Integrand size = 21, antiderivative size = 307 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt {-1+c x} \sqrt {1+c x}}{315 c^9}-\frac {b \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^9}-\frac {b e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^9}-\frac {b e^2 \left (27 c^2 d+28 e\right ) (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^9}-\frac {b e^3 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^9}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x)) \] Output:
-1/315*b*(105*c^6*d^3+189*c^4*d^2*e+135*c^2*d*e^2+35*e^3)*(c*x-1)^(1/2)*(c *x+1)^(1/2)/c^9-1/945*b*(105*c^6*d^3+378*c^4*d^2*e+405*c^2*d*e^2+140*e^3)* (c*x-1)^(3/2)*(c*x+1)^(3/2)/c^9-1/525*b*e*(63*c^4*d^2+135*c^2*d*e+70*e^2)* (c*x-1)^(5/2)*(c*x+1)^(5/2)/c^9-1/441*b*e^2*(27*c^2*d+28*e)*(c*x-1)^(7/2)* (c*x+1)^(7/2)/c^9-1/81*b*e^3*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c^9+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*arcc osh(c*x))+1/9*e^3*x^9*(a+b*arccosh(c*x))
Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.77 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^3+80 c^2 e^2 \left (243 d+28 e x^2\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+2 c^6 \left (11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right )+c^8 \left (11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right )\right )}{c^9}+315 b x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right ) \text {arccosh}(c x)}{99225} \] Input:
Integrate[x^2*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
(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*Sqr t[-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
Time = 1.00 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 2113, 2331, 2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^2 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 e^3 \left (c^2 x^2-1\right )^{7/2}}{c^8}+\frac {5 e^2 \left (27 d c^2+28 e\right ) \left (c^2 x^2-1\right )^{5/2}}{c^8}+\frac {3 e \left (63 d^2 c^4+135 d e c^2+70 e^2\right ) \left (c^2 x^2-1\right )^{3/2}}{c^8}+\frac {\left (105 d^3 c^6+378 d^2 e c^4+405 d e^2 c^2+140 e^3\right ) \sqrt {c^2 x^2-1}}{c^8}+\frac {105 d^3 c^6+189 d^2 e c^4+135 d e^2 c^2+35 e^3}{c^8 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {10 e^2 \left (c^2 x^2-1\right )^{7/2} \left (27 c^2 d+28 e\right )}{7 c^{10}}+\frac {70 e^3 \left (c^2 x^2-1\right )^{9/2}}{9 c^{10}}+\frac {6 e \left (c^2 x^2-1\right )^{5/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{5 c^{10}}+\frac {2 \left (c^2 x^2-1\right )^{3/2} \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right )}{3 c^{10}}+\frac {2 \sqrt {c^2 x^2-1} \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{c^{10}}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x^2*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
-1/630*(b*c*Sqrt[-1 + c^2*x^2]*((2*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2* d*e^2 + 35*e^3)*Sqrt[-1 + c^2*x^2])/c^10 + (2*(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)^(3/2))/(3*c^10) + (6*e*(63*c^4* d^2 + 135*c^2*d*e + 70*e^2)*(-1 + c^2*x^2)^(5/2))/(5*c^10) + (10*e^2*(27*c ^2*d + 28*e)*(-1 + c^2*x^2)^(7/2))/(7*c^10) + (70*e^3*(-1 + c^2*x^2)^(9/2) )/(9*c^10)))/(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
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(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] && !IntegerQ[m]
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])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[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]
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]}, Sim p[(a + b*ArcCosh[c*x]) u, x] - Simp[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] && Le Q[m + p, 0]))
Time = 0.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{3} x^{9}+\frac {3}{7} d \,e^{2} x^{7}+\frac {3}{5} d^{2} e \,x^{5}+\frac {1}{3} d^{3} x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arccosh}\left (c x \right ) e^{3} x^{9}}{9}+\frac {3 c^{3} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{7}}{7}+\frac {3 c^{3} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) c^{3} x^{3} d^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} e^{3} x^{4}+22050 c^{6} d^{3}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} e^{3} x^{2}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225 c^{6}}\right )}{c^{3}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} c^{9} d^{3} x^{3}+\frac {3}{5} c^{9} d^{2} e \,x^{5}+\frac {3}{7} c^{9} d \,e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{3} x^{3}}{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} e \,x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d \,e^{2} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} e^{3} x^{4}+22050 c^{6} d^{3}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} e^{3} x^{2}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225}\right )}{c^{6}}}{c^{3}}\) | \(289\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} c^{9} d^{3} x^{3}+\frac {3}{5} c^{9} d^{2} e \,x^{5}+\frac {3}{7} c^{9} d \,e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{3} x^{3}}{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} e \,x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{9} d \,e^{2} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} e^{3} x^{4}+22050 c^{6} d^{3}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} e^{3} x^{2}+19440 c^{2} d \,e^{2}+4480 e^{3}\right )}{99225}\right )}{c^{6}}}{c^{3}}\) | \(289\) |
