3.167 \(\int \frac{(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{e r \left (a^2+2 a b n+2 b^2 n^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b e r (a+b n) \log \left (c x^n\right )}{x}-\frac{2 b e n r (a+b n)}{x}-\frac{b^2 e r \log ^2\left (c x^n\right )}{x}-\frac{2 b^2 e n r \log \left (c x^n\right )}{x}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b^2 e n^2 r}{x} \]

[Out]

(-2*b^2*e*n^2*r)/x - (2*b*e*n*(a + b*n)*r)/x - (e*(a^2 + 2*a*b*n + 2*b^2*n^2)*r)/x - (2*b^2*e*n*r*Log[c*x^n])/
x - (2*b*e*(a + b*n)*r*Log[c*x^n])/x - (b^2*e*r*Log[c*x^n]^2)/x - (2*b^2*n^2*(d + e*Log[f*x^r]))/x - (2*b*n*(a
+ b*Log[c*x^n])*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x

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Rubi [A]  time = 0.192719, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2305, 2304, 2366, 14} \[ -\frac{e r \left (a^2+2 a b n+2 b^2 n^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b e r (a+b n) \log \left (c x^n\right )}{x}-\frac{2 b e n r (a+b n)}{x}-\frac{b^2 e r \log ^2\left (c x^n\right )}{x}-\frac{2 b^2 e n r \log \left (c x^n\right )}{x}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b^2 e n^2 r}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*e*n^2*r)/x - (2*b*e*n*(a + b*n)*r)/x - (e*(a^2 + 2*a*b*n + 2*b^2*n^2)*r)/x - (2*b^2*e*n*r*Log[c*x^n])/
x - (2*b*e*(a + b*n)*r*Log[c*x^n])/x - (b^2*e*r*Log[c*x^n]^2)/x - (2*b^2*n^2*(d + e*Log[f*x^r]))/x - (2*b*n*(a
+ b*Log[c*x^n])*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx &=-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \frac{-a^2 \left (1+\frac{2 b n (a+b n)}{a^2}\right )-2 b (a+b n) \log \left (c x^n\right )-b^2 \log ^2\left (c x^n\right )}{x^2} \, dx\\ &=-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \left (\frac{-a^2-2 a b n-2 b^2 n^2}{x^2}-\frac{2 b (a+b n) \log \left (c x^n\right )}{x^2}-\frac{b^2 \log ^2\left (c x^n\right )}{x^2}\right ) \, dx\\ &=-\frac{e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (b^2 e r\right ) \int \frac{\log ^2\left (c x^n\right )}{x^2} \, dx+(2 b e (a+b n) r) \int \frac{\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac{2 b e n (a+b n) r}{x}-\frac{e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac{2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac{b^2 e r \log ^2\left (c x^n\right )}{x}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (2 b^2 e n r\right ) \int \frac{\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac{2 b^2 e n^2 r}{x}-\frac{2 b e n (a+b n) r}{x}-\frac{e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac{2 b^2 e n r \log \left (c x^n\right )}{x}-\frac{2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac{b^2 e r \log ^2\left (c x^n\right )}{x}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.148259, size = 138, normalized size = 0.76 \[ -\frac{e \left (a^2+2 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+a^2 d+a^2 e r+2 b \log \left (c x^n\right ) \left (e (a+b n) \log \left (f x^r\right )+a (d+e r)+b n (d+2 e r)\right )+2 a b d n+4 a b e n r+b^2 \log ^2\left (c x^n\right ) \left (d+e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+6 b^2 e n^2 r}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*d + 2*a*b*d*n + 2*b^2*d*n^2 + a^2*e*r + 4*a*b*e*n*r + 6*b^2*e*n^2*r + e*(a^2 + 2*a*b*n + 2*b^2*n^2)*Log
[f*x^r] + b^2*Log[c*x^n]^2*(d + e*r + e*Log[f*x^r]) + 2*b*Log[c*x^n]*(a*(d + e*r) + b*n*(d + 2*e*r) + e*(a + b
*n)*Log[f*x^r]))/x)

