### 3.83 $$\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x} \, dx$$

Optimal. Leaf size=955 $\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x$

[Out]

(2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/3 + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*S
qrt[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)])/(3*a*e*Sqrt[(a*(
d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*d*(a*d + b*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)])/(3*a*e*Sqrt[d + e*x]*(c + b*x + a*x^2)) + (4*Sqrt[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)])/(3*a*Sqrt[d + e*x]*(c + b*x + a*x^
2)) - (Sqrt[2]*c*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)])/(Sqrt[a]*(c + b*x + a*x^
2))

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Rubi [A]  time = 3.31272, antiderivative size = 955, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.423, Rules used = {1449, 918, 6742, 718, 419, 934, 169, 538, 537, 843, 424} $\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/3 + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*S
qrt[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)])/(3*a*e*Sqrt[(a*(
d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*d*(a*d + b*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)])/(3*a*e*Sqrt[d + e*x]*(c + b*x + a*x^2)) + (4*Sqrt[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)])/(3*a*Sqrt[d + e*x]*(c + b*x + a*x^
2)) - (Sqrt[2]*c*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)])/(Sqrt[a]*(c + b*x + a*x^
2))

Rule 1449

Int[((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[((d + e*x^n)
^q*(c + b*x^n + a*x^(2*n))^p)/x^(2*n*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn
2, 2*mn] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n]

Rule 918

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[(2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(e*(2*m + 5)), x] - Dist[1/(e*(2*m + 5)), Int[(
(d + e*x)^m*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*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, m}, 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 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 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x} \, 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} \, dx}{\sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{-3 c d-2 (b d+c e) x-(a d+b e) x^2}{x \sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \left (-\frac{2 (b d+c e)}{\sqrt{d+e x} \sqrt{c+b x+a x^2}}-\frac{3 c d}{x \sqrt{d+e x} \sqrt{c+b x+a x^2}}-\frac{(a d+b e) x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}}\right ) \, dx}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\left (c d \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}{\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{x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 \sqrt{c+b x+a x^2}}+\frac{\left (2 (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}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\left (c d \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}{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{\sqrt{d+e x}}{\sqrt{c+b x+a x^2}} \, dx}{3 e \sqrt{c+b x+a x^2}}+\frac{\left (d (-a d-b 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}{3 e \sqrt{c+b x+a x^2}}+\frac{\left (4 \sqrt{2} \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 )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{4 \sqrt{2} \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 )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \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 (\sqrt{2} \sqrt{b^2-4 a c} (-a d-b 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 )}{3 a e \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 (2 \sqrt{2} \sqrt{b^2-4 a c} d (-a d-b 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 )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b 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 )}{3 a e \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{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \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 )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \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{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b 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 )}{3 a e \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{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \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 )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \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{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b 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 )}{3 a e \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{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b 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 )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \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 )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\sqrt{2} c \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{a} \left (c+b x+a x^2\right )}\\ \end{align*}

Mathematica [C]  time = 10.6806, size = 1258, normalized size = 1.32 $\frac{x \sqrt{a+\frac{c+b x}{x^2}} \left (\frac{4 (a d+b 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)) e^2}{(d+e x)^2}+\frac{6 i \sqrt{2} a c \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \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 ) e^2}{\sqrt{d+e x}}-\frac{i \sqrt{2} (a d+b e) \left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 )}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left (b e \left (\sqrt{\left (b^2-4 a c\right ) e^2}-b e\right )+a \left (-2 c e^2+3 b d e+d \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \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 )}{\sqrt{d+e x}}\right ) (d+e x)^{3/2}}{6 a e^2 \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))}+\frac{2}{3} x \sqrt{a+\frac{c+b x}{x^2}} \sqrt{d+e x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*x*Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2])/3 + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*((4*e^2*(a*d + b*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)))/(d + e*x)^2 - (I
*Sqrt[2]*(a*d + b*e)*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt
[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x
+ b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*El
lipticE[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]))])/Sqrt[d + e*x] + (I*
Sqrt[2]*(b*e*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2]) + a*(3*b*d*e - 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*Sqrt[(-2*
c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]
)*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e +
Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*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]))])/Sqrt[d + e*x] + ((6*I)*Sqrt[2]*a*c*e^2*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[
(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x +
b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Ell
ipticPi[(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]))])/Sqrt[d + e*x]))/(6*a*e^2*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)))

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Maple [B]  time = 0.048, size = 3023, normalized size = 3.2 \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)

[Out]

1/3*((a*x^2+b*x+c)/x^2)^(1/2)*x*(e*x+d)^(1/2)*(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)*a*d^2*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)*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)*b*d*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))*(-4*a*c+b^2)^(1/2)*c*e^3+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)*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))*a*b*d^
2*e+6*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))*a*c*d*e^2-3*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)*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))*b^2*d*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)*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))*a^2*d^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)*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))*a*c*d*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)*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))*b^2*d*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)*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))*b*c*e^3+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)*c*e^3-6*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))*a*c*d*e^2+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))*b*c*e^3+2
*x^3*a^2*e^3+2*x^2*a^2*d*e^2+2*x^2*a*b*e^3+2*x*a*b*d*e^2+2*x*a*c*e^3+2*a*c*d*e^2)/a/e^2/(a*e*x^3+a*d*x^2+b*e*x
^2+b*d*x+c*e*x+c*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{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, algorithm="maxima")

[Out]

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

[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, algorithm="giac")

[Out]

Timed out