3.85 \(\int \frac{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x}}{x^2} \, dx\)

Optimal. Leaf size=1287 \[ \text{result too large to display} \]

[Out]

-((b*d + c*e)*Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(4*c*d) - (Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(2*x) + (Sq
rt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]
*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(
2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(4*Sqrt[2]*c*d*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c
+ b*x + a*x^2)) + (3*Sqrt[b^2 - 4*a*c]*e*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 -
4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)
/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[2]*Sqrt[d +
e*x]*(c + b*x + a*x^2)) - (Sqrt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 -
4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(
2*Sqrt[2]*c*Sqrt[d + e*x]*(c + b*x + a*x^2)) - ((a*d + b*e)*Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[a + c
/x^2 + b/x]*x*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e*x))/(2*a*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d), ArcSin[(Sqrt[2]*Sqrt[a]*Sq
rt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]], (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b + Sqrt[b^2 - 4*a
*c] - (2*a*d)/e)])/(Sqrt[2]*Sqrt[a]*d*(c + b*x + a*x^2)) + ((b*d + c*e)^2*Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])
*e]*Sqrt[a + c/x^2 + b/x]*x*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e
*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d), ArcSin[(Sqrt
[2]*Sqrt[a]*Sqrt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]], (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b +
Sqrt[b^2 - 4*a*c] - (2*a*d)/e)])/(4*Sqrt[2]*Sqrt[a]*c*d^2*(c + b*x + a*x^2))

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Rubi [A]  time = 5.30239, antiderivative size = 1287, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {1573, 916, 6742, 718, 419, 939, 934, 169, 538, 537, 843, 424} \[ \frac{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) (b d+c e)^2}{4 \sqrt{2} \sqrt{a} c d^2 \left (a x^2+b x+c\right )}+\frac{\sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) (b d+c e)}{4 \sqrt{2} c d \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{\sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) (b d+c e)}{2 \sqrt{2} c \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} (b d+c e)}{4 c d}+\frac{3 \sqrt{b^2-4 a c} e \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{2} \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{(a d+b e) \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right )}{\sqrt{2} \sqrt{a} d \left (a x^2+b x+c\right )}-\frac{\sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x}}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d + c*e)*Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(4*c*d) - (Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(2*x) + (Sq
rt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]
*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(
2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(4*Sqrt[2]*c*d*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c
+ b*x + a*x^2)) + (3*Sqrt[b^2 - 4*a*c]*e*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 -
4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)
/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[2]*Sqrt[d +
e*x]*(c + b*x + a*x^2)) - (Sqrt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 -
4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(
2*Sqrt[2]*c*Sqrt[d + e*x]*(c + b*x + a*x^2)) - ((a*d + b*e)*Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[a + c
/x^2 + b/x]*x*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e*x))/(2*a*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d), ArcSin[(Sqrt[2]*Sqrt[a]*Sq
rt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]], (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b + Sqrt[b^2 - 4*a
*c] - (2*a*d)/e)])/(Sqrt[2]*Sqrt[a]*d*(c + b*x + a*x^2)) + ((b*d + c*e)^2*Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])
*e]*Sqrt[a + c/x^2 + b/x]*x*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e
*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d), ArcSin[(Sqrt
[2]*Sqrt[a]*Sqrt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]], (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b +
Sqrt[b^2 - 4*a*c] - (2*a*d)/e)])/(4*Sqrt[2]*Sqrt[a]*c*d^2*(c + b*x + a*x^2))

Rule 1573

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol
] :> Dist[(x^(2*n*FracPart[p])*(a + b/x^n + c/x^(2*n))^FracPart[p])/(c + b*x^n + a*x^(2*n))^FracPart[p], Int[x
^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[m
n, -n] && EqQ[mn2, 2*mn] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n]

Rule 916

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[((d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(e*(m + 1)), x] - Dist[1/(2*e*(m + 1)), Int[((d +
e*x)^(m + 1)*Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x]
/; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0
] && IntegerQ[2*m] && LtQ[m, -1]

Rule 6742

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

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

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 939

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :>
Simp[(e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)
), x] + Dist[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x)^(m + 1)*Simp[2*d*(c*e*f - c*d*g
+ b*e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b*g)*(m + 2))*x - c*e^2*g*(2*m +
5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g
, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 538

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

Rule 537

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 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]

