3.169 \(\int \frac{(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^4} \, dx\)

Optimal. Leaf size=205 \[ -\frac{e r \left (9 a^2+6 a b n+2 b^2 n^2\right )}{81 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac{2 b e r (3 a+b n) \log \left (c x^n\right )}{27 x^3}-\frac{2 b e n r (3 a+b n)}{81 x^3}-\frac{b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac{2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b^2 e n^2 r}{81 x^3} \]

[Out]

(-2*b^2*e*n^2*r)/(81*x^3) - (2*b*e*n*(3*a + b*n)*r)/(81*x^3) - (e*(9*a^2 + 6*a*b*n + 2*b^2*n^2)*r)/(81*x^3) -
(2*b^2*e*n*r*Log[c*x^n])/(27*x^3) - (2*b*e*(3*a + b*n)*r*Log[c*x^n])/(27*x^3) - (b^2*e*r*Log[c*x^n]^2)/(9*x^3)
- (2*b^2*n^2*(d + e*Log[f*x^r]))/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*L
og[c*x^n])^2*(d + e*Log[f*x^r]))/(3*x^3)

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Rubi [A]  time = 0.211357, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2305, 2304, 2366, 12, 14} \[ -\frac{e r \left (9 a^2+6 a b n+2 b^2 n^2\right )}{81 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac{2 b e r (3 a+b n) \log \left (c x^n\right )}{27 x^3}-\frac{2 b e n r (3 a+b n)}{81 x^3}-\frac{b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac{2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b^2 e n^2 r}{81 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*e*n^2*r)/(81*x^3) - (2*b*e*n*(3*a + b*n)*r)/(81*x^3) - (e*(9*a^2 + 6*a*b*n + 2*b^2*n^2)*r)/(81*x^3) -
(2*b^2*e*n*r*Log[c*x^n])/(27*x^3) - (2*b*e*(3*a + b*n)*r*Log[c*x^n])/(27*x^3) - (b^2*e*r*Log[c*x^n]^2)/(9*x^3)
- (2*b^2*n^2*(d + e*Log[f*x^r]))/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*L
og[c*x^n])^2*(d + e*Log[f*x^r]))/(3*x^3)

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^4} \, dx &=-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-(e r) \int \frac{-9 a^2 \left (1+\frac{2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{27 x^4} \, dx\\ &=-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac{1}{27} (e r) \int \frac{-9 a^2 \left (1+\frac{2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{x^4} \, dx\\ &=-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac{1}{27} (e r) \int \left (\frac{-9 a^2-6 a b n-2 b^2 n^2}{x^4}-\frac{6 b (3 a+b n) \log \left (c x^n\right )}{x^4}-\frac{9 b^2 \log ^2\left (c x^n\right )}{x^4}\right ) \, dx\\ &=-\frac{e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac{1}{3} \left (b^2 e r\right ) \int \frac{\log ^2\left (c x^n\right )}{x^4} \, dx+\frac{1}{9} (2 b e (3 a+b n) r) \int \frac{\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac{2 b e n (3 a+b n) r}{81 x^3}-\frac{e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac{2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac{b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac{1}{9} \left (2 b^2 e n r\right ) \int \frac{\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac{2 b^2 e n^2 r}{81 x^3}-\frac{2 b e n (3 a+b n) r}{81 x^3}-\frac{e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac{2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac{2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac{b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac{2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.156018, size = 155, normalized size = 0.76 \[ -\frac{e \left (9 a^2+6 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+9 a^2 d+3 a^2 e r+2 b \log \left (c x^n\right ) \left (3 e (3 a+b n) \log \left (f x^r\right )+9 a d+3 a e r+3 b d n+2 b e n r\right )+6 a b d n+4 a b e n r+3 b^2 \log ^2\left (c x^n\right ) \left (3 d+3 e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+2 b^2 e n^2 r}{27 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(9*a^2*d + 6*a*b*d*n + 2*b^2*d*n^2 + 3*a^2*e*r + 4*a*b*e*n*r + 2*b^2*e*n^2*r + e*(9*a^2 + 6*a*b*n + 2*b^2*n^2
)*Log[f*x^r] + 3*b^2*Log[c*x^n]^2*(3*d + e*r + 3*e*Log[f*x^r]) + 2*b*Log[c*x^n]*(9*a*d + 3*b*d*n + 3*a*e*r + 2
*b*e*n*r + 3*e*(3*a + b*n)*Log[f*x^r]))/(27*x^3)

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Maple [C]  time = 0.71, size = 8407, normalized size = 41. \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^4,x)

