Integrand size = 21, antiderivative size = 278 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{24 c^8}+\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{72 c^8}-\frac {b e \left (8 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{120 c^8}+\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{7/2}}{56 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right ) \]
1/4*d^2*x^4*(a+b*arcsech(c*x))+1/3*d*e*x^6*(a+b*arcsech(c*x))+1/8*e^2*x^8* (a+b*arcsech(c*x))+1/72*b*(6*c^4*d^2+16*c^2*d*e+9*e^2)*(-c^2*x^2+1)^(3/2)* (1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^8-1/120*b*e*(8*c^2*d+9*e)*(-c^2*x^2+1)^( 5/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^8+1/56*b*e^2*(-c^2*x^2+1)^(7/2)*(1/ (c*x+1))^(1/2)*(c*x+1)^(1/2)/c^8-1/24*b*(6*c^4*d^2+8*c^2*d*e+3*e^2)*(1/(c* x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^8
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.60 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{24} \left (6 a d^2 x^4+8 a d e x^6+3 a e^2 x^8-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (144 e^2+8 c^2 e \left (56 d+9 e x^2\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{105 c^8}+b x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \text {sech}^{-1}(c x)\right ) \]
(6*a*d^2*x^4 + 8*a*d*e*x^6 + 3*a*e^2*x^8 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) + c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/(105*c^8) + b*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x])/24
Time = 0.50 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6855, 27, 1578, 1195, 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.
| \(\displaystyle \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {1-c^2 x^2}}dx+\frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (-\frac {3 e^2 \left (1-c^2 x^2\right )^{5/2}}{c^6}+\frac {e \left (8 d c^2+9 e\right ) \left (1-c^2 x^2\right )^{3/2}}{c^6}+\frac {\left (-6 d^2 c^4-16 d e c^2-9 e^2\right ) \sqrt {1-c^2 x^2}}{c^6}+\frac {6 d^2 c^4+8 d e c^2+3 e^2}{c^6 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {2 e \left (1-c^2 x^2\right )^{5/2} \left (8 c^2 d+9 e\right )}{5 c^8}+\frac {6 e^2 \left (1-c^2 x^2\right )^{7/2}}{7 c^8}+\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{3 c^8}-\frac {2 \sqrt {1-c^2 x^2} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{c^8}\right )\) |
(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-2*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2) *Sqrt[1 - c^2*x^2])/c^8 + (2*(6*c^4*d^2 + 16*c^2*d*e + 9*e^2)*(1 - c^2*x^2 )^(3/2))/(3*c^8) - (2*e*(8*c^2*d + 9*e)*(1 - c^2*x^2)^(5/2))/(5*c^8) + (6* e^2*(1 - c^2*x^2)^(7/2))/(7*c^8)))/48 + (d^2*x^4*(a + b*ArcSech[c*x]))/4 + (d*e*x^6*(a + b*ArcSech[c*x]))/3 + (e^2*x^8*(a + b*ArcSech[c*x]))/8
3.2.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[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] && Int egerQ[(m - 1)/2]
Int[((a_.) + ArcSech[(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]}, Si mp[(a + b*ArcSech[c*x]) u, x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; Fre eQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2 *p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.74 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} x^{4} d^{2}\right )+\frac {b \left (\frac {c^{4} \operatorname {arcsech}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arcsech}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) d^{2} x^{4} c^{4}}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 c^{4} d^{2}+72 c^{2} e^{2} x^{2}+448 c^{2} d e +144 e^{2}\right )}{2520 c^{3}}\right )}{c^{4}}\) | \(198\) |
| derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\operatorname {arcsech}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\operatorname {arcsech}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \operatorname {arcsech}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-105 c^{8} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+168 c^{6} d \,e^{3} \sqrt {-c^{2} x^{2}+1}\, x^{4}+45 e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}+420 c^{4} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}+224 \sqrt {-c^{2} x^{2}+1}\, c^{4} d \,e^{3} x^{2}+54 e^{4} c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {-c^{2} x^{2}+1}+72 e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+144 e^{4} \sqrt {-c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{4}}\) | \(395\) |
