3.129 \(\int \frac {(d+e x^2)^{3/2} (a+b \sec ^{-1}(c x))}{x^6} \, dx\)

Optimal. Leaf size=416 \[ -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 b c \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}} \]

[Out]

-1/5*(e*x^2+d)^(5/2)*(a+b*arcsec(c*x))/d/x^5+1/25*b*c*(e*x^2+d)^(3/2)*(c^2*x^2-1)^(1/2)/x^4/(c^2*x^2)^(1/2)+1/
75*b*c*(8*c^4*d^2+23*c^2*d*e+23*e^2)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/(c^2*x^2)^(1/2)+4/75*b*c*(c^2*d+2*e)*
(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/x^2/(c^2*x^2)^(1/2)-1/75*b*c^2*(8*c^4*d^2+23*c^2*d*e+23*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/(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*d+e)*(8*c^4*d^2+19*c^2*d*e+15*e^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/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.54, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {264, 5238, 12, 474, 580, 583, 524, 427, 426, 424, 421, 419} \[ -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {b c \sqrt {c^2 x^2-1} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}+\frac {4 b c \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^6,x]

[Out]

(b*c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d*Sqrt[c^2*x^2]) + (4*b*c*(c^2*
d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*x^2*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2
))/(25*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*d*x^5) - (b*c^2*(8*c^4*d^2 + 23*c^2*d*e
+ 23*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[
-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(8*c^4*d^2 + 19*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt
[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*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 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 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 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 474

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

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int -\frac {\left (d+e x^2\right )^{5/2}}{5 d x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{5 d \sqrt {c^2 x^2}}\\ &=\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-4 d \left (c^2 d+2 e\right )-e \left (c^2 d+5 e\right ) x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{25 d \sqrt {c^2 x^2}}\\ &=\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d \left (8 c^4 d^2+23 c^2 d e+23 e^2\right )+e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right )-c^2 d e \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{75 d \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\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 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {b c^2 \left (8 c^4 d^2+23 c^2 d e+23 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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.68, size = 303, normalized size = 0.73 \[ \frac {\sqrt {d+e x^2} \left (-15 a \left (d+e x^2\right )^2+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (d e x^2 \left (23 c^2 x^2+11\right )+d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+23 e^2 x^4\right )-15 b \sec ^{-1}(c x) \left (d+e x^2\right )^2\right )}{75 d x^5}-\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (8 c^6 d^3+27 c^4 d^2 e+34 c^2 d e^2+15 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{75 \sqrt {-c^2} d \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^6,x]

[Out]

(Sqrt[d + e*x^2]*(-15*a*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(23*e^2*x^4 + d*e*x^2*(11 + 23*c^2*x^2) +
d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(d + e*x^2)^2*ArcSec[c*x]))/(75*d*x^5) - ((I/75)*b*c*Sqrt[1 - 1/(c^2*x
^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^
2*d))] - (8*c^6*d^3 + 27*c^4*d^2*e + 34*c^2*d*e^2 + 15*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))
/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

________________________________________________________________________________________

fricas [F]  time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/x^6,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/x^6,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arcsec(c*x) + a)/x^6, x)

________________________________________________________________________________________

maple [F]  time = 4.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/x^6,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/x^6,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/x^6,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^(3/2)*(a + b*acos(1/(c*x))))/x^6,x)

[Out]

int(((d + e*x^2)^(3/2)*(a + b*acos(1/(c*x))))/x^6, x)

________________________________________________________________________________________

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*asec(c*x))/x**6,x)

[Out]

Timed out

________________________________________________________________________________________