[Out]

result too large to display

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Maxima [A]  time = 1.23511, size = 311, normalized size = 1.52 \begin{align*} -\frac{1}{9} \, b^{2} e{\left (\frac{r}{x^{3}} + \frac{3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right )^{2} - \frac{2}{9} \, a b e{\left (\frac{r}{x^{3}} + \frac{3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right ) - \frac{2}{27} \, b^{2} e{\left (\frac{{\left (r \log \left (x\right ) + r + \log \left (f\right )\right )} n^{2}}{x^{3}} + \frac{n{\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{2}{27} \, b^{2} d{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{2 \, a b e n{\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )}}{27 \, x^{3}} - \frac{b^{2} d \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac{2 \, a b d n}{9 \, x^{3}} - \frac{a^{2} e r}{9 \, x^{3}} - \frac{2 \, a b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a^{2} e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac{a^{2} d}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

-1/9*b^2*e*(r/x^3 + 3*log(f*x^r)/x^3)*log(c*x^n)^2 - 2/9*a*b*e*(r/x^3 + 3*log(f*x^r)/x^3)*log(c*x^n) - 2/27*b^
2*e*((r*log(x) + r + log(f))*n^2/x^3 + n*(2*r + 3*log(f) + 3*log(x^r))*log(c*x^n)/x^3) - 2/27*b^2*d*(n^2/x^3 +
3*n*log(c*x^n)/x^3) - 2/27*a*b*e*n*(2*r + 3*log(f) + 3*log(x^r))/x^3 - 1/3*b^2*d*log(c*x^n)^2/x^3 - 2/9*a*b*d
*n/x^3 - 1/9*a^2*e*r/x^3 - 2/3*a*b*d*log(c*x^n)/x^3 - 1/3*a^2*e*log(f*x^r)/x^3 - 1/3*a^2*d/x^3

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Fricas [A]  time = 0.832708, size = 801, normalized size = 3.91 \begin{align*} -\frac{9 \, b^{2} e n^{2} r \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 6 \, a b d n + 9 \, a^{2} d + 3 \,{\left (b^{2} e r + 3 \, b^{2} d\right )} \log \left (c\right )^{2} + 9 \,{\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 (b^{2} e n^{2} + 2 \, a b e n\right )} r\right )} \log \left (x\right )^{2} +{\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \,{\left (3 \, b^{2} d n + 9 \, a b d +{\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \left (c\right ) +{\left (2 \, b^{2} e n^{2} + 9 \, b^{2} e \log \left (c\right )^{2} + 6 \, a b e n + 9 \, a^{2} e + 6 \,{\left (b^{2} e n + 3 \, a b e\right )} \log \left (c\right )\right )} \log \left (f\right ) + 3 \,{\left (3 \, b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 6 \, a b d n +{\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \,{\left (3 \, b^{2} d n +{\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \left (c\right ) + 2 \,{\left (b^{2} e n^{2} + 3 \, b^{2} e n \log \left (c\right ) + 3 \, a b e n\right )} \log \left (f\right )\right )} \log \left (x\right )}{27 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

-1/27*(9*b^2*e*n^2*r*log(x)^3 + 2*b^2*d*n^2 + 6*a*b*d*n + 9*a^2*d + 3*(b^2*e*r + 3*b^2*d)*log(c)^2 + 9*(2*b^2*
e*n*r*log(c) + b^2*e*n^2*log(f) + b^2*d*n^2 + (b^2*e*n^2 + 2*a*b*e*n)*r)*log(x)^2 + (2*b^2*e*n^2 + 4*a*b*e*n +
3*a^2*e)*r + 2*(3*b^2*d*n + 9*a*b*d + (2*b^2*e*n + 3*a*b*e)*r)*log(c) + (2*b^2*e*n^2 + 9*b^2*e*log(c)^2 + 6*a
*b*e*n + 9*a^2*e + 6*(b^2*e*n + 3*a*b*e)*log(c))*log(f) + 3*(3*b^2*e*r*log(c)^2 + 2*b^2*d*n^2 + 6*a*b*d*n + (2
*b^2*e*n^2 + 4*a*b*e*n + 3*a^2*e)*r + 2*(3*b^2*d*n + (2*b^2*e*n + 3*a*b*e)*r)*log(c) + 2*(b^2*e*n^2 + 3*b^2*e*
n*log(c) + 3*a*b*e*n)*log(f))*log(x))/x^3

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Sympy [B]  time = 34.7083, size = 656, normalized size = 3.2 \begin{align*} \text{result too large to display} \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**4,x)

[Out]

