\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx\) [264]

Optimal result

Integrand size = 44, antiderivative size = 150 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(e f-d g) (f+g x)}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{(e f-d g)^{3/2} \sqrt {c d f-a e g}} \] Output:

(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-d*g+e*f)/(g*x+f)-(-a*e^2+c*d^2)* 
arctanh((-a*e*g+c*d*f)^(1/2)*(e*x+d)/(-d*g+e*f)^(1/2)/(a*d*e+(a*e^2+c*d^2) 
*x+c*d*e*x^2)^(1/2))/(-d*g+e*f)^(3/2)/(-a*e*g+c*d*f)^(1/2)
 

Mathematica [A] (verified)

Time = 10.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx=\sqrt {(a e+c d x) (d+e x)} \left (\frac {1}{(e f-d g) (f+g x)}+\frac {\left (-c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e f-d g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(e f-d g)^{3/2} \sqrt {c d f-a e g} \sqrt {a e+c d x} \sqrt {d+e x}}\right ) \] Input:

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

Output:

Sqrt[(a*e + c*d*x)*(d + e*x)]*(1/((e*f - d*g)*(f + g*x)) + ((-(c*d^2) + a* 
e^2)*ArcTanh[(Sqrt[e*f - d*g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d*f - a*e*g]*Sqrt 
[d + e*x])])/((e*f - d*g)^(3/2)*Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x]*Sqrt 
[d + e*x]))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1215, 1228, 1154, 219}

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.

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

\(\Big \downarrow \) 1215

\(\displaystyle \int \frac {a e+c d x}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}dx\)

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(f+g x) (e f-d g)}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{2 (e f-d g)^{3/2} \sqrt {c d f-a e g}}\)

Input:

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

Output:

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((e*f - d*g)*(f + g*x)) - ((c* 
d^2 - a*e^2)*ArcTanh[(c*d^2*f + a*e*(e*f - 2*d*g) - (a*e^2*g - c*d*(2*e*f 
- d*g))*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*d*f - a*e*g]*Sqrt[a*d*e + (c*d^2 + a* 
e^2)*x + c*d*e*x^2])])/(2*(e*f - d*g)^(3/2)*Sqrt[c*d*f - a*e*g])
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1747\) vs. \(2(138)=276\).

Time = 2.51 (sec) , antiderivative size = 1748, normalized size of antiderivative = 11.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1748\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)/(g*x+f)^2,x,method=_RE 
TURNVERBOSE)
 

Output:

e/(d*g-e*f)^2*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/2*(a*e^2-c* 
d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2 
+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2))+1/g/(d*g-e*f)*(-1/(a*d*e*g^2 
-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)*g^2/(x+f/g)*(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^ 
2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(3 
/2)+1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)*g/(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e 
*f^2)*((c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a 
*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)+1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g 
*ln((1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g+d*e*c*(x+f/g))/(d*e*c)^(1/2)+(c*d*( 
x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^ 
2*f*g+c*d*e*f^2)/g^2)^(1/2))/(d*e*c)^(1/2)-(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+ 
c*d*e*f^2)/g^2/((a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)*ln((2 
*(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2+(a*e^2*g+c*d^2*g-2*c*d*e*f) 
/g*(x+f/g)+2*((a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)*(c*d*(x 
+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2 
*f*g+c*d*e*f^2)/g^2)^(1/2))/(x+f/g)))+2*d*e*c/(a*d*e*g^2-a*e^2*f*g-c*d^2*f 
*g+c*d*e*f^2)*g^2*(1/4*(2*d*e*c*(x+f/g)+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g)/d/e 
/c*(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2 
*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)+1/8*(4*d*e*c*(a*d*e*g^2-a*e^2*f*g-c*d 
^2*f*g+c*d*e*f^2)/g^2-(a*e^2*g+c*d^2*g-2*c*d*e*f)^2/g^2)/d/e/c*ln((1/2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (138) = 276\).

Time = 0.97 (sec) , antiderivative size = 1113, normalized size of antiderivative = 7.42 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="fricas")
 

Output:

[1/4*(sqrt(c*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*((c*d^2 - a*e^2)*g 
*x + (c*d^2 - a*e^2)*f)*log((8*a^2*d^2*e^2*g^2 + (c^2*d^4 + 6*a*c*d^2*e^2 
+ a^2*e^4)*f^2 - 8*(a*c*d^3*e + a^2*d*e^3)*f*g + (8*c^2*d^2*e^2*f^2 - 8*(c 
^2*d^3*e + a*c*d*e^3)*f*g + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*g^2)*x^2 + 
 4*sqrt(c*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*sqrt(c*d*e*x^2 + a*d* 
e + (c*d^2 + a*e^2)*x)*(2*a*d*e*g - (c*d^2 + a*e^2)*f - (2*c*d*e*f - (c*d^ 
2 + a*e^2)*g)*x) + 2*(4*(c^2*d^3*e + a*c*d*e^3)*f^2 - (3*c^2*d^4 + 10*a*c* 
d^2*e^2 + 3*a^2*e^4)*f*g + 4*(a*c*d^3*e + a^2*d*e^3)*g^2)*x)/(g^2*x^2 + 2* 
f*g*x + f^2)) + 4*(c*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*sqrt(c*d*e 
*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*f^4 - a*d^2*e*f*g^3 - (2*c*d^2 
*e + a*e^3)*f^3*g + (c*d^3 + 2*a*d*e^2)*f^2*g^2 + (c*d*e^2*f^3*g - a*d^2*e 
*g^4 - (2*c*d^2*e + a*e^3)*f^2*g^2 + (c*d^3 + 2*a*d*e^2)*f*g^3)*x), -1/2*( 
sqrt(-c*d*e*f^2 - a*d*e*g^2 + (c*d^2 + a*e^2)*f*g)*((c*d^2 - a*e^2)*g*x + 
(c*d^2 - a*e^2)*f)*arctan(1/2*sqrt(-c*d*e*f^2 - a*d*e*g^2 + (c*d^2 + a*e^2 
)*f*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e*g - (c*d^2 + a 
*e^2)*f - (2*c*d*e*f - (c*d^2 + a*e^2)*g)*x)/(a*c*d^2*e^2*f^2 + a^2*d^2*e^ 
2*g^2 - (a*c*d^3*e + a^2*d*e^3)*f*g + (c^2*d^2*e^2*f^2 + a*c*d^2*e^2*g^2 - 
 (c^2*d^3*e + a*c*d*e^3)*f*g)*x^2 + ((c^2*d^3*e + a*c*d*e^3)*f^2 - (c^2*d^ 
4 + 2*a*c*d^2*e^2 + a^2*e^4)*f*g + (a*c*d^3*e + a^2*d*e^3)*g^2)*x)) - 2*(c 
*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*sqrt(c*d*e*x^2 + a*d*e + (c...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (138) = 276\).

Time = 0.73 (sec) , antiderivative size = 744, normalized size of antiderivative = 4.96 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^2,x, alg 
orithm="giac")
 

Output:

1/2*(2*sqrt(c*d*e - 2*c*d*e*f/(g*x + f) + c*d*e*f^2/(g*x + f)^2 + c*d^2*g/ 
(g*x + f) + a*e^2*g/(g*x + f) - c*d^2*f*g/(g*x + f)^2 - a*e^2*f*g/(g*x + f 
)^2 + a*d*e*g^2/(g*x + f)^2)*g^2*sgn(1/(g*x + f))*sgn(g)/(e*f*g^5 - d*g^6) 
 - (c*d^2*g^2*log(abs(2*c*d*e*f*g - c*d^2*g^2 - a*e^2*g^2 - 2*sqrt(c*d*e*f 
^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*sqrt(c*d*e)*abs(g))) - a*e^2*g^2*l 
og(abs(2*c*d*e*f*g - c*d^2*g^2 - a*e^2*g^2 - 2*sqrt(c*d*e*f^2 - c*d^2*f*g 
- a*e^2*f*g + a*d*e*g^2)*sqrt(c*d*e)*abs(g))) + 2*sqrt(c*d*e*f^2 - c*d^2*f 
*g - a*e^2*f*g + a*d*e*g^2)*sqrt(c*d*e)*abs(g))*sgn(1/(g*x + f))*sgn(g)/(s 
qrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*e*f*g^3*abs(g) - sqrt(c 
*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*d*g^4*abs(g)) + (c*d^2*sgn(1 
/(g*x + f))*sgn(g) - a*e^2*sgn(1/(g*x + f))*sgn(g))*log(abs(2*c*d*e*f*g - 
c*d^2*g^2 - a*e^2*g^2 - 2*sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g 
^2)*(sqrt(c*d*e - 2*c*d*e*f/(g*x + f) + c*d*e*f^2/(g*x + f)^2 + c*d^2*g/(g 
*x + f) + a*e^2*g/(g*x + f) - c*d^2*f*g/(g*x + f)^2 - a*e^2*f*g/(g*x + f)^ 
2 + a*d*e*g^2/(g*x + f)^2) + sqrt(c*d*e*f^2*g^2 - c*d^2*f*g^3 - a*e^2*f*g^ 
3 + a*d*e*g^4)/((g*x + f)*g))*abs(g)))/(sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2 
*f*g + a*d*e*g^2)*(e*f*g - d*g^2)*abs(g)))*g^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1453, normalized size of antiderivative = 9.69 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d 
*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + 
 a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x)) 
*a*e**2*f - sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a 
*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - 
c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d 
 + e*x))*a*e**2*g*x + sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt 
(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqr 
t(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt 
(c)*sqrt(d + e*x))*c*d**2*f + sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt 
(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - 
e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt 
(d)*sqrt(c)*sqrt(d + e*x))*c*d**2*g*x - sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f 
)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sq 
rt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*e**2*f - sqrt(d*g - e*f)*sqrt(a*e*g 
 - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(e)*sqrt(d)*s 
qrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e 
*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*e**2*g*x + sqrt(d*g - e*f)* 
sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt...