Integrand size = 23, antiderivative size = 329 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac {2 b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^2} \]
-1/3*d*(e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/e^2+1/5*(e*x^2+d)^(5/2)*(a+b*arc sech(c*x))/e^2+1/120*b*(15*c^4*d^2-10*c^2*d*e-9*e^2)*arctan(e^(1/2)*(-c^2* x^2+1)^(1/2)/c/(e*x^2+d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^5/e^(3/2 )+2/15*b*d^(5/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c *x+1))^(1/2)*(c*x+1)^(1/2)/e^2-1/20*b*(e*x^2+d)^(3/2)*(1/(c*x+1))^(1/2)*(c *x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/e-1/120*b*(c^2*d+9*e)*(1/(c*x+1))^(1/2) *(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/c^4/e
Time = 23.45 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.11 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {\sqrt {d+e x^2} \left (8 a c^4 \left (2 d^2-d e x^2-3 e^2 x^4\right )+b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (7 d+6 e x^2\right )\right )+8 b c^4 \left (2 d^2-d e x^2-3 e^2 x^4\right ) \text {sech}^{-1}(c x)\right )}{120 c^4 e^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+16 c^7 d^{5/2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{120 c^7 e^2 (-1+c x) \sqrt {d+e x^2}} \]
-1/120*(Sqrt[d + e*x^2]*(8*a*c^4*(2*d^2 - d*e*x^2 - 3*e^2*x^4) + b*e*Sqrt[ (1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(7*d + 6*e*x^2)) + 8*b*c^4*(2*d^ 2 - d*e*x^2 - 3*e^2*x^4)*ArcSech[c*x]))/(c^4*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(15*c^4*d^ 2 - 10*c^2*d*e - 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt [e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 16*c^7*d^(5/2)*S qrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(12 0*c^7*e^2*(-1 + c*x)*Sqrt[d + e*x^2])
Time = 0.58 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.85, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6855, 27, 435, 171, 27, 171, 27, 175, 66, 104, 218, 220}
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.
| \(\displaystyle \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {\left (2 d-3 e x^2\right ) \left (e x^2+d\right )^{3/2}}{15 e^2 x \sqrt {1-c^2 x^2}}dx+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (2 d-3 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x \sqrt {1-c^2 x^2}}dx}{15 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (2 d-3 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x^2 \sqrt {1-c^2 x^2}}dx^2}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}-\frac {\int -\frac {\sqrt {e x^2+d} \left (8 c^2 d^2-e \left (d c^2+9 e\right ) x^2\right )}{2 x^2 \sqrt {1-c^2 x^2}}dx^2}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {\sqrt {e x^2+d} \left (8 c^2 d^2-e \left (d c^2+9 e\right ) x^2\right )}{x^2 \sqrt {1-c^2 x^2}}dx^2}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}-\frac {\int -\frac {16 d^3 c^4+e \left (15 d^2 c^4-10 d e c^2-9 e^2\right ) x^2}{2 x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\int \frac {16 d^3 c^4+e \left (15 d^2 c^4-10 d e c^2-9 e^2\right ) x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {16 c^4 d^3 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {16 c^4 d^3 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+2 e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {32 c^4 d^3 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}+2 e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {32 c^4 d^3 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {e} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {-\frac {2 \sqrt {e} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}-32 c^4 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^2}\) |
-1/3*(d*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/e^2 + ((d + e*x^2)^(5/2)*( a + b*ArcSech[c*x]))/(5*e^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((3*e *Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + ((e*(c^2*d + 9*e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/c^2 + ((-2*Sqrt[e]*(15*c^4*d^2 - 10*c^2*d*e - 9 *e^2)*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/c - 32*c^4* d^(5/2)*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*c^2))/(4* c^2)))/(30*e^2)
3.2.31.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(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]))
\[\int x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
Time = 0.74 (sec) , antiderivative size = 1669, normalized size of antiderivative = 5.07 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
[1/480*(16*b*c^5*d^(5/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 - 4*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt (-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) + (15*b*c^4*d^2 - 10*b*c^2*d*e - 9*b*e^2)*sqrt(-e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c ^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d - c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(-e) *sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2) + 32*(3*b*c^5*e^2*x^4 + b*c^5*d*e*x ^2 - 2*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(24*a*c^5*e^2*x^4 + 8*a*c^5*d*e*x^2 - 16*a*c^5*d^2 - (6*b* c^4*e^2*x^3 + (7*b*c^4*d*e + 9*b*c^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2) ))*sqrt(e*x^2 + d))/(c^5*e^2), 1/240*(8*b*c^5*d^(5/2)*log(((c^4*d^2 - 6*c^ 2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 - 4*((c^3*d - c*e)*x^3 - 2*c*d*x) *sqrt(e*x^2 + d)*sqrt(d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) + (1 5*b*c^4*d^2 - 10*b*c^2*d*e - 9*b*e^2)*sqrt(e)*arctan(1/2*(2*c^2*e*x^3 + (c ^2*d - e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))/(c^2*e ^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e)) + 16*(3*b*c^5*e^2*x^4 + b*c^5*d*e*x^2 - 2*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(24*a*c^5*e^2*x^4 + 8*a*c^5*d*e*x^2 - 16*a*c^5*d^2 - (6*b*c^ 4*e^2*x^3 + (7*b*c^4*d*e + 9*b*c^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))) *sqrt(e*x^2 + d))/(c^5*e^2), 1/480*(32*b*c^5*sqrt(-d)*d^2*arctan(-1/2*((c^ 3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt(-(c^2*x^2 - 1)/...
\[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Exception generated. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^3\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]