3.149 \(\int \frac {(d+e x^2)^{3/2} (a+b \text {sech}^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=556 \[ \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 x}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {\frac {e x^2}{d}+1}} \]

[Out]

-1/7*(e*x^2+d)^(5/2)*(a+b*arcsech(c*x))/d/x^7+2/35*e*(e*x^2+d)^(5/2)*(a+b*arcsech(c*x))/d^2/x^5+1/1225*b*(30*c
^2*d+11*e)*(e*x^2+d)^(3/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/x^5+1/49*b*(e*x^2+d)^(5/2)*(1/
(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/x^7+1/3675*b*(120*c^4*d^2+159*c^2*d*e-37*e^2)*(1/(c*x+1))^(1
/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d/x^3+1/3675*b*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-2
47*e^3)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/x+1/3675*b*c*(240*c^6*d^3+528*c
^4*d^2*e+193*c^2*d*e^2-247*e^3)*EllipticE(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x^2+d)^(1/2
)/d^2/(1+e*x^2/d)^(1/2)-2/3675*b*(c^2*d+e)*(120*c^6*d^3+204*c^4*d^2*e+17*c^2*d*e^2-105*e^3)*EllipticF(c*x,(-e/
c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(1+e*x^2/d)^(1/2)/c/d^2/(e*x^2+d)^(1/2)

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Rubi [A]  time = 0.76, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {271, 264, 6301, 12, 580, 583, 524, 426, 424, 421, 419} \[ \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3675 d x^3}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d+e\right ) \left (204 c^4 d^2 e+120 c^6 d^3+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {\frac {e x^2}{d}+1}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^8,x]

[Out]

(b*(120*c^4*d^2 + 159*c^2*d*e - 37*e^2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/
(3675*d*x^3) + (b*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*S
qrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(3675*d^2*x) + (b*(30*c^2*d + 11*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[
1 - c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*x^5) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*
x^2)^(5/2))/(49*d*x^7) - ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*Ar
cSech[c*x]))/(35*d^2*x^5) + (b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[(1 + c*x)^(-1)]*
Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2
*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 105*e^3)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 + (e*
x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*c*d^2*Sqrt[d + e*x^2])

Rule 12

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 6301

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]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[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]] /; FreeQ[{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]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8 \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 )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8 \sqrt {1-c^2 x^2}} \, dx}{35 d^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (-d \left (30 c^2 d+11 e\right )-\left (5 c^2 d-14 e\right ) e x^2\right )}{x^6 \sqrt {1-c^2 x^2}} \, dx}{245 d^2}\\ &=\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2} \left (-d \left (120 c^4 d^2+159 c^2 d e-37 e^2\right )-2 e \left (15 c^4 d^2+18 c^2 d e-35 e^2\right ) x^2\right )}{x^4 \sqrt {1-c^2 x^2}} \, dx}{1225 d^2}\\ &=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right )-e \left (120 c^6 d^3+249 c^4 d^2 e+71 c^2 d e^2-210 e^3\right ) x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^2}\\ &=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {d e \left (120 c^6 d^3+249 c^4 d^2 e+71 c^2 d e^2-210 e^3\right )-c^2 d e \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x^2}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^3}\\ &=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{3675 d^2}-\frac {\left (2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^2}\\ &=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {\left (b c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{3675 d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3675 d^2 \sqrt {d+e x^2}}\\ &=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3675 c d^2 \sqrt {d+e x^2}}\\ \end {align*}

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Mathematica [C]  time = 7.81, size = 728, normalized size = 1.31 \[ \frac {\frac {105 a \left (2 e x^2-5 d\right ) \left (d+e x^2\right )^3}{x^7}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \left (d e^2 x^4 \left (193 c^2 x^2+71\right )+3 d^2 e x^2 \left (176 c^4 x^4+83 c^2 x^2+61\right )+15 d^3 \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )-247 e^3 x^6\right )}{x^7}+\frac {b \sqrt {\frac {1-c x}{c x+1}} \left (-\left (c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \left (d+e x^2\right )\right )-\frac {i (c x+1) \left (c \sqrt {d}-i \sqrt {e}\right )^2 \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}-i \sqrt {e}\right )}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}} \left (\left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+2 \sqrt {e} \left (240 i c^5 d^{5/2}-360 c^4 d^2 \sqrt {e}+48 i c^3 d^{3/2} e-207 c^2 d e^{3/2}-173 i c \sqrt {d} e^2+210 e^{5/2}\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {(c x-1) \left (c \sqrt {d}-i \sqrt {e}\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}}}\right )}{c}+\frac {105 b \text {sech}^{-1}(c x) \left (2 e x^2-5 d\right ) \left (d+e x^2\right )^3}{x^7}}{3675 d^2 \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^8,x]

[Out]

((105*a*(d + e*x^2)^3*(-5*d + 2*e*x^2))/x^7 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)*(-247*e^3*x^6
+ d*e^2*x^4*(71 + 193*c^2*x^2) + 3*d^2*e*x^2*(61 + 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(5 + 6*c^2*x^2 + 8*c^4*
x^4 + 16*c^6*x^6)))/x^7 + (105*b*(d + e*x^2)^3*(-5*d + 2*e*x^2)*ArcSech[c*x])/x^7 + (b*Sqrt[(1 - c*x)/(1 + c*x
)]*(-(c^2*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*(d + e*x^2)) - (I*(c*Sqrt[d] - I*Sqrt[e])^2*
(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x
))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*EllipticE[I*A
rcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt
[d] - I*Sqrt[e])^2] + 2*Sqrt[e]*((240*I)*c^5*d^(5/2) - 360*c^4*d^2*Sqrt[e] + (48*I)*c^3*d^(3/2)*e - 207*c^2*d*
e^(3/2) - (173*I)*c*Sqrt[d]*e^2 + 210*e^(5/2))*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] +
I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[
e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(3675*d^2*Sqrt[d + e*x^2])

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arsech}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/x^8,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arcsech(c*x))*sqrt(e*x^2 + d)/x^8, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arcsech(c*x) + a)/x^8, x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/x^8,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/x^8,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{35} \, a {\left (\frac {2 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e}{d^{2} x^{5}} - \frac {5 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} - \frac {1}{105} \, b {\left (\frac {{\left (8 \, e^{3} x^{7} - 4 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + 15 \, d^{3} x - 7 \, {\left (2 \, e^{3} x^{5} - d e^{2} x^{3} - 3 \, d^{2} e x\right )} x^{2}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{d^{2} x^{8}} + 105 \, \int \frac {{\left (105 \, c^{2} d^{3} x^{2} \log \relax (c) - 105 \, d^{3} \log \relax (c) + 105 \, {\left (c^{2} d^{2} e x^{2} \log \relax (c) - d^{2} e \log \relax (c)\right )} x^{2} - {\left (8 \, c^{2} e^{3} x^{8} - 4 \, c^{2} d e^{2} x^{6} + 3 \, c^{2} d^{2} e x^{4} - 15 \, {\left (7 \, d^{3} \log \relax (c) - d^{3}\right )} c^{2} x^{2} + 105 \, d^{3} \log \relax (c) - 7 \, {\left (2 \, c^{2} e^{3} x^{6} - c^{2} d e^{2} x^{4} + 3 \, {\left (5 \, d^{2} e \log \relax (c) - d^{2} e\right )} c^{2} x^{2} - 15 \, d^{2} e \log \relax (c)\right )} x^{2} - 210 \, {\left (c^{2} d^{3} x^{2} - d^{3} + {\left (c^{2} d^{2} e x^{2} - d^{2} e\right )} x^{2}\right )} \log \left (\sqrt {x}\right )\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} + 210 \, {\left (c^{2} d^{3} x^{2} - d^{3} + {\left (c^{2} d^{2} e x^{2} - d^{2} e\right )} x^{2}\right )} \log \left (\sqrt {x}\right )\right )} \sqrt {e x^{2} + d}}{105 \, {\left ({\left (c^{2} d^{2} x^{2} - d^{2}\right )} x^{8} + {\left (c^{2} d^{2} x^{2} - d^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right ) + 8 \, \log \relax (x)\right )}\right )}}\,{d x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/x^8,x, algorithm="maxima")

[Out]

1/35*a*(2*(e*x^2 + d)^(5/2)*e/(d^2*x^5) - 5*(e*x^2 + d)^(5/2)/(d*x^7)) - 1/105*b*((8*e^3*x^7 - 4*d*e^2*x^5 + 3
*d^2*e*x^3 + 15*d^3*x - 7*(2*e^3*x^5 - d*e^2*x^3 - 3*d^2*e*x)*x^2)*sqrt(e*x^2 + d)*log(sqrt(c*x + 1)*sqrt(-c*x
+ 1) + 1)/(d^2*x^8) + 105*integrate(1/105*(105*c^2*d^3*x^2*log(c) - 105*d^3*log(c) + 105*(c^2*d^2*e*x^2*log(c
) - d^2*e*log(c))*x^2 - (8*c^2*e^3*x^8 - 4*c^2*d*e^2*x^6 + 3*c^2*d^2*e*x^4 - 15*(7*d^3*log(c) - d^3)*c^2*x^2 +
105*d^3*log(c) - 7*(2*c^2*e^3*x^6 - c^2*d*e^2*x^4 + 3*(5*d^2*e*log(c) - d^2*e)*c^2*x^2 - 15*d^2*e*log(c))*x^2
- 210*(c^2*d^3*x^2 - d^3 + (c^2*d^2*e*x^2 - d^2*e)*x^2)*log(sqrt(x)))*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)
) + 210*(c^2*d^3*x^2 - d^3 + (c^2*d^2*e*x^2 - d^2*e)*x^2)*log(sqrt(x)))*sqrt(e*x^2 + d)/((c^2*d^2*x^2 - d^2)*x
^8 + (c^2*d^2*x^2 - d^2)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1) + 8*log(x))), x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^(3/2)*(a + b*acosh(1/(c*x))))/x^8,x)

[Out]

int(((d + e*x^2)^(3/2)*(a + b*acosh(1/(c*x))))/x^8, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*asech(c*x))/x**8,x)

[Out]

Timed out

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