\(\int \frac {a+b \sec ^{-1}(c x)}{x^2 (d+e x^2)^{5/2}} \, dx\) [160]

Optimal result

Integrand size = 23, antiderivative size = 631 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

(-a-b*arcsec(c*x))/d/x/(e*x^2+d)^(3/2)-4/3*e*x*(a+b*arcsec(c*x))/d^2/(e*x^2+d)^(3/2)-8/3*e*x*(a+b*arcsec(c*x))
/d^3/(e*x^2+d)^(1/2)-b*c*e*(c^2*x^2-1)^(1/2)/d^2/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)-4/3*b*c*e^2*x^2*(c^
2*x^2-1)^(1/2)/d^3/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)+b*c*(c^2*d+2*e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)
/d^3/(c^2*d+e)/(c^2*x^2)^(1/2)+4/3*b*c^2*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*d+e)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-b*c^2*(c^2*d+2*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*d+e)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)
+b*c^2*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)+8/3*b*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] (verified)

Time = 1.03 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {277, 198, 197, 5346, 12, 6874, 425, 21, 438, 437, 435, 483, 597, 538, 432, 430, 482, 434} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {c^2 x^2-1}}{3 d^3 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{d^3 \sqrt {c^2 x^2} \left (c^2 d+e\right )}-\frac {b c e \sqrt {c^2 x^2-1}}{d^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]

[In]

Int[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(5/2)),x]

[Out]

-((b*c*e*Sqrt[-1 + c^2*x^2])/(d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) - (4*b*c*e^2*x^2*Sqrt[-1 + c^2*x
^2])/(3*d^3*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]
)/(d^3*(c^2*d + e)*Sqrt[c^2*x^2]) - (a + b*ArcSec[c*x])/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*ArcSec[c*x]))/
(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (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*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*S
qrt[1 + (e*x^2)/d]) - (b*c^2*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2
*d))])/(d^3*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqr
t[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])
+ (8*b*e*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]*Sq
rt[-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 197

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 198

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

Rule 277

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 430

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

Rule 432

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

Rule 434

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 435

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

Rule 437

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

Rule 438

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

Rule 482

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

Rule 483

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

Rule 538

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 597

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 5346

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 6874

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{3 d^3 x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \left (-\frac {12 d e}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 d^2}{x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}-\frac {8 e^2 x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}\right ) \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e 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^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{d \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e 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+2 e-c^2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {c^2 d+c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(8 b c e x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}+\frac {(b c x) \int \frac {-c^2 d e-c^2 e \left (c^2 d+2 e\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {\left (b c^3 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (c^2 d+2 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 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 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (8 b c 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^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (b c^3 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 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 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 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}{d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c 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^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (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 (b c^3 \left (c^2 d+2 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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 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}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (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] (verified)

Result contains complex when optimal does not.

Time = 7.59 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\frac {-a \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (3 c^2 d \left (d+e x^2\right )+e \left (3 d+2 e x^2\right )\right )-b \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right ) \sec ^{-1}(c x)}{3 d^3 \left (c^2 d+e\right ) x \left (d+e x^2\right )^{3/2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (3 c^2 d+2 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (3 c^4 d^2+11 c^2 d e+8 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^3 \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

Integrate[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(5/2)),x]

[Out]

(-(a*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)*(3*c^2*d*(d + e*x
^2) + e*(3*d + 2*e*x^2)) - b*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)*ArcSec[c*x])/(3*d^3*(c^2*d + e)*x*(d
+ e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(3*c^2*d + 2*e)*EllipticE[I*A
rcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (3*c^4*d^2 + 11*c^2*d*e + 8*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/
(c^2*d))]))/(Sqrt[-c^2]*d^3*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

Maple [F]

\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

int((a+b*arcsec(c*x))/x^2/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arcsec(c*x))/x^2/(e*x^2+d)^(5/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, a c^{3} d^{4} + 3 \, a c d^{3} e + 8 \, {\left (a c^{3} d^{2} e^{2} + a c d e^{3}\right )} x^{4} + 12 \, {\left (a c^{3} d^{3} e + a c d^{2} e^{2}\right )} x^{2} + {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + 8 \, {\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{4} + 12 \, {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + {\left (3 \, b c^{3} d^{2} e^{2} + 2 \, b c d e^{3}\right )} x^{4} + {\left (6 \, b c^{3} d^{3} e + 5 \, b c d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left ({\left (3 \, b c^{6} d^{2} e^{2} + 2 \, b c^{4} d e^{3}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + 2 \, b c^{4} d^{2} e^{2}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + 2 \, b c^{4} d^{3} e\right )} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (3 \, b c^{6} d^{2} e^{2} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{2} e^{2} + 8 \, b d e^{3}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{3} e + 8 \, b d^{2} e^{2}\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left ({\left (c^{3} d^{5} e^{2} + c d^{4} e^{3}\right )} x^{5} + 2 \, {\left (c^{3} d^{6} e + c d^{5} e^{2}\right )} x^{3} + {\left (c^{3} d^{7} + c d^{6} e\right )} x\right )}} \]

[In]

integrate((a+b*arcsec(c*x))/x^2/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

-1/3*((3*a*c^3*d^4 + 3*a*c*d^3*e + 8*(a*c^3*d^2*e^2 + a*c*d*e^3)*x^4 + 12*(a*c^3*d^3*e + a*c*d^2*e^2)*x^2 + (3
*b*c^3*d^4 + 3*b*c*d^3*e + 8*(b*c^3*d^2*e^2 + b*c*d*e^3)*x^4 + 12*(b*c^3*d^3*e + b*c*d^2*e^2)*x^2)*arcsec(c*x)
- (3*b*c^3*d^4 + 3*b*c*d^3*e + (3*b*c^3*d^2*e^2 + 2*b*c*d*e^3)*x^4 + (6*b*c^3*d^3*e + 5*b*c*d^2*e^2)*x^2)*sqr
t(c^2*x^2 - 1))*sqrt(e*x^2 + d) - (((3*b*c^6*d^2*e^2 + 2*b*c^4*d*e^3)*x^5 + 2*(3*b*c^6*d^3*e + 2*b*c^4*d^2*e^2
)*x^3 + (3*b*c^6*d^4 + 2*b*c^4*d^3*e)*x)*elliptic_e(arcsin(c*x), -e/(c^2*d)) - ((3*b*c^6*d^2*e^2 + (2*b*c^4 +
9*b*c^2)*d*e^3 + 8*b*e^4)*x^5 + 2*(3*b*c^6*d^3*e + (2*b*c^4 + 9*b*c^2)*d^2*e^2 + 8*b*d*e^3)*x^3 + (3*b*c^6*d^4
+ (2*b*c^4 + 9*b*c^2)*d^3*e + 8*b*d^2*e^2)*x)*elliptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/((c^3*d^5*e^2 +
c*d^4*e^3)*x^5 + 2*(c^3*d^6*e + c*d^5*e^2)*x^3 + (c^3*d^7 + c*d^6*e)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*asec(c*x))/x**2/(e*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsec(c*x))/x^2/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arcsec(c*x))/x^2/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)/((e*x^2 + d)^(5/2)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((a + b*acos(1/(c*x)))/(x^2*(d + e*x^2)^(5/2)),x)

[Out]

int((a + b*acos(1/(c*x)))/(x^2*(d + e*x^2)^(5/2)), x)