\(\int \frac {\sqrt {g x}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) [53]

Optimal result

Integrand size = 29, antiderivative size = 384 \[ \int \frac {\sqrt {g x}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{c e \sqrt {g x} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} d g \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{e \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {g x} \sqrt {a+b x+c x^2}} \] Output:

2*2^(1/2)*(-4*a*c+b^2)^(1/2)*g*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(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^(1/2)*((-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)))^(1/2 
))/c/e/(g*x)^(1/2)/(c*x^2+b*x+a)^(1/2)-4*2^(1/2)*(-4*a*c+b^2)^(1/2)*d*g*(- 
c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*El 
lipticPi(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),2^(1/2)*((-4*a*c+b^2)^(1/2)/(b+ 
(-4*a*c+b^2)^(1/2)))^(1/2))/e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(g*x)^(1/2) 
/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {g x}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {i \sqrt {2} \sqrt {g x} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \sqrt {x} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1284, 1171, 1170, 1279, 187, 25, 413, 413, 412, 1416}

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.

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

\(\Big \downarrow \) 1284

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

\(\Big \downarrow \) 1171

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

\(\Big \downarrow \) 1170

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

\(\Big \downarrow \) 1279

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

\(\Big \downarrow \) 187

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 1416

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1170

Int[(x_)^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 
 Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, b, c}, x] && EqQ[m^2, 1/4]
 

rule 1171

Int[((e_)*(x_))^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> S 
imp[(e*x)^m/x^m   Int[x^m/Sqrt[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, 
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]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

Maple [A] (verified)

Time = 1.94 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {g x}\, \sqrt {x g \left (c \,x^{2}+b x +a \right )}\, \left (\frac {g \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{e c \sqrt {c g \,x^{3}+b g \,x^{2}+a g x}}-\frac {d g \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, -\frac {b +\sqrt {-4 a c +b^{2}}}{2 c \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{e^{2} c \sqrt {c g \,x^{3}+b g \,x^{2}+a g x}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{g x \sqrt {c \,x^{2}+b x +a}}\)	\(638\)
default	\(-\frac {2 \left (2 \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) a c e -\operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b^{2} e +\operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b c d -\sqrt {-4 a c +b^{2}}\, \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b e +\sqrt {-4 a c +b^{2}}\, \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c d -\operatorname {EllipticPi}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b c d -\sqrt {-4 a c +b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c d \right ) \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {g x}}{\sqrt {c \,x^{2}+b x +a}\, c e \left (\sqrt {-4 a c +b^{2}}\, e +b e -2 c d \right ) x}\)	\(717\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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