\(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [327]

Optimal result

Integrand size = 26, antiderivative size = 821 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}} \] Output:

-1/4*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f/(e*(-f)^(1/2)+d*g^(1/2))/((-f)^(1/2 
)-g^(1/2)*x)-1/4*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f/(e*(-f)^(1/2)-d*g^(1/2) 
)/((-f)^(1/2)+g^(1/2)*x)-1/2*b*e*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)- 
g^(1/2)*x)/(e*(-f)^(1/2)+d*g^(1/2)))/f/(e*(-f)^(1/2)+d*g^(1/2))/g^(1/2)-1/ 
4*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)-g^(1/2)*x)/(e*(-f)^(1/2)+d*g^(1 
/2)))/(-f)^(3/2)/g^(1/2)-1/2*b*e*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+ 
g^(1/2)*x)/(e*(-f)^(1/2)-d*g^(1/2)))/(e*(-f)^(3/2)+d*f*g^(1/2))/g^(1/2)+1/ 
4*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)+g^(1/2)*x)/(e*(-f)^(1/2)-d*g^(1 
/2)))/(-f)^(3/2)/g^(1/2)-1/2*b^2*e*n^2*polylog(2,-g^(1/2)*(e*x+d)/(e*(-f)^ 
(1/2)-d*g^(1/2)))/(e*(-f)^(3/2)+d*f*g^(1/2))/g^(1/2)+1/2*b*n*(a+b*ln(c*(e* 
x+d)^n))*polylog(2,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(3/2)/g 
^(1/2)-1/2*b^2*e*n^2*polylog(2,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/f 
/(e*(-f)^(1/2)+d*g^(1/2))/g^(1/2)-1/2*b*n*(a+b*ln(c*(e*x+d)^n))*polylog(2, 
g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(3/2)/g^(1/2)-1/2*b^2*n^2*p 
olylog(3,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(3/2)/g^(1/2)+1/2 
*b^2*n^2*polylog(3,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(3/2)/g^ 
(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.51 (sec) , antiderivative size = 1143, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((2*Sqrt[f]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) 
 + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x 
)^n])^2)/Sqrt[g] + (2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(( 
Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + I*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[ 
I*Sqrt[f] - Sqrt[g]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x 
)) + (Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + e*((-I)*Sqrt[f] - Sqrt[g]* 
x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sq 
rt[g]*x)) - I*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I 
*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt[g 
])]) + I*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sq 
rt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])])))/S 
qrt[g] + (b^2*n^2*(-((Sqrt[f]*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2) + 2*e*( 
I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqr 
t[f] + I*d*Sqrt[g])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*( 
d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] 
 - I*Sqrt[g]*x))) + (Sqrt[f]*(Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d + e*x] 
 + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt 
[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, (Sqrt[g] 
*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[ 
f] + I*Sqrt[g]*x)) + I*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I...
 

Rubi [A] (verified)

Time = 2.14 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2856, 2009}

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.

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

\(\Big \downarrow \) 2856

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^2 e \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}\)

Input:

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

Output:

-1/4*((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(f*(e*Sqrt[-f] + d*Sqrt[g])* 
(Sqrt[-f] - Sqrt[g]*x)) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f*(e 
*Sqrt[-f] - d*Sqrt[g])*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*n*(a + b*Log[c*(d + 
e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*(e 
*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sq 
rt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g]) - ( 
b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f 
] - d*Sqrt[g])])/(2*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + ((a + b*Log[c* 
(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/ 
(4*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sq 
rt[-f] - d*Sqrt[g]))])/(2*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + (b*n*(a 
+ b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*S 
qrt[g]))])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, (Sqrt[g]*(d + e* 
x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - (b 
*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + 
 d*Sqrt[g])])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + 
 e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*Sqrt[g]) + (b^2*n^2*PolyL 
og[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)*Sqrt[g] 
)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) 
^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g \,x^{2}+f \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*log((e*x + d)^n*c)^2 + 2*a*b*log((e*x + d)^n*c) + a^2)/(g^2* 
x^4 + 2*f*g*x^2 + f^2), x)
 

Sympy [F(-1)]

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

integrate((a+b*ln(c*(e*x+d)**n))**2/(g*x**2+f)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*a^2*(x/(f*g*x^2 + f^2) + arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f)) + integr 
ate((b^2*log((e*x + d)^n)^2 + b^2*log(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) 
+ a*b)*log((e*x + d)^n))/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {\sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a^{2} f +\sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a^{2} g \,x^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{g^{2} x^{4}+2 f g \,x^{2}+f^{2}}d x \right ) b^{2} f^{3} g +2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{g^{2} x^{4}+2 f g \,x^{2}+f^{2}}d x \right ) b^{2} f^{2} g^{2} x^{2}+4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{g^{2} x^{4}+2 f g \,x^{2}+f^{2}}d x \right ) a b \,f^{3} g +4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{g^{2} x^{4}+2 f g \,x^{2}+f^{2}}d x \right ) a b \,f^{2} g^{2} x^{2}+a^{2} f g x}{2 f^{2} g \left (g \,x^{2}+f \right )} \] Input:

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

Output:

(sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*f + sqrt(g)*sqrt(f)*at 
an((g*x)/(sqrt(g)*sqrt(f)))*a**2*g*x**2 + 2*int(log((d + e*x)**n*c)**2/(f* 
*2 + 2*f*g*x**2 + g**2*x**4),x)*b**2*f**3*g + 2*int(log((d + e*x)**n*c)**2 
/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*b**2*f**2*g**2*x**2 + 4*int(log((d + e 
*x)**n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*a*b*f**3*g + 4*int(log((d + e 
*x)**n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*a*b*f**2*g**2*x**2 + a**2*f*g 
*x)/(2*f**2*g*(f + g*x**2))