3.140 \(\int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx\)

Optimal. Leaf size=1006 \[ -\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b e \left (2 d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{45 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{75 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b \left (8 c^2 d-e\right ) \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{75 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}-\frac {8 b e \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{45 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}+\frac {8 b e^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^3 \sqrt {c^2 x^2}}+\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \left (4 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{25 d x^4 \sqrt {c^2 x^2}}-\frac {8 e^2 \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {4 e \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {8 b e^2 \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}} \]

[Out]

-1/5*(a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/d/x^5+4/15*e*(a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/d^2/x^3-8/15*e^2*(a+b*ar
csec(c*x))*(e*x^2+d)^(1/2)/d^3/x+8/15*b*c*e^2*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^3/(c^2*x^2)^(1/2)-4/45*b*c*e
*(2*c^2*d+e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^3/(c^2*x^2)^(1/2)+1/75*b*c*(8*c^4*d^2+3*c^2*d*e-2*e^2)*(c^2*x
^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^3/(c^2*x^2)^(1/2)+1/25*b*c*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^4/(c^2*x^2)^(1/
2)-4/45*b*c*e*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/x^2/(c^2*x^2)^(1/2)+1/75*b*c*(4*c^2*d+e)*(c^2*x^2-1)^(1/2)
*(e*x^2+d)^(1/2)/d^2/x^2/(c^2*x^2)^(1/2)-8/15*b*c^2*e^2*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(
e*x^2+d)^(1/2)/d^3/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)+4/45*b*c^2*e*(2*c^2*d+e)*x*EllipticE(c*
x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^3/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)
-1/75*b*c^2*(8*c^4*d^2+3*c^2*d*e-2*e^2)*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d
^3/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)+1/75*b*c^2*(8*c^2*d-e)*(c^2*d+e)*x*EllipticF(c*x,(-e/c^
2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)-8/45*b*
c^2*e*(c^2*d+e)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^
2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)+8/15*b*e^2*(c^2*d+e)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*
x^2/d)^(1/2)/d^3/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.78, antiderivative size = 1006, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {271, 264, 5238, 12, 6742, 475, 583, 524, 427, 426, 424, 421, 419, 21, 423} \[ -\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b e \left (2 d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{45 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{75 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b \left (8 c^2 d-e\right ) \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{75 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}-\frac {8 b e \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right ) c^2}{45 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}+\frac {8 b e^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^3 \sqrt {c^2 x^2}}+\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \left (4 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{25 d x^4 \sqrt {c^2 x^2}}-\frac {8 e^2 \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {4 e \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {8 b e^2 \left (d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSec[c*x])/(x^6*Sqrt[d + e*x^2]),x]

[Out]

(8*b*c*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(15*d^3*Sqrt[c^2*x^2]) - (4*b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x
^2]*Sqrt[d + e*x^2])/(45*d^3*Sqrt[c^2*x^2]) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d +
e*x^2])/(75*d^3*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(25*d*x^4*Sqrt[c^2*x^2]) - (4*b*c*e
*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*x^2*Sqrt[c^2*x^2]) + (b*c*(4*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d
+ e*x^2])/(75*d^2*x^2*Sqrt[c^2*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(5*d*x^5) + (4*e*Sqrt[d + e*x^2]
*(a + b*ArcSec[c*x]))/(15*d^2*x^3) - (8*e^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(15*d^3*x) - (8*b*c^2*e^2*x*S
qrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(15*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2
]*Sqrt[1 + (e*x^2)/d]) + (4*b*c^2*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -
(e/(c^2*d))])/(45*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*(8*c^4*d^2 + 3*c^2*d*e -
2*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d^3*Sqrt[c^2*x^2]*Sqrt[-1
+ c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*El
lipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (8*b*c^2*e*(c^
2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(45*d^2*Sqrt[c^2*x^2]*S
qrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) + (8*b*e^2*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[Ar
cSin[c*x], -(e/(c^2*d))])/(15*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 423

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

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 427

Int[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[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 475

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

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 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 5238

Int[((a_.) + ArcSec[(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*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{15 d^3 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \left (-\frac {3 d^2 \sqrt {d+e x^2}}{x^6 \sqrt {-1+c^2 x^2}}+\frac {4 d e \sqrt {d+e x^2}}{x^4 \sqrt {-1+c^2 x^2}}-\frac {8 e^2 \sqrt {d+e x^2}}{x^2 \sqrt {-1+c^2 x^2}}\right ) \, dx}{15 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2}}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{5 d \sqrt {c^2 x^2}}-\frac {(4 b c e x) \int \frac {\sqrt {d+e x^2}}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{15 d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 x\right ) \int \frac {\sqrt {d+e x^2}}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-4 c^2 d-e-3 c^2 e x^2}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{25 d \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {-2 c^2 d-e-c^2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^2 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {-e+c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-8 c^4 d^2-3 c^2 d e+2 e^2-c^2 e \left (4 c^2 d+e\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {-c^2 d e+c^2 e \left (2 c^2 d+e\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^3 x\right ) \int \frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-c^2 d e \left (4 c^2 d+e\right )+c^2 e \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e^2 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}+\frac {\left (4 b c^3 e \left (2 c^2 d+e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {\left (b c^3 \left (8 c^2 d-e\right ) \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e^2 x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (4 b c^3 e \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 b c^3 e \left (c^2 d+e\right ) 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}{45 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c e^2 \left (c^2 d+e\right ) 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}{15 d^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 b c^3 e^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (4 b c^3 e \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 \left (8 c^2 d-e\right ) \left (c^2 d+e\right ) 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}{75 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (8 b c^3 e \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \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}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c e^2 \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \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}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {8 b c^2 e^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {4 b c^2 e \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {8 b c^2 e \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e^2 \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 \left (8 c^2 d-e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \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}{75 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {8 b c^2 e^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {4 b c^2 e \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 \left (8 c^2 d-e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b c^2 e \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e^2 \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.74, size = 329, normalized size = 0.33 \[ \frac {\sqrt {d+e x^2} \left (-15 a \left (3 d^2-4 d e x^2+8 e^2 x^4\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (-d e x^2 \left (31 c^2 x^2+17\right )+3 d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+94 e^2 x^4\right )-15 b \sec ^{-1}(c x) \left (3 d^2-4 d e x^2+8 e^2 x^4\right )\right )}{225 d^3 x^5}-\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (24 c^4 d^2-31 c^2 d e+94 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (24 c^6 d^3-19 c^4 d^2 e+77 c^2 d e^2+120 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^3 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSec[c*x])/(x^6*Sqrt[d + e*x^2]),x]

[Out]

(Sqrt[d + e*x^2]*(-15*a*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(94*e^2*x^4 - d*e*x^2*(1
7 + 31*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4)*ArcSec[c*x]))/(225
*d^3*x^5) - ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 - 31*c^2*d*e + 94*e^2)
*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (24*c^6*d^3 - 19*c^4*d^2*e + 77*c^2*d*e^2 + 120*e^3)*Ellip
ticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^3*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{e x^{8} + d x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^6/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)/(e*x^8 + d*x^6), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsec}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^6/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)/(sqrt(e*x^2 + d)*x^6), x)

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maple [F]  time = 6.31, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{x^{6} \sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsec(c*x))/x^6/(e*x^2+d)^(1/2),x)

[Out]

int((a+b*arcsec(c*x))/x^6/(e*x^2+d)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^6/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/15*a*(8*sqrt(e*x^2 + d)*e^2/(d^3*x) - 4*sqrt(e*x^2 + d)*e/(d^2*x^3) + 3*sqrt(e*x^2 + d)/(d*x^5)) - 1/15*(15
*d^3*x^5*integrate((15*c^2*d^3*x^2*log(c) - 15*d^3*log(c) - (8*c^2*e^3*x^8 + 4*c^2*d*e^2*x^6 - c^2*d^2*e*x^4 -
3*(5*c^2*log(c) - c^2)*d^3*x^2 + 15*d^3*log(c))*e^(log(c*x + 1) + log(c*x - 1)) + 15*(c^2*d^3*x^2 - d^3 + (c^
2*d^3*x^2 - d^3)*e^(log(c*x + 1) + log(c*x - 1)))*log(x))/((c^2*d^3*x^8 - d^3*x^6 + (c^2*d^3*x^8 - d^3*x^6)*e^
(log(c*x + 1) + log(c*x - 1)))*sqrt(e*x^2 + d)), x) + (8*e^2*x^4 - 4*d*e*x^2 + 3*d^2)*sqrt(e*x^2 + d)*arctan(s
qrt(c*x + 1)*sqrt(c*x - 1)))*b/(d^3*x^5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^6\,\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(1/(c*x)))/(x^6*(d + e*x^2)^(1/2)),x)

[Out]

int((a + b*acos(1/(c*x)))/(x^6*(d + e*x^2)^(1/2)), 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((a+b*asec(c*x))/x**6/(e*x**2+d)**(1/2),x)

[Out]

Timed out

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