3.168 \(\int \frac{(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^3} \, dx\)

Optimal. Leaf size=204 \[ -\frac{e r \left (2 a^2+2 a b n+b^2 n^2\right )}{8 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{b e r (2 a+b n) \log \left (c x^n\right )}{4 x^2}-\frac{b e n r (2 a+b n)}{8 x^2}-\frac{b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac{b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b^2 e n^2 r}{8 x^2} \]

[Out]

-(b^2*e*n^2*r)/(8*x^2) - (b*e*n*(2*a + b*n)*r)/(8*x^2) - (e*(2*a^2 + 2*a*b*n + b^2*n^2)*r)/(8*x^2) - (b^2*e*n*
r*Log[c*x^n])/(4*x^2) - (b*e*(2*a + b*n)*r*Log[c*x^n])/(4*x^2) - (b^2*e*r*Log[c*x^n]^2)/(4*x^2) - (b^2*n^2*(d
+ e*Log[f*x^r]))/(4*x^2) - (b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(2*x^2) - ((a + b*Log[c*x^n])^2*(d + e*
Log[f*x^r]))/(2*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.205904, antiderivative size = 204, 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} \[ -\frac{e r \left (2 a^2+2 a b n+b^2 n^2\right )}{8 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{b e r (2 a+b n) \log \left (c x^n\right )}{4 x^2}-\frac{b e n r (2 a+b n)}{8 x^2}-\frac{b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac{b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b^2 e n^2 r}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x^3,x]

[Out]

-(b^2*e*n^2*r)/(8*x^2) - (b*e*n*(2*a + b*n)*r)/(8*x^2) - (e*(2*a^2 + 2*a*b*n + b^2*n^2)*r)/(8*x^2) - (b^2*e*n*
r*Log[c*x^n])/(4*x^2) - (b*e*(2*a + b*n)*r*Log[c*x^n])/(4*x^2) - (b^2*e*r*Log[c*x^n]^2)/(4*x^2) - (b^2*n^2*(d
+ e*Log[f*x^r]))/(4*x^2) - (b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(2*x^2) - ((a + b*Log[c*x^n])^2*(d + e*
Log[f*x^r]))/(2*x^2)

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^3} \, dx &=-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-(e r) \int \frac{-2 a^2 \left (1+\frac{b n (2 a+b n)}{2 a^2}\right )-2 b (2 a+b n) \log \left (c x^n\right )-2 b^2 \log ^2\left (c x^n\right )}{4 x^3} \, dx\\ &=-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{1}{4} (e r) \int \frac{-2 a^2 \left (1+\frac{b n (2 a+b n)}{2 a^2}\right )-2 b (2 a+b n) \log \left (c x^n\right )-2 b^2 \log ^2\left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{1}{4} (e r) \int \left (\frac{-2 a^2-2 a b n-b^2 n^2}{x^3}-\frac{2 b (2 a+b n) \log \left (c x^n\right )}{x^3}-\frac{2 b^2 \log ^2\left (c x^n\right )}{x^3}\right ) \, dx\\ &=-\frac{e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac{1}{2} \left (b^2 e r\right ) \int \frac{\log ^2\left (c x^n\right )}{x^3} \, dx+\frac{1}{2} (b e (2 a+b n) r) \int \frac{\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b e n (2 a+b n) r}{8 x^2}-\frac{e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac{b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac{b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac{1}{2} \left (b^2 e n r\right ) \int \frac{\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b^2 e n^2 r}{8 x^2}-\frac{b e n (2 a+b n) r}{8 x^2}-\frac{e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac{b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac{b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac{b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac{b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.154521, size = 151, normalized size = 0.74 \[ -\frac{2 e \left (2 a^2+2 a b n+b^2 n^2\right ) \log \left (f x^r\right )+4 a^2 d+2 a^2 e r+4 b \log \left (c x^n\right ) \left (e (2 a+b n) \log \left (f x^r\right )+2 a d+a e r+b d n+b e n r\right )+4 a b d n+4 a b e n r+2 b^2 \log ^2\left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+3 b^2 e n^2 r}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x^3,x]

[Out]

-(4*a^2*d + 4*a*b*d*n + 2*b^2*d*n^2 + 2*a^2*e*r + 4*a*b*e*n*r + 3*b^2*e*n^2*r + 2*e*(2*a^2 + 2*a*b*n + b^2*n^2
)*Log[f*x^r] + 2*b^2*Log[c*x^n]^2*(2*d + e*r + 2*e*Log[f*x^r]) + 4*b*Log[c*x^n]*(2*a*d + b*d*n + a*e*r + b*e*n
*r + e*(2*a + b*n)*Log[f*x^r]))/(8*x^2)

________________________________________________________________________________________

Maple [C]  time = 0.671, size = 8407, normalized size = 41.2 \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^3,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [A]  time = 1.21626, size = 302, normalized size = 1.48 \begin{align*} -\frac{1}{4} \, b^{2} e{\left (\frac{r}{x^{2}} + \frac{2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right )^{2} - \frac{1}{2} \, a b e{\left (\frac{r}{x^{2}} + \frac{2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right ) - \frac{1}{8} \, b^{2} e{\left (\frac{{\left (2 \, r \log \left (x\right ) + 3 \, r + 2 \, \log \left (f\right )\right )} n^{2}}{x^{2}} + \frac{4 \, n{\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{1}{4} \, b^{2} d{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{a b e n{\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )}}{2 \, x^{2}} - \frac{b^{2} d \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac{a b d n}{2 \, x^{2}} - \frac{a^{2} e r}{4 \, x^{2}} - \frac{a b d \log \left (c x^{n}\right )}{x^{2}} - \frac{a^{2} e \log \left (f x^{r}\right )}{2 \, x^{2}} - \frac{a^{2} d}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

-1/4*b^2*e*(r/x^2 + 2*log(f*x^r)/x^2)*log(c*x^n)^2 - 1/2*a*b*e*(r/x^2 + 2*log(f*x^r)/x^2)*log(c*x^n) - 1/8*b^2
*e*((2*r*log(x) + 3*r + 2*log(f))*n^2/x^2 + 4*n*(r + log(f) + log(x^r))*log(c*x^n)/x^2) - 1/4*b^2*d*(n^2/x^2 +
2*n*log(c*x^n)/x^2) - 1/2*a*b*e*n*(r + log(f) + log(x^r))/x^2 - 1/2*b^2*d*log(c*x^n)^2/x^2 - 1/2*a*b*d*n/x^2
- 1/4*a^2*e*r/x^2 - a*b*d*log(c*x^n)/x^2 - 1/2*a^2*e*log(f*x^r)/x^2 - 1/2*a^2*d/x^2

________________________________________________________________________________________

Fricas [A]  time = 0.967525, size = 791, normalized size = 3.88 \begin{align*} -\frac{4 \, b^{2} e n^{2} r \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 4 \, a b d n + 4 \, a^{2} d + 2 \,{\left (b^{2} e r + 2 \, b^{2} d\right )} \log \left (c\right )^{2} + 2 \,{\left (4 \, b^{2} e n r \log \left (c\right ) + 2 \, b^{2} e n^{2} \log \left (f\right ) + 2 \, b^{2} d n^{2} +{\left (3 \, b^{2} e n^{2} + 4 \, a b e n\right )} r\right )} \log \left (x\right )^{2} +{\left (3 \, b^{2} e n^{2} + 4 \, a b e n + 2 \, a^{2} e\right )} r + 4 \,{\left (b^{2} d n + 2 \, a b d +{\left (b^{2} e n + a b e\right )} r\right )} \log \left (c\right ) + 2 \,{\left (b^{2} e n^{2} + 2 \, b^{2} e \log \left (c\right )^{2} + 2 \, a b e n + 2 \, a^{2} e + 2 \,{\left (b^{2} e n + 2 \, a b e\right )} \log \left (c\right )\right )} \log \left (f\right ) + 2 \,{\left (2 \, b^{2} e r \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 4 \, a b d n +{\left (3 \, b^{2} e n^{2} + 4 \, a b e n + 2 \, a^{2} e\right )} r + 4 \,{\left (b^{2} d n +{\left (b^{2} e n + a b e\right )} r\right )} \log \left (c\right ) + 2 \,{\left (b^{2} e n^{2} + 2 \, b^{2} e n \log \left (c\right ) + 2 \, a b e n\right )} \log \left (f\right )\right )} \log \left (x\right )}{8 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

-1/8*(4*b^2*e*n^2*r*log(x)^3 + 2*b^2*d*n^2 + 4*a*b*d*n + 4*a^2*d + 2*(b^2*e*r + 2*b^2*d)*log(c)^2 + 2*(4*b^2*e
*n*r*log(c) + 2*b^2*e*n^2*log(f) + 2*b^2*d*n^2 + (3*b^2*e*n^2 + 4*a*b*e*n)*r)*log(x)^2 + (3*b^2*e*n^2 + 4*a*b*
e*n + 2*a^2*e)*r + 4*(b^2*d*n + 2*a*b*d + (b^2*e*n + a*b*e)*r)*log(c) + 2*(b^2*e*n^2 + 2*b^2*e*log(c)^2 + 2*a*
b*e*n + 2*a^2*e + 2*(b^2*e*n + 2*a*b*e)*log(c))*log(f) + 2*(2*b^2*e*r*log(c)^2 + 2*b^2*d*n^2 + 4*a*b*d*n + (3*
b^2*e*n^2 + 4*a*b*e*n + 2*a^2*e)*r + 4*(b^2*d*n + (b^2*e*n + a*b*e)*r)*log(c) + 2*(b^2*e*n^2 + 2*b^2*e*n*log(c
) + 2*a*b*e*n)*log(f))*log(x))/x^2

________________________________________________________________________________________

Sympy [B]  time = 12.4679, size = 602, normalized size = 2.95 \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**3,x)