-a**2*d/(3*x**3) - a**2*e*r*log(x)/(3*x**3) - a**2*e*r/(9*x**3) - a**2*e*log(f)/(3*x**3) - 2*a*b*d*n*log(x)/(3
*x**3) - 2*a*b*d*n/(9*x**3) - 2*a*b*d*log(c)/(3*x**3) - 2*a*b*e*n*r*log(x)**2/(3*x**3) - 4*a*b*e*n*r*log(x)/(9
*x**3) - 4*a*b*e*n*r/(27*x**3) - 2*a*b*e*n*log(f)*log(x)/(3*x**3) - 2*a*b*e*n*log(f)/(9*x**3) - 2*a*b*e*r*log(
c)*log(x)/(3*x**3) - 2*a*b*e*r*log(c)/(9*x**3) - 2*a*b*e*log(c)*log(f)/(3*x**3) - b**2*d*n**2*log(x)**2/(3*x**
3) - 2*b**2*d*n**2*log(x)/(9*x**3) - 2*b**2*d*n**2/(27*x**3) - 2*b**2*d*n*log(c)*log(x)/(3*x**3) - 2*b**2*d*n*
log(c)/(9*x**3) - b**2*d*log(c)**2/(3*x**3) - b**2*e*n**2*r*log(x)**3/(3*x**3) - b**2*e*n**2*r*log(x)**2/(3*x*
*3) - 2*b**2*e*n**2*r*log(x)/(9*x**3) - 2*b**2*e*n**2*r/(27*x**3) - b**2*e*n**2*log(f)*log(x)**2/(3*x**3) - 2*
b**2*e*n**2*log(f)*log(x)/(9*x**3) - 2*b**2*e*n**2*log(f)/(27*x**3) - 2*b**2*e*n*r*log(c)*log(x)**2/(3*x**3) -
4*b**2*e*n*r*log(c)*log(x)/(9*x**3) - 4*b**2*e*n*r*log(c)/(27*x**3) - 2*b**2*e*n*log(c)*log(f)*log(x)/(3*x**3
) - 2*b**2*e*n*log(c)*log(f)/(9*x**3) - b**2*e*r*log(c)**2*log(x)/(3*x**3) - b**2*e*r*log(c)**2/(9*x**3) - b**
2*e*log(c)**2*log(f)/(3*x**3)

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Giac [B]  time = 1.27977, size = 544, normalized size = 2.65 \begin{align*} -\frac{9 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 9 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 18 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + 9 \, 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 ) + 12 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + 9 \, b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 6 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 18 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + 9 \, b^{2} d n^{2} \log \left (x\right )^{2} + 18 \, a b n r e \log \left (x\right )^{2} + 2 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + 3 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 6 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + 9 \, b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 6 \, b^{2} d n^{2} \log \left (x\right ) + 12 \, a b n r e \log \left (x\right ) + 18 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 18 \, a b r e \log \left (c\right ) \log \left (x\right ) + 18 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 6 \, b^{2} d n \log \left (c\right ) + 6 \, a b r e \log \left (c\right ) + 9 \, b^{2} d \log \left (c\right )^{2} + 6 \, a b n e \log \left (f\right ) + 18 \, a b e \log \left (c\right ) \log \left (f\right ) + 18 \, a b d n \log \left (x\right ) + 9 \, a^{2} r e \log \left (x\right ) + 6 \, a b d n + 3 \, a^{2} r e + 18 \, a b d \log \left (c\right ) + 9 \, a^{2} e \log \left (f\right ) + 9 \, a^{2} d}{27 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

-1/27*(9*b^2*n^2*r*e*log(x)^3 + 9*b^2*n^2*r*e*log(x)^2 + 18*b^2*n*r*e*log(c)*log(x)^2 + 9*b^2*n^2*e*log(f)*log
(x)^2 + 6*b^2*n^2*r*e*log(x) + 12*b^2*n*r*e*log(c)*log(x) + 9*b^2*r*e*log(c)^2*log(x) + 6*b^2*n^2*e*log(f)*log
(x) + 18*b^2*n*e*log(c)*log(f)*log(x) + 9*b^2*d*n^2*log(x)^2 + 18*a*b*n*r*e*log(x)^2 + 2*b^2*n^2*r*e + 4*b^2*n
*r*e*log(c) + 3*b^2*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 6*b^2*n*e*log(c)*log(f) + 9*b^2*e*log(c)^2*log(f) + 6*
b^2*d*n^2*log(x) + 12*a*b*n*r*e*log(x) + 18*b^2*d*n*log(c)*log(x) + 18*a*b*r*e*log(c)*log(x) + 18*a*b*n*e*log(
f)*log(x) + 2*b^2*d*n^2 + 4*a*b*n*r*e + 6*b^2*d*n*log(c) + 6*a*b*r*e*log(c) + 9*b^2*d*log(c)^2 + 6*a*b*n*e*log
(f) + 18*a*b*e*log(c)*log(f) + 18*a*b*d*n*log(x) + 9*a^2*r*e*log(x) + 6*a*b*d*n + 3*a^2*r*e + 18*a*b*d*log(c)
+ 9*a^2*e*log(f) + 9*a^2*d)/x^3