3.1.85 \(\int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx\) [85]

Optimal. Leaf size=1287 \[ -\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 )} \]

[Out]

-1/4*(b*d+c*e)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/c/d-1/2*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x+1/8*(b*d+c*e)*x*E
llipticE(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)/c/d/(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)+3/2*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))*(-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)*2^(1/2)/(e*x+d)^(1/2)-1/4*(b*d+c*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))*(-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)/c/(a*x^2+b*x+c)*2^(1/2)/(e*x+d)^(1/2)-1/2*(a*d+b*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)+1/8*(b*d+c*e)^2*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)/c/d^2/(a*x^2+b*x+c)
*2^(1/2)/a^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 3.57, antiderivative size = 1287, normalized size of antiderivative = 1.00, 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 = {1587, 930, 6874, 732, 430, 953, 948, 175, 552, 551, 857, 435} \begin {gather*} \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};\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 ) (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 (\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 ) (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 (\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 ) (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 (\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 {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};\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 )}-\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {d+e x}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((b*d + c*e)*Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(c*d) - (Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/(2*x) + (
Sqrt[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]*
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)) + ((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[(Sq
rt[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 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 953

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)/(Sqrt[f + g*x]*Sqrt[a +
b*x + c*x^2]))*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], 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 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^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 ) \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 {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 ) \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 ((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 ) \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}\\ &=-\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 ) \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}\\ &=-\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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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] Result contains complex when optimal does not.
time = 31.66, size = 1392, normalized size = 1.08 \begin {gather*} \frac {1}{16} x \sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}} \left (-\frac {4 (2 c d+b d x+c e x)}{c d x^2}+\frac {(d+e x) \left (\frac {4 d e^2 (b d+c 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}}} (c+x (b+a x))}{(d+e x)^2}-\frac {i \sqrt {2} d (b d+c e) \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (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+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-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 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 (b^2 d^2 e+b d \left (-5 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (4 a d^2+2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (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+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 \left (b^2 d^2-2 b c d e+c \left (-4 a d^2+c e^2\right )\right ) \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (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+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-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 (-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 )}{c d^2 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}}} (c+x (b+a x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4956\) vs. \(2(1127)=2254\).
time = 0.24, size = 4957, normalized size = 3.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(1597\)
default	\(\text {Expression too large to display}\)	\(4957\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [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**2,x)

[Out]

Timed out

________________________________________________________________________________________

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^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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^2} \,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^2,x)

[Out]

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

________________________________________________________________________________________