\(\int (a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [99]

Optimal result

Integrand size = 32, antiderivative size = 253 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b}+\frac {B (b c-a d)^2 g^2 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b d^2}+\frac {B (b c-a d)^3 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b d^3}+\frac {2 B^2 (b c-a d)^3 g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3} \] Output:

-1/3*B*(-a*d+b*c)*g^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d+1/3*g^2*(b 
*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/3*B*(-a*d+b*c)^2*g^2*(b*x+a)*(2* 
A+B+2*B*ln(e*(b*x+a)/(d*x+c)))/b/d^2+1/3*B*(-a*d+b*c)^3*g^2*ln((-a*d+b*c)/ 
b/(d*x+c))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/b/d^3+2/3*B^2*(-a*d+b*c)^3* 
g^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^3
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B (b c-a d) \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{3 b} \] Input:

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

Output:

(g^2*((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(b*c - a*d)* 
(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)*Log[(e*(a + b*x))/(c 
+ d*x)] - d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c 
- a*d)^2*Log[c + d*x] - 2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x) 
])*Log[c + d*x] + B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + B* 
(b*c - a*d)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c 
+ d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^3))/(3*b)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2950, 2781, 2784, 2784, 2754, 2838}

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.

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

\(\Big \downarrow \) 2950

\(\displaystyle g^2 (b c-a d)^3 \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2781

\(\displaystyle g^2 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^2 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{3 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^2 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{3 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle g^2 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {2 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{d}}{d}}{2 d}\right )}{3 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle g^2 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{d}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{3 b}\right )\)

Input:

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

Output:

(b*c - a*d)^3*g^2*(((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3 
*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*(((a + b*x)^2*(A + 
B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x))/(c + 
d*x))^2) - (((a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c 
 + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((2*A + 3*B + 2*B*Log[(e*(a + 
b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (2*B*PolyLog[ 
2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/(2*d)))/(3*b))
 

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2781

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a 
+ b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1)))   Int[(f*x) 
^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] 
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="frica 
s")
 

Output:

integral(A^2*b^2*g^2*x^2 + 2*A^2*a*b*g^2*x + A^2*a^2*g^2 + (B^2*b^2*g^2*x^ 
2 + 2*B^2*a*b*g^2*x + B^2*a^2*g^2)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B 
*b^2*g^2*x^2 + 2*A*B*a*b*g^2*x + A*B*a^2*g^2)*log((b*e*x + a*e)/(d*x + c)) 
, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1165 vs. \(2 (242) = 484\).

Time = 0.13 (sec) , antiderivative size = 1165, normalized size of antiderivative = 4.60 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxim 
a")
 

Output:

1/3*A^2*b^2*g^2*x^3 + A^2*a*b*g^2*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d* 
x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*a^2*g^2 + 2*(x^2*log(b* 
e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d 
^2 - (b*c - a*d)*x/(b*d))*A*B*a*b*g^2 + 1/3*(2*x^3*log(b*e*x/(d*x + c) + a 
*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c* 
d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*b^2*g^2 + A^2*a 
^2*g^2*x - 1/3*((2*g^2*log(e) + 3*g^2)*b^2*c^3 - (6*g^2*log(e) + 7*g^2)*a* 
b*c^2*d + 2*(3*g^2*log(e) + 2*g^2)*a^2*c*d^2)*B^2*log(d*x + c)/d^3 - 2/3*( 
b^3*c^3*g^2 - 3*a*b^2*c^2*d*g^2 + 3*a^2*b*c*d^2*g^2 - a^3*d^3*g^2)*(log(b* 
x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a* 
d)))*B^2/(b*d^3) + 1/3*(B^2*b^3*d^3*g^2*x^3*log(e)^2 - (b^3*c*d^2*g^2*log( 
e) - (3*g^2*log(e)^2 + g^2*log(e))*a*b^2*d^3)*B^2*x^2 + ((2*g^2*log(e) + g 
^2)*b^3*c^2*d - 2*(3*g^2*log(e) + g^2)*a*b^2*c*d^2 + (3*g^2*log(e)^2 + 4*g 
^2*log(e) + g^2)*a^2*b*d^3)*B^2*x + (B^2*b^3*d^3*g^2*x^3 + 3*B^2*a*b^2*d^3 
*g^2*x^2 + 3*B^2*a^2*b*d^3*g^2*x + B^2*a^3*d^3*g^2)*log(b*x + a)^2 + (B^2* 
b^3*d^3*g^2*x^3 + 3*B^2*a*b^2*d^3*g^2*x^2 + 3*B^2*a^2*b*d^3*g^2*x + (b^3*c 
^3*g^2 - 3*a*b^2*c^2*d*g^2 + 3*a^2*b*c*d^2*g^2)*B^2)*log(d*x + c)^2 + (2*B 
^2*b^3*d^3*g^2*x^3*log(e) - (b^3*c*d^2*g^2 - (6*g^2*log(e) + g^2)*a*b^2*d^ 
3)*B^2*x^2 + 2*(b^3*c^2*d*g^2 - 3*a*b^2*c*d^2*g^2 + (3*g^2*log(e) + 2*g^2) 
*a^2*b*d^3)*B^2*x + (2*a*b^2*c^2*d*g^2 - 5*a^2*b*c*d^2*g^2 + (2*g^2*log...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g**2*(2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x 
**2),x)*a**3*b**2*d**4 - 6*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d 
*x + b*c*x + b*d*x**2),x)*a**2*b**3*c*d**3 + 6*int((log((a*e + b*e*x)/(c + 
 d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**4*c**2*d**2 - 2*int((lo 
g((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**5*c** 
3*d + 2*log(c + d*x)*a**4*d**3 - 6*log(c + d*x)*a**3*b*c*d**2 + 3*log(c + 
d*x)*a**3*b*d**3 + 6*log(c + d*x)*a**2*b**2*c**2*d - 9*log(c + d*x)*a**2*b 
**2*c*d**2 - 2*log(c + d*x)*a*b**3*c**3 + 9*log(c + d*x)*a*b**3*c**2*d - 3 
*log(c + d*x)*b**4*c**3 + 2*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*c*d* 
*2 + 3*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*d**3*x - log((a*e + b*e*x 
)/(c + d*x))**2*a*b**3*c**2*d + 3*log((a*e + b*e*x)/(c + d*x))**2*a*b**3*d 
**3*x**2 + log((a*e + b*e*x)/(c + d*x))**2*b**4*d**3*x**3 + 2*log((a*e + b 
*e*x)/(c + d*x))*a**4*d**3 + 6*log((a*e + b*e*x)/(c + d*x))*a**3*b*d**3*x 
+ 3*log((a*e + b*e*x)/(c + d*x))*a**3*b*d**3 - 5*log((a*e + b*e*x)/(c + d* 
x))*a**2*b**2*c*d**2 + 6*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*d**3*x**2 
+ 4*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*d**3*x + 2*log((a*e + b*e*x)/(c 
 + d*x))*a*b**3*c**2*d - 6*log((a*e + b*e*x)/(c + d*x))*a*b**3*c*d**2*x + 
2*log((a*e + b*e*x)/(c + d*x))*a*b**3*d**3*x**3 + log((a*e + b*e*x)/(c + d 
*x))*a*b**3*d**3*x**2 + 2*log((a*e + b*e*x)/(c + d*x))*b**4*c**2*d*x - log 
((a*e + b*e*x)/(c + d*x))*b**4*c*d**2*x**2 + 3*a**4*d**3*x + 3*a**3*b*d...