3.150 \(\int \frac{a+b \sec ^{-1}(c x)}{x^4 (d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=701 \[ -\frac{8 b e^2 x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (2 c^2 d-e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{2 b c \sqrt{c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]

[Out]

(2*b*c*(c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^3*Sqrt[c^2*x^2]) - (4*b*c*e*Sqrt[-1 + c^2*x^2]*Sqr
t[d + e*x^2])/(3*d^3*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*x^2*Sqrt[c^2*x^2]) - (a
+ b*ArcSec[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcSec[c*x]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a
+ b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) - (2*b*c^2*(c^2*d - e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*Elliptic
E[ArcSin[c*x], -(e/(c^2*d))])/(9*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (4*b*c^2*e*x*Sqrt
[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sq
rt[1 + (e*x^2)/d]) + (b*c^2*(2*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))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (4*b*c^2*e*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (
e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (8*
b*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[
-1 + c^2*x^2]*Sqrt[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.39178, antiderivative size = 701, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {271, 191, 5238, 12, 6742, 421, 419, 480, 583, 524, 427, 426, 424, 493} \[ \frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{8 b e^2 x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 b c \sqrt{c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (2 c^2 d-e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSec[c*x])/(x^4*(d + e*x^2)^(3/2)),x]

[Out]

(2*b*c*(c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^3*Sqrt[c^2*x^2]) - (4*b*c*e*Sqrt[-1 + c^2*x^2]*Sqr
t[d + e*x^2])/(3*d^3*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*x^2*Sqrt[c^2*x^2]) - (a
+ b*ArcSec[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcSec[c*x]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a
+ b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) - (2*b*c^2*(c^2*d - e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*Elliptic
E[ArcSin[c*x], -(e/(c^2*d))])/(9*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (4*b*c^2*e*x*Sqrt
[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sq
rt[1 + (e*x^2)/d]) + (b*c^2*(2*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))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (4*b*c^2*e*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (
e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (8*
b*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[
-1 + c^2*x^2]*Sqrt[d + e*x^2])

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 191

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

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 12

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

Rule 6742

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

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 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 480

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 + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

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 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 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 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 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 493

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2}}\\ &=-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c x) \int \left (\frac{8 e^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2}{x^4 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}+\frac{4 d e}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\right ) \, dx}{3 d^3 \sqrt{c^2 x^2}}\\ &=-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{1}{x^4 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d \sqrt{c^2 x^2}}-\frac{(4 b c e x) \int \frac{1}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^2 \sqrt{c^2 x^2}}-\frac{\left (8 b c e^2 x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2}}\\ &=-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-2 \left (c^2 d-e\right )-c^2 e x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}+\frac{(4 b c e x) \int \frac{c^2 e x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2}}-\frac{\left (8 b c e^2 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}{3 d^3 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-c^2 d e+2 c^2 \left (c^2 d-e\right ) e x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^3 \sqrt{c^2 x^2}}+\frac{\left (4 b c^3 e^2 x\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2}}-\frac{\left (8 b c e^2 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}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{8 b e^2 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 )}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{\left (2 b c^3 \left (c^2 d-e\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{9 d^3 \sqrt{c^2 x^2}}+\frac{\left (b c^3 \left (2 c^2 d-e\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}+\frac{\left (4 b c^3 e x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2}}-\frac{\left (4 b c^3 e x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^2 \sqrt{c^2 x^2}}\\ &=\frac{2 b c \left (c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{8 b e^2 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 )}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{\left (2 b c^3 \left (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}{9 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (4 b c^3 e x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (b c^3 \left (2 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}{9 d^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{\left (4 b c^3 e 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}{3 d^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{8 b e^2 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 )}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{\left (2 b c^3 \left (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}{9 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 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}{3 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 (2 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}{9 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{\left (4 b c^3 e 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}{3 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{2 b c^2 \left (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 )}{9 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 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 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 (2 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 )}{9 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{4 b c^2 e 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 )}{3 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}-\frac{8 b e^2 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 )}{3 d^3 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.71654, size = 292, normalized size = 0.42 \[ \frac{-3 a \left (d^2-4 d e x^2-8 e^2 x^4\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (d^2 \left (2 c^2 x^2+1\right )+d e x^2 \left (2 c^2 x^2-13\right )-14 e^2 x^4\right )-3 b \sec ^{-1}(c x) \left (d^2-4 d e x^2-8 e^2 x^4\right )}{9 d^3 x^3 \sqrt{d+e x^2}}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (\left (-2 c^4 d^2+13 c^2 d e+24 e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )+2 c^2 d \left (c^2 d-7 e\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )\right )}{9 \sqrt{-c^2} d^3 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSec[c*x])/(x^4*(d + e*x^2)^(3/2)),x]

[Out]

(-3*a*(d^2 - 4*d*e*x^2 - 8*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-14*e^2*x^4 + d*e*x^2*(-13 + 2*c^2*x^2) + d
^2*(1 + 2*c^2*x^2)) - 3*b*(d^2 - 4*d*e*x^2 - 8*e^2*x^4)*ArcSec[c*x])/(9*d^3*x^3*Sqrt[d + e*x^2]) - ((I/9)*b*c*
Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 7*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2
*d))] + (-2*c^4*d^2 + 13*c^2*d*e + 24*e^2)*EllipticF[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])

________________________________________________________________________________________

Maple [F]  time = 1.481, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsec} \left (cx\right )}{{x}^{4}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsec(c*x))/x^4/(e*x^2+d)^(3/2),x)

[Out]

int((a+b*arcsec(c*x))/x^4/(e*x^2+d)^(3/2),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^4/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{e^{2} x^{8} + 2 \, d e x^{6} + d^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^4/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)/(e^2*x^8 + 2*d*e*x^6 + d^2*x^4), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asec(c*x))/x**4/(e*x**2+d)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))/x^4/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)/((e*x^2 + d)^(3/2)*x^4), x)