3.63 \(\int \frac{(a+b \log (c x^n))^3}{x^3} \, dx\)
Optimal. Leaf size=77 \[ -\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{3 b^3 n^3}{8 x^2} \]
[Out]
(-3*b^3*n^3)/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n]))/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2)/(4*x^2) - (a + b*
Log[c*x^n])^3/(2*x^2)
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Rubi [A] time = 0.0593631, antiderivative size = 77, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{2305, 2304} \[ -\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{3 b^3 n^3}{8 x^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Log[c*x^n])^3/x^3,x]
[Out]
(-3*b^3*n^3)/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n]))/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2)/(4*x^2) - (a + b*
Log[c*x^n])^3/(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
]
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{1}{2} (3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\\ &=-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{1}{2} \left (3 b^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{3 b^3 n^3}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0206204, size = 60, normalized size = 0.78 \[ -\frac{4 \left (a+b \log \left (c x^n\right )\right )^3+3 b n \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+2 b \log \left (c x^n\right )+b n\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Log[c*x^n])^3/x^3,x]
[Out]
-(4*(a + b*Log[c*x^n])^3 + 3*b*n*(2*(a + b*Log[c*x^n])^2 + b*n*(2*a + b*n + 2*b*Log[c*x^n])))/(8*x^2)
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Maple [C] time = 0.237, size = 2673, normalized size = 34.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))^3/x^3,x)
[Out]
-1/2*b^3/x^2*ln(x^n)^3-3/4*(I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-
I*Pi*b^3*csgn(I*c*x^n)^3+I*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)*b^3+b^3*n+2*a*b^2)/x^2*ln(x^n)^2-3/8*(4*a^
2*b+2*b^3*n^2-2*I*n*Pi*b^3*csgn(I*c*x^n)^3+8*ln(c)*a*b^2+4*n*ln(c)*b^3+2*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^3*
csgn(I*c)^2-4*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^3-4*I*Pi*a*b^2*csgn(I*c*x^n)^3+4*a*b^2*n+2*Pi^2*b^3*csgn(I*x^n)^2*c
sgn(I*c*x^n)^3*csgn(I*c)-Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^
n)^4*csgn(I*c)+4*ln(c)^2*b^3-Pi^2*b^3*csgn(I*c*x^n)^6-Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*Pi^2*b^3*csgn(I
*x^n)*csgn(I*c*x^n)^5+2*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)-Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2-2*I*n*Pi*b^3*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b^2*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)+4*I*Pi*a*b^2*csgn(I*c*x^n)^2*csgn(I*c)+4*I*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2
+4*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I*n*Pi*b^3*csgn(I*x^n)*
csgn(I*c*x^n)^2+2*I*n*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c))/x^2*ln(x^n)-1/16*(8*a^3+12*a*b^2*n^2+12*a^2*b*n-6*I*Pi
*b^3*n^2*csgn(I*c*x^n)^3-6*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+8*ln(c)^3*b^3-24*Pi^2*a*b^2*csgn(I*x^n)*cs
gn(I*c*x^n)^4*csgn(I*c)-3*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8-3*I*Pi^3*b^3*csgn(I*c*x^n)^8*csgn(I*c)+3*I*Pi
^3*b^3*csgn(I*c*x^n)^7*csgn(I*c)^2-I*Pi^3*b^3*csgn(I*c*x^n)^6*csgn(I*c)^3-12*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^3+
6*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b^3*n^2*csgn(I*c*x^n)^2*csgn(I*c)+6*Pi^2*b^3*n*csgn(I*x^n)^2
*csgn(I*c*x^n)^3*csgn(I*c)-12*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*I*ln(c)*Pi*b^3*n*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)-3*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-12*Pi^2*b^3*n*csgn(I*x^n)*csgn
(I*c*x^n)^4*csgn(I*c)-6*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+12*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^
