac {3}{7} c^{9} d \,e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{3} x^{3}}{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} e \,x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d \,e^{2} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} x^{4} e^{3}+22050 d^{3} c^{6}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} x^{2} e^{3}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225}\right )}{c^{6}}}{c^{3}}\)	\(289\)
default	\(\frac {\frac {a \left (\frac {1}{3} c^{9} d^{3} x^{3}+\frac {3}{5} c^{9} d^{2} e \,x^{5}+\frac {3}{7} c^{9} d \,e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{3} x^{3}}{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} e \,x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d \,e^{2} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} x^{4} e^{3}+22050 d^{3} c^{6}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} x^{2} e^{3}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225}\right )}{c^{6}}}{c^{3}}\)	\(289\)

[In]

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

[Out]

a*(1/9*e^3*x^9+3/7*d*e^2*x^7+3/5*d^2*e*x^5+1/3*d^3*x^3)+b/c^3*(1/9*c^3*arccosh(c*x)*e^3*x^9+3/7*c^3*arccosh(c*
x)*d*e^2*x^7+3/5*c^3*arccosh(c*x)*d^2*e*x^5+1/3*arccosh(c*x)*c^3*x^3*d^3-1/99225/c^6*(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+194
40*c^2*d*e^2+4480*e^3))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.79 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{3} x^{9} + 42525 \, a c^{9} d e^{2} x^{7} + 59535 \, a c^{9} d^{2} e x^{5} + 33075 \, a c^{9} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{3} x^{8} + 22050 \, b c^{6} d^{3} + 31752 \, b c^{4} d^{2} e + 25 \, {\left (243 \, b c^{8} d e^{2} + 56 \, b c^{6} e^{3}\right )} x^{6} + 19440 \, b c^{2} d e^{2} + 3 \, {\left (3969 \, b c^{8} d^{2} e + 2430 \, b c^{6} d e^{2} + 560 \, b c^{4} e^{3}\right )} x^{4} + 4480 \, b e^{3} + {\left (11025 \, b c^{8} d^{3} + 15876 \, b c^{6} d^{2} e + 9720 \, b c^{4} d e^{2} + 2240 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \]

[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

Sympy [F]

\[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.02 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} e + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e^{2} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{3} \]

[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

Giac [F(-2)]

Exception generated. \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]

[In]

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

[Out]

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