\(\int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx\) [92]

Optimal result

Integrand size = 27, antiderivative size = 404 \[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.45 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=-\frac {i \sqrt {2} \sqrt {\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right ),\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-\operatorname {EllipticPi}\left (1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right ),\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )\right )}{\sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.11 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1284, 1172, 321, 1279, 187, 413, 413, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1284

\(\displaystyle e \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+d \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}+d \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 321

\(\displaystyle d \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {d \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \int \frac {1}{x \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 187

\(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 d \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 d \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2} \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 d \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int -\frac {1}{e x \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2} \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}} \sqrt {\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticPi}\left (\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {c} \sqrt {a+b x+c x^2} \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}} \sqrt {\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}\)

Input:

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

Output:

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 
 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin 
[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqr 
t[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*Sqrt[d + e*x]*S 
qrt[a + b*x + c*x^2]) - (Sqrt[2]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*S 
qrt[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqr 
t[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*( 
d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*c*d - b*e + S 
qrt[b^2 - 4*a*c]*e)/(2*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2 
*c*d - (b - Sqrt[b^2 - 4*a*c])*e]], (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2 
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[c]*Sqrt[a + b*x + c*x^2]*Sqrt[b 
- Sqrt[b^2 - 4*a*c] - (2*c*d)/e + (2*c*(d + e*x))/e]*Sqrt[b + Sqrt[b^2 - 4 
*a*c] - (2*c*d)/e + (2*c*(d + e*x))/e])
 

Defintions of rubi rules used

rule 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(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] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[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] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[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 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[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] && EqQ[m^2, 1/4]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[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]
 

rule 1284

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2]), x_Symbol] :> Simp[g/e   Int[1/(Sqrt[f + g*x]*Sqrt[a + b*x 
 + c*x^2]), x], x] + Simp[(e*f - d*g)/e   Int[1/((d + e*x)*Sqrt[f + g*x]*Sq 
rt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
 

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}-\frac {2 \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, e \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, -\frac {\left (-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) e}{d}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(638\)
default	\(\frac {\left (-\operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) e \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, -\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 d c}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) e -\operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) e b +2 d \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) c +\operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, -\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 d c}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) b e -2 \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, -\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 d c}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {\frac {\left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) e}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {\frac {\left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}}{c \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right )}\)	\(807\)

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{x \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{x\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d+e x}}{x \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {e x +d}}{x \sqrt {c \,x^{2}+b x +a}}d x \] Input:

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

Output:

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