Rubi steps

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

Mathematica [C]  time = 12.6526, size = 811, normalized size = 0.63 \[ \frac{x \sqrt{a+\frac{c+b x}{x^2}} \left (-\frac{8 c d^3}{x^2}-\frac{8 c e d^2}{x}-\frac{4 (b d+c e) d^2}{x}-\frac{i (d+e x)^{3/2} \sqrt{1-\frac{2 \left (a d^2+e (c e-b d)\right )}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{4 \left (a d^2+e (c e-b d)\right )}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+2} \left (d (b d+c e) \left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-\left (b^2 e d^2+b \left (d \sqrt{\left (b^2-4 a c\right ) e^2}-5 c e^2\right ) d+c e \left (4 a d^2+\sqrt{\left (b^2-4 a c\right ) e^2} d+2 c e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right ),-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+2 e \left (b^2 d^2-2 b c e d+c \left (c e^2-4 a d^2\right )\right ) \Pi \left (\frac{d \left (2 a d-b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (c e-b d)\right )};i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}\right )}{16 c d^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[a + (c + b*x)/x^2]*((-8*c*d^3)/x^2 - (8*c*d^2*e)/x - (4*d^2*(b*d + c*e))/x - (I*(d + e*x)^(3/2)*Sqrt[1
- (2*(a*d^2 + e*(-(b*d) + c*e)))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(a*d^2 + e*
(-(b*d) + c*e)))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(d*(b*d + c*e)*(2*a*d - b*e + Sqrt[(b^2
- 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e
^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] -
(b^2*d^2*e + b*d*(-5*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(4*a*d^2 + 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))
*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d +
e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + 2*e*(b^2*d^2 -
2*b*c*d*e + c*(-4*a*d^2 + c*e^2))*EllipticPi[(d*(2*a*d - b*e - Sqrt[(b^2 - 4*a*c)*e^2]))/(2*(a*d^2 + e*(-(b*d)
+ c*e))), I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d +
e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e*Sqrt[(a*d^2
+ e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))))/(16*c*d^2*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [B]  time = 0.051, size = 4957, normalized size = 3.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*x^2+b*x+c)/x^2)^(1/2)*(e*x+d)^(1/2)*(-2*x^3*a*b*c*d*e^3-2*x^4*a^2*b*d^2*e^2-2*x^4*a^2*c*d*e^3-2*x^3*a^
2*b*d^3*e-6*x^3*a^2*c*d^2*e^2-2*x^3*a*b^2*d^2*e^2-2*x^2*a*c^2*d*e^3-6*x*a*c^2*d^2*e^2-2*x^2*a*b^2*d^3*e-4*x^2*
a^2*c*d^3*e-4*a*c^2*d^3*e+2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2
)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d
+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)
-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)
*x^2*b*c*d*e^3-5*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/
(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2
)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*
(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a*c*d^2*e^2+2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(
1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-
4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-
2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/
2)*x^2*a*b*d^3*e-5*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b
)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1
/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(
e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*b*c*d^2*e^2+4*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*
e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1
/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^
(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d
-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a*c*d^2*e^2+2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)
*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(
-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e
*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*c^2*d*e^3+2*2^(1/2)*(-a*(e*x+d)/
(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/
2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(
-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*
e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*c^2*d*e^3+2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+
b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^
(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)
)^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a
*d-b*e))^(1/2))*x^2*b^2*c*d*e^3-2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c
+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2
*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(
1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(
1/2)*x^2*b^2*d^2*e^2-8*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/
2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)
)^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*
d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a^2*c*d^3*e+2*2^(1/
2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+
2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(
-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)
^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*b^2*d^3*e-2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^
2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*
x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1
/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a^2*c*d^
3*e+2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+
b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE
(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2
)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*b^2*d^3*e-2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-
2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c
+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a
*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*c^2*d*e^3+12*2^(1/2)*(-a*(e*x+d)/(e*(-
4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e
*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c
+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*
a^2*c*d^3*e-2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-
4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*Ell
ipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a
*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a*b^2*d^3*e-6*x*a*b*c*d^3*e-8*x^2*a*b*c*d^2*e^2-2^(1/2)*(-a*(e*x+d)/(e*(-
4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e
*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*
c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e
*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*b*c^2*e^4-2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(
1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/
(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),
(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*a^2*b*d^4-2^(1/2)*(-a*(e*x+d)/
(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/
2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(
-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*
e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*x^2*b^3*d^2*e^2-2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*
e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1
/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^
(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d
-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*c^2*e^4)/x/a/e/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)/c/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x, algorithm="fricas")

[Out]

Timed out

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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+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2)/x**2,x)

[Out]

Timed out

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Giac [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+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x, algorithm="giac")

[Out]

Timed out