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Maple [C]  time = 0.657, size = 8407, normalized size = 46.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]  time = 1.21231, size = 298, normalized size = 1.65 \begin{align*} -b^{2} e{\left (\frac{r}{x} + \frac{\log \left (f x^{r}\right )}{x}\right )} \log \left (c x^{n}\right )^{2} - 2 \, a b e{\left (\frac{r}{x} + \frac{\log \left (f x^{r}\right )}{x}\right )} \log \left (c x^{n}\right ) - 2 \,{\left (\frac{{\left (r \log \left (x\right ) + 3 \, r + \log \left (f\right )\right )} n^{2}}{x} + \frac{n{\left (2 \, r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x}\right )} b^{2} e - 2 \, b^{2} d{\left (\frac{n^{2}}{x} + \frac{n \log \left (c x^{n}\right )}{x}\right )} - \frac{2 \, a b e n{\left (2 \, r + \log \left (f\right ) + \log \left (x^{r}\right )\right )}}{x} - \frac{b^{2} d \log \left (c x^{n}\right )^{2}}{x} - \frac{2 \, a b d n}{x} - \frac{a^{2} e r}{x} - \frac{2 \, a b d \log \left (c x^{n}\right )}{x} - \frac{a^{2} e \log \left (f x^{r}\right )}{x} - \frac{a^{2} d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2*e*(r/x + log(f*x^r)/x)*log(c*x^n)^2 - 2*a*b*e*(r/x + log(f*x^r)/x)*log(c*x^n) - 2*((r*log(x) + 3*r + log(
f))*n^2/x + n*(2*r + log(f) + log(x^r))*log(c*x^n)/x)*b^2*e - 2*b^2*d*(n^2/x + n*log(c*x^n)/x) - 2*a*b*e*n*(2*
r + log(f) + log(x^r))/x - b^2*d*log(c*x^n)^2/x - 2*a*b*d*n/x - a^2*e*r/x - 2*a*b*d*log(c*x^n)/x - a^2*e*log(f
*x^r)/x - a^2*d/x

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Fricas [A]  time = 0.831248, size = 743, normalized size = 4.1 \begin{align*} -\frac{b^{2} e n^{2} r \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 2 \, a b d n + a^{2} d +{\left (b^{2} e r + b^{2} d\right )} \log \left (c\right )^{2} +{\left (2 \, b^{2} e n r \log \left (c\right ) + b^{2} e n^{2} \log \left (f\right ) + b^{2} d n^{2} +{\left (3 \, b^{2} e n^{2} + 2 \, a b e n\right )} r\right )} \log \left (x\right )^{2} +{\left (6 \, b^{2} e n^{2} + 4 \, a b e n + a^{2} e\right )} r + 2 \,{\left (b^{2} d n + a b d +{\left (2 \, b^{2} e n + a b e\right )} r\right )} \log \left (c\right ) +{\left (2 \, b^{2} e n^{2} + b^{2} e \log \left (c\right )^{2} + 2 \, a b e n + a^{2} e + 2 \,{\left (b^{2} e n + a b e\right )} \log \left (c\right )\right )} \log \left (f\right ) +{\left (b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 2 \, a b d n +{\left (6 \, b^{2} e n^{2} + 4 \, a b e n + a^{2} e\right )} r + 2 \,{\left (b^{2} d n +{\left (2 \, b^{2} e n + a b e\right )} r\right )} \log \left (c\right ) + 2 \,{\left (b^{2} e n^{2} + b^{2} e n \log \left (c\right ) + a b e n\right )} \log \left (f\right )\right )} \log \left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*e*n^2*r*log(x)^3 + 2*b^2*d*n^2 + 2*a*b*d*n + a^2*d + (b^2*e*r + b^2*d)*log(c)^2 + (2*b^2*e*n*r*log(c) +
b^2*e*n^2*log(f) + b^2*d*n^2 + (3*b^2*e*n^2 + 2*a*b*e*n)*r)*log(x)^2 + (6*b^2*e*n^2 + 4*a*b*e*n + a^2*e)*r + 2
*(b^2*d*n + a*b*d + (2*b^2*e*n + a*b*e)*r)*log(c) + (2*b^2*e*n^2 + b^2*e*log(c)^2 + 2*a*b*e*n + a^2*e + 2*(b^2
*e*n + a*b*e)*log(c))*log(f) + (b^2*e*r*log(c)^2 + 2*b^2*d*n^2 + 2*a*b*d*n + (6*b^2*e*n^2 + 4*a*b*e*n + a^2*e)
*r + 2*(b^2*d*n + (2*b^2*e*n + a*b*e)*r)*log(c) + 2*(b^2*e*n^2 + b^2*e*n*log(c) + a*b*e*n)*log(f))*log(x))/x