[Out]

-a**2*d/(2*x**2) - a**2*e*r*log(x)/(2*x**2) - a**2*e*r/(4*x**2) - a**2*e*log(f)/(2*x**2) - a*b*d*n*log(x)/x**2
- a*b*d*n/(2*x**2) - a*b*d*log(c)/x**2 - a*b*e*n*r*log(x)**2/x**2 - a*b*e*n*r*log(x)/x**2 - a*b*e*n*r/(2*x**2
) - a*b*e*n*log(f)*log(x)/x**2 - a*b*e*n*log(f)/(2*x**2) - a*b*e*r*log(c)*log(x)/x**2 - a*b*e*r*log(c)/(2*x**2
) - a*b*e*log(c)*log(f)/x**2 - b**2*d*n**2*log(x)**2/(2*x**2) - b**2*d*n**2*log(x)/(2*x**2) - b**2*d*n**2/(4*x
**2) - b**2*d*n*log(c)*log(x)/x**2 - b**2*d*n*log(c)/(2*x**2) - b**2*d*log(c)**2/(2*x**2) - b**2*e*n**2*r*log(
x)**3/(2*x**2) - 3*b**2*e*n**2*r*log(x)**2/(4*x**2) - 3*b**2*e*n**2*r*log(x)/(4*x**2) - 3*b**2*e*n**2*r/(8*x**
2) - b**2*e*n**2*log(f)*log(x)**2/(2*x**2) - b**2*e*n**2*log(f)*log(x)/(2*x**2) - b**2*e*n**2*log(f)/(4*x**2)
- b**2*e*n*r*log(c)*log(x)**2/x**2 - b**2*e*n*r*log(c)*log(x)/x**2 - b**2*e*n*r*log(c)/(2*x**2) - b**2*e*n*log
(c)*log(f)*log(x)/x**2 - b**2*e*n*log(c)*log(f)/(2*x**2) - b**2*e*r*log(c)**2*log(x)/(2*x**2) - b**2*e*r*log(c
)**2/(4*x**2) - b**2*e*log(c)**2*log(f)/(2*x**2)

________________________________________________________________________________________

Giac [B]  time = 1.26367, size = 544, normalized size = 2.67 \begin{align*} -\frac{4 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 6 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 8 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + 4 \, 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 ) + 8 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + 4 \, b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 4 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 8 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + 4 \, b^{2} d n^{2} \log \left (x\right )^{2} + 8 \, a b n r e \log \left (x\right )^{2} + 3 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + 2 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 4 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + 4 \, b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 4 \, b^{2} d n^{2} \log \left (x\right ) + 8 \, a b n r e \log \left (x\right ) + 8 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 8 \, a b r e \log \left (c\right ) \log \left (x\right ) + 8 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 4 \, b^{2} d n \log \left (c\right ) + 4 \, a b r e \log \left (c\right ) + 4 \, b^{2} d \log \left (c\right )^{2} + 4 \, a b n e \log \left (f\right ) + 8 \, a b e \log \left (c\right ) \log \left (f\right ) + 8 \, a b d n \log \left (x\right ) + 4 \, a^{2} r e \log \left (x\right ) + 4 \, a b d n + 2 \, a^{2} r e + 8 \, a b d \log \left (c\right ) + 4 \, a^{2} e \log \left (f\right ) + 4 \, a^{2} d}{8 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

-1/8*(4*b^2*n^2*r*e*log(x)^3 + 6*b^2*n^2*r*e*log(x)^2 + 8*b^2*n*r*e*log(c)*log(x)^2 + 4*b^2*n^2*e*log(f)*log(x
)^2 + 6*b^2*n^2*r*e*log(x) + 8*b^2*n*r*e*log(c)*log(x) + 4*b^2*r*e*log(c)^2*log(x) + 4*b^2*n^2*e*log(f)*log(x)
+ 8*b^2*n*e*log(c)*log(f)*log(x) + 4*b^2*d*n^2*log(x)^2 + 8*a*b*n*r*e*log(x)^2 + 3*b^2*n^2*r*e + 4*b^2*n*r*e*
log(c) + 2*b^2*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 4*b^2*n*e*log(c)*log(f) + 4*b^2*e*log(c)^2*log(f) + 4*b^2*d
*n^2*log(x) + 8*a*b*n*r*e*log(x) + 8*b^2*d*n*log(c)*log(x) + 8*a*b*r*e*log(c)*log(x) + 8*a*b*n*e*log(f)*log(x)
+ 2*b^2*d*n^2 + 4*a*b*n*r*e + 4*b^2*d*n*log(c) + 4*a*b*r*e*log(c) + 4*b^2*d*log(c)^2 + 4*a*b*n*e*log(f) + 8*a
*b*e*log(c)*log(f) + 8*a*b*d*n*log(x) + 4*a^2*r*e*log(x) + 4*a*b*d*n + 2*a^2*r*e + 8*a*b*d*log(c) + 4*a^2*e*lo
g(f) + 4*a^2*d)/x^2