n)^5+12*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)+6*Pi^2*b^3*n*csgn(I*c*x^n)^5*csgn(I*c)-3*Pi^2*b^3*n*csgn(I*c*
x^n)^4*csgn(I*c)^2-24*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+6*b^3*n^3+12*Pi^2*a*b^2*csgn(I*x^n)
*csgn(I*c*x^n)^5-12*I*Pi*a^2*b*csgn(I*c*x^n)^3-I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6+3*I*Pi^3*b^3*csgn(I*x^
n)^2*csgn(I*c*x^n)^7+12*ln(c)^2*b^3*n+12*ln(c)*b^3*n^2+24*ln(c)*a^2*b+24*ln(c)^2*a*b^2+I*Pi^3*b^3*csgn(I*c*x^n
)^9+12*Pi^2*a*b^2*csgn(I*c*x^n)^5*csgn(I*c)-6*Pi^2*a*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-6*ln(c)*Pi^2*b^3*csgn(I*c
*x^n)^4*csgn(I*c)^2-3*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4+6*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5-3*Pi^2
*b^3*n*csgn(I*c*x^n)^6-6*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^6-6*Pi^2*a*b^2*csgn(I*c*x^n)^6+12*ln(c)*Pi^2*b^3*csgn(I*
x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+12*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-6*ln(c)*Pi^2*b^3*csgn(I
*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-24*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+6*Pi^2*b^3*n*csgn(
I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+12*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-12*I*ln(c)^2*Pi*b
^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-6*Pi^2*a*b^2*csgn
(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+12*I*Pi*a^2*b*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*
c*x^n)^3*csgn(I*c)^3+3*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^3+12*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn
(I*c*x^n)^2+12*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)-24*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^3+12*I*Pi*a^2*b*cs
gn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)^3+9*I*Pi^3*b^3*csgn(I*x^n)*csgn
(I*c*x^n)^7*csgn(I*c)-9*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^6*csgn(I*c)^2+3*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c
*x^n)^5*csgn(I*c)-3*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^4*csgn(I*c)^2-9*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x
^n)^6*csgn(I*c)+9*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*csgn(I*c)^2+24*ln(c)*a*b^2*n+24*I*ln(c)*Pi*a*b^2*cs
gn(I*x^n)*csgn(I*c*x^n)^2+24*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^2*csgn(I*c)+12*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I
*c*x^n)^2+12*I*Pi*a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-6*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*n*
Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+12*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c)-12*I*ln(c)*Pi*b^3*n*csgn(I*
c*x^n)^3-12*I*Pi*a*b^2*n*csgn(I*c*x^n)^3-12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/x^2
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Maxima [A] time = 1.17584, size = 182, normalized size = 2.36 \begin{align*} -\frac{3}{8} \,{\left (n{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} + \frac{2 \, n \log \left (c x^{n}\right )^{2}}{x^{2}}\right )} b^{3} - \frac{3}{4} \, a b^{2}{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{b^{3} \log \left (c x^{n}\right )^{3}}{2 \, x^{2}} - \frac{3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac{3 \, a^{2} b n}{4 \, x^{2}} - \frac{3 \, a^{2} b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3/x^3,x, algorithm="maxima")
[Out]
-3/8*(n*(n^2/x^2 + 2*n*log(c*x^n)/x^2) + 2*n*log(c*x^n)^2/x^2)*b^3 - 3/4*a*b^2*(n^2/x^2 + 2*n*log(c*x^n)/x^2)
- 1/2*b^3*log(c*x^n)^3/x^2 - 3/2*a*b^2*log(c*x^n)^2/x^2 - 3/4*a^2*b*n/x^2 - 3/2*a^2*b*log(c*x^n)/x^2 - 1/2*a^3
/x^2