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Sympy [B]  time = 12.4992, size = 536, normalized size = 2.96 \begin{align*} - \frac{a^{2} d}{x} - \frac{a^{2} e r \log{\left (x \right )}}{x} - \frac{a^{2} e r}{x} - \frac{a^{2} e \log{\left (f \right )}}{x} - \frac{2 a b d n \log{\left (x \right )}}{x} - \frac{2 a b d n}{x} - \frac{2 a b d \log{\left (c \right )}}{x} - \frac{2 a b e n r \log{\left (x \right )}^{2}}{x} - \frac{4 a b e n r \log{\left (x \right )}}{x} - \frac{4 a b e n r}{x} - \frac{2 a b e n \log{\left (f \right )} \log{\left (x \right )}}{x} - \frac{2 a b e n \log{\left (f \right )}}{x} - \frac{2 a b e r \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{2 a b e r \log{\left (c \right )}}{x} - \frac{2 a b e \log{\left (c \right )} \log{\left (f \right )}}{x} - \frac{b^{2} d n^{2} \log{\left (x \right )}^{2}}{x} - \frac{2 b^{2} d n^{2} \log{\left (x \right )}}{x} - \frac{2 b^{2} d n^{2}}{x} - \frac{2 b^{2} d n \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{2 b^{2} d n \log{\left (c \right )}}{x} - \frac{b^{2} d \log{\left (c \right )}^{2}}{x} - \frac{b^{2} e n^{2} r \log{\left (x \right )}^{3}}{x} - \frac{3 b^{2} e n^{2} r \log{\left (x \right )}^{2}}{x} - \frac{6 b^{2} e n^{2} r \log{\left (x \right )}}{x} - \frac{6 b^{2} e n^{2} r}{x} - \frac{b^{2} e n^{2} \log{\left (f \right )} \log{\left (x \right )}^{2}}{x} - \frac{2 b^{2} e n^{2} \log{\left (f \right )} \log{\left (x \right )}}{x} - \frac{2 b^{2} e n^{2} \log{\left (f \right )}}{x} - \frac{2 b^{2} e n r \log{\left (c \right )} \log{\left (x \right )}^{2}}{x} - \frac{4 b^{2} e n r \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{4 b^{2} e n r \log{\left (c \right )}}{x} - \frac{2 b^{2} e n \log{\left (c \right )} \log{\left (f \right )} \log{\left (x \right )}}{x} - \frac{2 b^{2} e n \log{\left (c \right )} \log{\left (f \right )}}{x} - \frac{b^{2} e r \log{\left (c \right )}^{2} \log{\left (x \right )}}{x} - \frac{b^{2} e r \log{\left (c \right )}^{2}}{x} - \frac{b^{2} e \log{\left (c \right )}^{2} \log{\left (f \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*d/x - a**2*e*r*log(x)/x - a**2*e*r/x - a**2*e*log(f)/x - 2*a*b*d*n*log(x)/x - 2*a*b*d*n/x - 2*a*b*d*log(
c)/x - 2*a*b*e*n*r*log(x)**2/x - 4*a*b*e*n*r*log(x)/x - 4*a*b*e*n*r/x - 2*a*b*e*n*log(f)*log(x)/x - 2*a*b*e*n*
log(f)/x - 2*a*b*e*r*log(c)*log(x)/x - 2*a*b*e*r*log(c)/x - 2*a*b*e*log(c)*log(f)/x - b**2*d*n**2*log(x)**2/x
- 2*b**2*d*n**2*log(x)/x - 2*b**2*d*n**2/x - 2*b**2*d*n*log(c)*log(x)/x - 2*b**2*d*n*log(c)/x - b**2*d*log(c)*
*2/x - b**2*e*n**2*r*log(x)**3/x - 3*b**2*e*n**2*r*log(x)**2/x - 6*b**2*e*n**2*r*log(x)/x - 6*b**2*e*n**2*r/x
- b**2*e*n**2*log(f)*log(x)**2/x - 2*b**2*e*n**2*log(f)*log(x)/x - 2*b**2*e*n**2*log(f)/x - 2*b**2*e*n*r*log(c
)*log(x)**2/x - 4*b**2*e*n*r*log(c)*log(x)/x - 4*b**2*e*n*r*log(c)/x - 2*b**2*e*n*log(c)*log(f)*log(x)/x - 2*b
**2*e*n*log(c)*log(f)/x - b**2*e*r*log(c)**2*log(x)/x - b**2*e*r*log(c)**2/x - b**2*e*log(c)**2*log(f)/x

