3.1.84 \(\int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x} \, dx\) [84]

Optimal. Leaf size=929 \[ -\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {3 \sqrt {b^2-4 a c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac {3 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(b d+c e) \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \Pi \left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {2} \sqrt {a} d \left (c+b x+a x^2\right )} \]

[Out]

-(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)+3/2*x*EllipticE(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)
*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1
/2)*(e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)/(a*x^2+b*x+c)*2^(1/2)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b
^2)^(1/2))))^(1/2)-3*d*x*EllipticF(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(
-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1/2)*(-a*
(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/(a*x^2+b*x+c)/(e*x+d)^(1/
2)+2*(a*d+b*e)*x*EllipticF(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b
^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1/2)*(-a*(a*x^2+b
*x+c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/a/(a*x^2+b*x+c)/(e*x+d)^(1/2)-1/2
*(b*d+c*e)*x*EllipticPi(2^(1/2)*a^(1/2)*(e*x+d)^(1/2)/(2*a*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2),1/2*(2*a*d-b*e+e*
(-4*a*c+b^2)^(1/2))/a/d,((b-2*a*d/e-(-4*a*c+b^2)^(1/2))/(b-2*a*d/e+(-4*a*c+b^2)^(1/2)))^(1/2))*(a+c/x^2+b/x)^(
1/2)*(1-2*a*(e*x+d)/(2*a*d-e*(b-(-4*a*c+b^2)^(1/2))))^(1/2)*(2*a*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1-2*a*(e*x
+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/d/(a*x^2+b*x+c)*2^(1/2)/a^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.87, antiderivative size = 929, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1587, 930, 6874, 732, 430, 948, 175, 552, 551, 857, 435} \begin {gather*} \frac {3 \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 (\text {ArcSin}\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 {\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 {3 \sqrt {2} \sqrt {b^2-4 a c} d \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 (\text {ArcSin}\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 {d+e x} \left (a x^2+b x+c\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (a d+b 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 (\text {ArcSin}\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 )}{a \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {(b d+c 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};\text {ArcSin}\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 )}-\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 175

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 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 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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 732

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 857

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 930

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)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x], 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 948

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 1587

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 6874

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{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^2} \, dx}{\sqrt {c+b x+a x^2}}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e 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 \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{2 \sqrt {c+b x+a x^2}}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \left (\frac {2 (a d+b e)}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}+\frac {b d+c e}{x \sqrt {d+e x} \sqrt {c+b x+a x^2}}+\frac {3 a e x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}}\right ) \, dx}{2 \sqrt {c+b x+a x^2}}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {\left (3 a 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}{2 \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}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{\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 \sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{2 \sqrt {c+b x+a x^2}}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {\left ((b d+c 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 (3 a \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}{2 \sqrt {c+b x+a x^2}}-\frac {\left (3 a d \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}{2 \sqrt {c+b x+a x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (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 ) \text {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 )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (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 )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left ((b d+c 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 ) \text {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 (3 \sqrt {b^2-4 a c} \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 ) \text {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 )}{\sqrt {2} \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 (3 \sqrt {2} \sqrt {b^2-4 a c} d \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 ) \text {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 {d+e x} \left (c+b x+a x^2\right )}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {3 \sqrt {b^2-4 a c} \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 )}{\sqrt {2} \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 {2} \sqrt {b^2-4 a c} d \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 {d+e x} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (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 )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left ((b d+c 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 ) \text {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}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {3 \sqrt {b^2-4 a c} \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 )}{\sqrt {2} \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 {2} \sqrt {b^2-4 a c} d \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 {d+e x} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (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 )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\left ((b d+c 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 ) \text {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}\\ &=-\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}+\frac {3 \sqrt {b^2-4 a c} \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 )}{\sqrt {2} \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 {2} \sqrt {b^2-4 a c} d \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 {d+e x} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (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 )}{a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {(b d+c 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 )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 29.59, size = 1372, normalized size = 1.48 \begin {gather*} -\sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}}+\frac {x (d+e x)^{3/2} \sqrt {a+\frac {c+b x}{x^2}} \left (12 d \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (a \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {c e}{d+e x}\right )}{d+e x}\right )-\frac {3 i \sqrt {2} d \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 c e^2}{d+e x}-2 a d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 c e^2}{d+e x}+2 a d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 (4 a d^2-b d e-2 c e^2+3 d \sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 c e^2}{d+e x}-2 a d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 c e^2}{d+e x}+2 a d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 {2 i \sqrt {2} e (b d+c e) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 c e^2}{d+e x}-2 a d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 c e^2}{d+e x}+2 a d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \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 (-b d+c e)\right )};i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+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 )}{4 d e \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {c+b x+a x^2} \sqrt {\frac {(d+e x)^2 \left (a \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {c e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2]) + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*(12*d*Sqrt[(a*d^2 + e*(-
(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (c
*e)/(d + e*x)))/(d + e*x)) - ((3*I)*Sqrt[2]*d*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)
*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^2
- 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e
)/(d + e*x)))/(-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]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(4*a*d^2 - b*d*e - 2*c*e^2 + 3*d*Sqrt[(b^2 -
4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)
/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*
a*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcS
inh[(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] + ((2*I)*Sqrt[2]*e*
(b*d + c*e)*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(
d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*
d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*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]))])/Sqrt[d + e*x]))/(4*d*e*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e
+ Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[c + b*x + a*x^2]*Sqrt[((d + e*x)^2*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(
d + e*x) + (c*e)/(d + e*x)))/(d + e*x)))/e^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3552\) vs. \(2(818)=1636\).
time = 0.23, size = 3553, normalized size = 3.82