| default | \(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\operatorname {arcsech}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\operatorname {arcsech}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \operatorname {arcsech}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-105 c^{8} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+168 c^{6} d \,e^{3} \sqrt {-c^{2} x^{2}+1}\, x^{4}+45 e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}+420 c^{4} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}+224 \sqrt {-c^{2} x^{2}+1}\, c^{4} d \,e^{3} x^{2}+54 e^{4} c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {-c^{2} x^{2}+1}+72 e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+144 e^{4} \sqrt {-c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{4}}\) | \(395\) |
a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*x^4*d^2)+b/c^4*(1/8*c^4*arcsech(c*x)*e^2*x^ 8+1/3*c^4*arcsech(c*x)*d*e*x^6+1/4*arcsech(c*x)*d^2*x^4*c^4-1/2520/c^3*(-( c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+21 0*c^6*d^2*x^2+54*c^4*e^2*x^4+224*c^4*d*e*x^2+420*c^4*d^2+72*c^2*e^2*x^2+44 8*c^2*d*e+144*e^2))
Time = 0.30 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.82 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \, {\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (45 \, b c^{6} e^{2} x^{7} + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} + 4 \, {\left (105 \, b c^{4} d^{2} + 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \]
1/2520*(315*a*c^7*e^2*x^8 + 840*a*c^7*d*e*x^6 + 630*a*c^7*d^2*x^4 + 105*(3 *b*c^7*e^2*x^8 + 8*b*c^7*d*e*x^6 + 6*b*c^7*d^2*x^4)*log((c*x*sqrt(-(c^2*x^ 2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (45*b*c^6*e^2*x^7 + 6*(28*b*c^6*d*e + 9*b* c^4*e^2)*x^5 + 2*(105*b*c^6*d^2 + 112*b*c^4*d*e + 36*b*c^2*e^2)*x^3 + 4*(1 05*b*c^4*d^2 + 112*b*c^2*d*e + 36*b*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) )/c^7
Time = 1.35 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.19 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {asech}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {asech}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {asech}{\left (c x \right )}}{8} - \frac {b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{15 c^{2}} - \frac {b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{56 c^{2}} - \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} - \frac {4 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {3 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{140 c^{4}} - \frac {8 b d e \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} - \frac {b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{6}} - \frac {2 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*asec h(c*x)/4 + b*d*e*x**6*asech(c*x)/3 + b*e**2*x**8*asech(c*x)/8 - b*d**2*x** 2*sqrt(-c**2*x**2 + 1)/(12*c**2) - b*d*e*x**4*sqrt(-c**2*x**2 + 1)/(15*c** 2) - b*e**2*x**6*sqrt(-c**2*x**2 + 1)/(56*c**2) - b*d**2*sqrt(-c**2*x**2 + 1)/(6*c**4) - 4*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(45*c**4) - 3*b*e**2*x**4 *sqrt(-c**2*x**2 + 1)/(140*c**4) - 8*b*d*e*sqrt(-c**2*x**2 + 1)/(45*c**6) - b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(35*c**6) - 2*b*e**2*sqrt(-c**2*x**2 + 1)/(35*c**8), Ne(c, 0)), ((a + oo*b)*(d**2*x**4/4 + d*e*x**6/3 + e**2*x**8 /8), True))
Time = 0.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.88 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arsech}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{7}}\right )} b e^{2} \]
1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arcsech(c*x) + (c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) - 1))/c^3)*b*d^2 + 1/45*(15*x^6*arcsech(c*x) - (3*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) - 10*c^2* x^3*(1/(c^2*x^2) - 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2) - 1))/c^5)*b*d*e + 1/2 80*(35*x^8*arcsech(c*x) + (5*c^6*x^7*(1/(c^2*x^2) - 1)^(7/2) - 21*c^4*x^5* (1/(c^2*x^2) - 1)^(5/2) + 35*c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) - 35*x*sqrt(1 /(c^2*x^2) - 1))/c^7)*b*e^2
\[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]