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Giac [B]  time = 1.33587, size = 529, normalized size = 2.92 \begin{align*} -\frac{b^{2} n^{2} r e \log \left (x\right )^{3} + 3 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 2 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + b^{2} n^{2} e \log \left (f\right ) \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e \log \left (x\right ) + 4 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 2 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + b^{2} d n^{2} \log \left (x\right )^{2} + 2 \, a b n r e \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 2 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, b^{2} d n^{2} \log \left (x\right ) + 4 \, a b n r e \log \left (x\right ) + 2 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 2 \, a b r e \log \left (c\right ) \log \left (x\right ) + 2 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 2 \, b^{2} d n \log \left (c\right ) + 2 \, a b r e \log \left (c\right ) + b^{2} d \log \left (c\right )^{2} + 2 \, a b n e \log \left (f\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) + 2 \, a b d n \log \left (x\right ) + a^{2} r e \log \left (x\right ) + 2 \, a b d n + a^{2} r e + 2 \, a b d \log \left (c\right ) + a^{2} e \log \left (f\right ) + a^{2} d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*n^2*r*e*log(x)^3 + 3*b^2*n^2*r*e*log(x)^2 + 2*b^2*n*r*e*log(c)*log(x)^2 + b^2*n^2*e*log(f)*log(x)^2 + 6*
b^2*n^2*r*e*log(x) + 4*b^2*n*r*e*log(c)*log(x) + b^2*r*e*log(c)^2*log(x) + 2*b^2*n^2*e*log(f)*log(x) + 2*b^2*n
*e*log(c)*log(f)*log(x) + b^2*d*n^2*log(x)^2 + 2*a*b*n*r*e*log(x)^2 + 6*b^2*n^2*r*e + 4*b^2*n*r*e*log(c) + b^2
*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 2*b^2*n*e*log(c)*log(f) + b^2*e*log(c)^2*log(f) + 2*b^2*d*n^2*log(x) + 4*
a*b*n*r*e*log(x) + 2*b^2*d*n*log(c)*log(x) + 2*a*b*r*e*log(c)*log(x) + 2*a*b*n*e*log(f)*log(x) + 2*b^2*d*n^2 +
4*a*b*n*r*e + 2*b^2*d*n*log(c) + 2*a*b*r*e*log(c) + b^2*d*log(c)^2 + 2*a*b*n*e*log(f) + 2*a*b*e*log(c)*log(f)
+ 2*a*b*d*n*log(x) + a^2*r*e*log(x) + 2*a*b*d*n + a^2*r*e + 2*a*b*d*log(c) + a^2*e*log(f) + a^2*d)/x