method	result	size
risch	\(-\sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, \sqrt {e x +d}+\frac {\left (\frac {3 a e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\right )}{2 a}\right )}{\sqrt {a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x +c e x +c d}}+\frac {2 a d \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\right )}{\sqrt {a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x +c e x +c d}}+\frac {2 e b \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\right )}{\sqrt {a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x +c e x +c d}}-\frac {\left (b d +c e \right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}\, e \EllipticPi \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}}, -\frac {\left (-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}\right ) e}{d}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}}\right )}{\sqrt {a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x +c e x +c d}\, d}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x \sqrt {\left (a \,x^{2}+b x +c \right ) \left (e x +d \right )}}{\left (a \,x^{2}+b x +c \right ) \sqrt {e x +d}}\)	\(1400\)
default	\(\text {Expression too large to display}\)	\(3553\)

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,x,method=_RETURNVERBOSE)

[Out]

1/2*((a*x^2+b*x+c)/x^2)^(1/2)*(e*x+d)^(1/2)*(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*
a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c
+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a
*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*d^2*e*x-2*2^(1/2)*(-(e*
x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2
)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*
a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))
^(1/2))*(-4*a*c+b^2)^(1/2)*b*d*e^2*x+4*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-
4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^
(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2
)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*d^3*x-2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2
)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c
+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a
*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*d^2*e*x+6*2^(1/
2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b
^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-
(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)
^(1/2)))^(1/2))*a*c*d*e^2*x-2*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2
)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a
*d+e*b))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2
*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*d*e^2*x-6*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a
*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)
^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*
b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*d^3*x+6*2^(1/2)*(-(e
*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/
2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)
*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2))
)^(1/2))*a*b*d^2*e*x-6*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)
-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)
)^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*
b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c*d*e^2*x+2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^
(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e
/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticPi(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2
),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)
))^(1/2))*(-4*a*c+b^2)^(1/2)*b*d*e^2*x+2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-
4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^
(1/2)-2*a*d+e*b))^(1/2)*EllipticPi(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),-1/2*(e*(-4*a*c
+b^2)^(1/2)-2*a*d+e*b)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c
+b^2)^(1/2)*c*e^3*x-2*2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-
b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))
^(1/2)*EllipticPi(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d
+e*b)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*d^2*e*x-2*2^(1/2)*(-
(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*a*d-e*b+e*(-4*a*c+b^2)^(
1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2)*EllipticPi(2^(1/2)*(-(e*x
+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+e*b)/a/d,(-(e*(-4*a*c+b^2)^(1/2
)-2*a*d+e*b)/(2*a*d-e*b+e*(-4*a*c+b^2)^(1/2)))^...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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,x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x}\, dx \end {gather*}

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,x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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