| orering | \(\frac {\left (20825 c^{10} e^{4} x^{12}+104600 c^{10} d \,e^{3} x^{10}+209466 c^{10} d^{2} e^{2} x^{8}+1400 c^{8} e^{4} x^{10}+204624 c^{10} d^{3} e \,x^{6}+10070 c^{8} d \,e^{3} x^{8}+55125 c^{10} d^{4} x^{4}+34182 c^{8} d^{2} e^{2} x^{6}+2240 c^{6} e^{4} x^{8}+96138 c^{8} d^{3} e \,x^{4}+20000 c^{6} d \,e^{3} x^{6}+22050 c^{8} d^{4} x^{2}+131868 c^{6} d^{2} e^{2} x^{4}+4480 c^{4} e^{4} x^{6}-144648 c^{6} d^{3} e \,x^{2}+78880 c^{4} d \,e^{3} x^{4}-44100 c^{6} d^{4}-234576 c^{4} d^{2} e^{2} x^{2}+17920 c^{2} e^{4} x^{4}-63504 c^{4} d^{3} e -151040 c^{2} d \,e^{3} x^{2}-38880 c^{2} d^{2} e^{2}-35840 e^{4} x^{2}-8960 d \,e^{3}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{99225 x \left (e \,x^{2}+d \right ) c^{10}}-\frac {\left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 c^{6} e^{3} x^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} e^{3} x^{4}+22050 c^{6} d^{3}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} e^{3} x^{2}+19440 c^{2} d \,e^{2}+4480 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 x \left (e \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+6 x^{3} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x^{2} \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{99225 c^{10} \left (e \,x^{2}+d \right )^{3} x^{2}}\) | \(546\) |
Input:
int(x^2*(e*x^2+d)^3*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
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*arc cosh(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+19440*c ^2*d*e^2+4480*e^3))
Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.94 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{3} x^{9} + 42525 \, a c^{9} d e^{2} x^{7} + 59535 \, a c^{9} d^{2} e x^{5} + 33075 \, a c^{9} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{3} x^{8} + 22050 \, b c^{6} d^{3} + 31752 \, b c^{4} d^{2} e + 25 \, {\left (243 \, b c^{8} d e^{2} + 56 \, b c^{6} e^{3}\right )} x^{6} + 19440 \, b c^{2} d e^{2} + 3 \, {\left (3969 \, b c^{8} d^{2} e + 2430 \, b c^{6} d e^{2} + 560 \, b c^{4} e^{3}\right )} x^{4} + 4480 \, b e^{3} + {\left (11025 \, b c^{8} d^{3} + 15876 \, b c^{6} d^{2} e + 9720 \, b c^{4} d e^{2} + 2240 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
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 + 1 89*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)) - (12 25*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
\[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \] Input:
integrate(x**2*(e*x**2+d)**3*(a+b*acosh(c*x)),x)
Output:
Integral(x**2*(a + b*acosh(c*x))*(d + e*x**2)**3, x)
Time = 0.04 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.22 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} e + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e^{2} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{3} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
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*sq rt(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
Exception generated. \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int(x^2*(a + b*acosh(c*x))*(d + e*x^2)^3,x)
Output:
int(x^2*(a + b*acosh(c*x))*(d + e*x^2)^3, x)
Time = 0.19 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.37 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {33075 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{3} x^{3}+59535 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{2} e \,x^{5}+42525 \mathit {acosh} \left (c x \right ) b \,c^{9} d \,e^{2} x^{7}+11025 \mathit {acosh} \left (c x \right ) b \,c^{9} e^{3} x^{9}-11025 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{3} x^{2}-11907 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{2} e \,x^{4}-6075 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d \,e^{2} x^{6}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} e^{3} x^{8}-22050 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{3}-15876 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} e \,x^{2}-7290 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d \,e^{2} x^{4}-1400 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{3} x^{6}-31752 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2} e -9720 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,e^{2} x^{2}-1680 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{3} x^{4}-19440 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d \,e^{2}-2240 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{3} x^{2}-4480 \sqrt {c^{2} x^{2}-1}\, b \,e^{3}+33075 a \,c^{9} d^{3} x^{3}+59535 a \,c^{9} d^{2} e \,x^{5}+42525 a \,c^{9} d \,e^{2} x^{7}+11025 a \,c^{9} e^{3} x^{9}}{99225 c^{9}} \] Input:
int(x^2*(e*x^2+d)^3*(a+b*acosh(c*x)),x)
Output:
(33075*acosh(c*x)*b*c**9*d**3*x**3 + 59535*acosh(c*x)*b*c**9*d**2*e*x**5 + 42525*acosh(c*x)*b*c**9*d*e**2*x**7 + 11025*acosh(c*x)*b*c**9*e**3*x**9 - 11025*sqrt(c**2*x**2 - 1)*b*c**8*d**3*x**2 - 11907*sqrt(c**2*x**2 - 1)*b* c**8*d**2*e*x**4 - 6075*sqrt(c**2*x**2 - 1)*b*c**8*d*e**2*x**6 - 1225*sqrt (c**2*x**2 - 1)*b*c**8*e**3*x**8 - 22050*sqrt(c**2*x**2 - 1)*b*c**6*d**3 - 15876*sqrt(c**2*x**2 - 1)*b*c**6*d**2*e*x**2 - 7290*sqrt(c**2*x**2 - 1)*b *c**6*d*e**2*x**4 - 1400*sqrt(c**2*x**2 - 1)*b*c**6*e**3*x**6 - 31752*sqrt (c**2*x**2 - 1)*b*c**4*d**2*e - 9720*sqrt(c**2*x**2 - 1)*b*c**4*d*e**2*x** 2 - 1680*sqrt(c**2*x**2 - 1)*b*c**4*e**3*x**4 - 19440*sqrt(c**2*x**2 - 1)* b*c**2*d*e**2 - 2240*sqrt(c**2*x**2 - 1)*b*c**2*e**3*x**2 - 4480*sqrt(c**2 *x**2 - 1)*b*e**3 + 33075*a*c**9*d**3*x**3 + 59535*a*c**9*d**2*e*x**5 + 42 525*a*c**9*d*e**2*x**7 + 11025*a*c**9*e**3*x**9)/(99225*c**9)