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Fricas [B] time = 0.938325, size = 436, normalized size = 5.66 \begin{align*} -\frac{4 \, b^{3} n^{3} \log \left (x\right )^{3} + 3 \, b^{3} n^{3} + 4 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 6 \, a^{2} b n + 4 \, a^{3} + 6 \,{\left (b^{3} n + 2 \, a b^{2}\right )} \log \left (c\right )^{2} + 6 \,{\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2} + 6 \,{\left (b^{3} n^{2} + 2 \, a b^{2} n + 2 \, a^{2} b\right )} \log \left (c\right ) + 6 \,{\left (b^{3} n^{3} + 2 \, b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 2 \, a^{2} b n + 2 \,{\left (b^{3} n^{2} + 2 \, a b^{2} n\right )} \log \left (c\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))^3/x^3,x, algorithm="fricas")
[Out]
-1/8*(4*b^3*n^3*log(x)^3 + 3*b^3*n^3 + 4*b^3*log(c)^3 + 6*a*b^2*n^2 + 6*a^2*b*n + 4*a^3 + 6*(b^3*n + 2*a*b^2)*
log(c)^2 + 6*(b^3*n^3 + 2*b^3*n^2*log(c) + 2*a*b^2*n^2)*log(x)^2 + 6*(b^3*n^2 + 2*a*b^2*n + 2*a^2*b)*log(c) +
6*(b^3*n^3 + 2*b^3*n*log(c)^2 + 2*a*b^2*n^2 + 2*a^2*b*n + 2*(b^3*n^2 + 2*a*b^2*n)*log(c))*log(x))/x^2
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Sympy [B] time = 1.69011, size = 338, normalized size = 4.39 \begin{align*} - \frac{a^{3}}{2 x^{2}} - \frac{3 a^{2} b n \log{\left (x \right )}}{2 x^{2}} - \frac{3 a^{2} b n}{4 x^{2}} - \frac{3 a^{2} b \log{\left (c \right )}}{2 x^{2}} - \frac{3 a b^{2} n^{2} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{3 a b^{2} n^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{3 a b^{2} n^{2}}{4 x^{2}} - \frac{3 a b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{3 a b^{2} n \log{\left (c \right )}}{2 x^{2}} - \frac{3 a b^{2} \log{\left (c \right )}^{2}}{2 x^{2}} - \frac{b^{3} n^{3} \log{\left (x \right )}^{3}}{2 x^{2}} - \frac{3 b^{3} n^{3} \log{\left (x \right )}^{2}}{4 x^{2}} - \frac{3 b^{3} n^{3} \log{\left (x \right )}}{4 x^{2}} - \frac{3 b^{3} n^{3}}{8 x^{2}} - \frac{3 b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{3 b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}}{2 x^{2}} - \frac{3 b^{3} n^{2} \log{\left (c \right )}}{4 x^{2}} - \frac{3 b^{3} n \log{\left (c \right )}^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{3 b^{3} n \log{\left (c \right )}^{2}}{4 x^{2}} - \frac{b^{3} \log{\left (c \right )}^{3}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*ln(c*x**n))**3/x**3,x)
[Out]
-a**3/(2*x**2) - 3*a**2*b*n*log(x)/(2*x**2) - 3*a**2*b*n/(4*x**2) - 3*a**2*b*log(c)/(2*x**2) - 3*a*b**2*n**2*l
og(x)**2/(2*x**2) - 3*a*b**2*n**2*log(x)/(2*x**2) - 3*a*b**2*n**2/(4*x**2) - 3*a*b**2*n*log(c)*log(x)/x**2 - 3
*a*b**2*n*log(c)/(2*x**2) - 3*a*b**2*log(c)**2/(2*x**2) - b**3*n**3*log(x)**3/(2*x**2) - 3*b**3*n**3*log(x)**2
/(4*x**2) - 3*b**3*n**3*log(x)/(4*x**2) - 3*b**3*n**3/(8*x**2) - 3*b**3*n**2*log(c)*log(x)**2/(2*x**2) - 3*b**
3*n**2*log(c)*log(x)/(2*x**2) - 3*b**3*n**2*log(c)/(4*x**2) - 3*b**3*n*log(c)**2*log(x)/(2*x**2) - 3*b**3*n*lo
g(c)**2/(4*x**2) - b**3*log(c)**3/(2*x**2)
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Giac [B] time = 1.22237, size = 274, normalized size = 3.56 \begin{align*} -\frac{b^{3} n^{3} \log \left (x\right )^{3}}{2 \, x^{2}} - \frac{3 \,{\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2}}{4 \, x^{2}} - \frac{3 \,{\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 4 \, a b^{2} n \log \left (c\right ) + 2 \, a^{2} b n\right )} \log \left (x\right )}{4 \, x^{2}} - \frac{3 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 6 \, b^{3} n \log \left (c\right )^{2} + 4 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 12 \, a b^{2} n \log \left (c\right ) + 12 \, a b^{2} \log \left (c\right )^{2} + 6 \, a^{2} b n + 12 \, a^{2} b \log \left (c\right ) + 4 \, a^{3}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3/x^3,x, algorithm="giac")
[Out]
-1/2*b^3*n^3*log(x)^3/x^2 - 3/4*(b^3*n^3 + 2*b^3*n^2*log(c) + 2*a*b^2*n^2)*log(x)^2/x^2 - 3/4*(b^3*n^3 + 2*b^3
*n^2*log(c) + 2*b^3*n*log(c)^2 + 2*a*b^2*n^2 + 4*a*b^2*n*log(c) + 2*a^2*b*n)*log(x)/x^2 - 1/8*(3*b^3*n^3 + 6*b
^3*n^2*log(c) + 6*b^3*n*log(c)^2 + 4*b^3*log(c)^3 + 6*a*b^2*n^2 + 12*a*b^2*n*log(c) + 12*a*b^2*log(c)^2 + 6*a^
2*b*n + 12*a^2*b*log(c) + 4*a^3)/x^2