3.58 \(\int x^2 (a+b \log (c x^n))^3 \, dx\)

Optimal. Leaf size=77 \[ \frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{27} b^3 n^3 x^3 \]

[Out]

(-2*b^3*n^3*x^3)/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/9 - (b*n*x^3*(a + b*Log[c*x^n])^2)/3 + (x^3*(a + b*Lo
g[c*x^n])^3)/3

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Rubi [A]  time = 0.0600692, 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} \[ \frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{27} b^3 n^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])^3,x]

[Out]

(-2*b^3*n^3*x^3)/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/9 - (b*n*x^3*(a + b*Log[c*x^n])^2)/3 + (x^3*(a + b*Lo
g[c*x^n])^3)/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
]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3-(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{3} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{2}{27} b^3 n^3 x^3+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0142996, size = 67, normalized size = 0.87 \[ \frac{1}{3} \left (x^3 \left (a+b \log \left (c x^n\right )\right )^3-b n \left (x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{9} b n x^3 \left (-3 a-3 b \log \left (c x^n\right )+b n\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])^3,x]

[Out]

(x^3*(a + b*Log[c*x^n])^3 - b*n*((2*b*n*x^3*(-3*a + b*n - 3*b*Log[c*x^n]))/9 + x^3*(a + b*Log[c*x^n])^2))/3

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Maple [C]  time = 0.332, size = 2650, normalized size = 34.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))^3,x)

[Out]

1/3*b^3*x^3*ln(x^n)^3+1/6*b^2*x^3*(3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)-3*I*b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)-2*b*n+6*a)*ln(x^n)^2+1/36*b*x^3*(
36*ln(c)^2*b^2-9*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-24*a*b*n-36*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-36*I*Pi*a*b*c
sgn(I*c*x^n)^3+12*I*Pi*b^2*n*csgn(I*c*x^n)^3+8*b^2*n^2+36*a^2+18*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c
)^2+18*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-3
6*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-9*Pi^2*b^2*csgn(I*c*x^n)^6+72*ln(c)*a*b-24*ln(c)*b^2*n+18*Pi^
2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+18*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+36*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn
(I*c)+36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b^2*n*csgn(I*x^n)*
csgn(I*c*x^n)^2-12*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-9*Pi^2*b
^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*I*Pi*b^2*n*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)-36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))*ln(x^n)+1/216*x^3*(72*a^3+48*a*b^2
*n^2-72*a^2*b*n-54*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+72*ln(c)^3*b^3-216*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c
*x^n)^4*csgn(I*c)-36*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+18*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x
^n)^2*csgn(I*c)^2+72*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-54*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c
*x^n)^4+108*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+108*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)-36*Pi^2*b^
3*n*csgn(I*c*x^n)^5*csgn(I*c)+18*Pi^2*b^3*n*csgn(I*c*x^n)^4*csgn(I*c)^2+72*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)+72*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-216*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)-16*b^3*n^3+9*I*Pi^3*b^3*csgn(I*c*x^n)^9+108*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-108*I*ln(c
)^2*Pi*b^3*csgn(I*c*x^n)^3-108*I*Pi*a^2*b*csgn(I*c*x^n)^3-24*I*Pi*b^3*n^2*csgn(I*c*x^n)^3-9*I*Pi^3*b^3*csgn(I*
x^n)^3*csgn(I*c*x^n)^6-27*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8-27*I*Pi^3*b^3*csgn(I*c*x^n)^8*csgn(I*c)+27*I*
Pi^3*b^3*csgn(I*c*x^n)^7*csgn(I*c)^2-9*I*Pi^3*b^3*csgn(I*c*x^n)^6*csgn(I*c)^3-72*ln(c)^2*b^3*n+48*ln(c)*b^3*n^
2+216*ln(c)*a^2*b+216*ln(c)^2*a*b^2+108*Pi^2*a*b^2*csgn(I*c*x^n)^5*csgn(I*c)-54*Pi^2*a*b^2*csgn(I*c*x^n)^4*csg
n(I*c)^2-54*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2+18*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4-36*Pi^2*b^3
*n*csgn(I*x^n)*csgn(I*c*x^n)^5+18*Pi^2*b^3*n*csgn(I*c*x^n)^6-54*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^6-54*Pi^2*a*b^2*c
sgn(I*c*x^n)^6+27*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^7+27*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^5*csgn(I*
c)-27*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)^3*csgn(I*c*x^n)^3*csgn(I*c
)^3-81*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^6*csgn(I*c)+108*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(
I*c)^2+108*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-54*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*
csgn(I*c)^2-216*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-36*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^3
*csgn(I*c)^2+108*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+108*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*
x^n)^3*csgn(I*c)-54*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-24*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)-72*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi
*b^3*n^2*csgn(I*c*x^n)^2*csgn(I*c)+72*I*Pi*a*b^2*n*csgn(I*c*x^n)^3-216*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^3+108*I*
Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2+108*I*Pi*a^2*b*csgn(I*c*x^n)^2*csgn(I*c)+72*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n
)^3-81*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^6*csgn(I*c)^2+27*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^
3+108*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+108*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)-144*ln(c)*a*
b^2*n+81*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*csgn(I*c)^2-27*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn
(I*c)^3+81*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)-72*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-10
8*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-108*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-72*I
*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c)+216*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+216*I*ln(c)*Pi*a*b^
2*csgn(I*c*x^n)^2*csgn(I*c)-72*I*Pi*a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c))

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Maxima [A]  time = 1.18369, size = 181, normalized size = 2.35 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \log \left (c x^{n}\right )^{3} + a b^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac{1}{3} \, a^{2} b n x^{3} + a^{2} b x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a^{3} x^{3} + \frac{2}{9} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} a b^{2} - \frac{1}{27} \,{\left (9 \, n x^{3} \log \left (c x^{n}\right )^{2} + 2 \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} n\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.954472, size = 512, normalized size = 6.65 \begin{align*} \frac{1}{3} \, b^{3} n^{3} x^{3} \log \left (x\right )^{3} + \frac{1}{3} \, b^{3} x^{3} \log \left (c\right )^{3} - \frac{1}{3} \,{\left (b^{3} n - 3 \, a b^{2}\right )} x^{3} \log \left (c\right )^{2} + \frac{1}{9} \,{\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b\right )} x^{3} \log \left (c\right ) - \frac{1}{27} \,{\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n - 9 \, a^{3}\right )} x^{3} + \frac{1}{3} \,{\left (3 \, b^{3} n^{2} x^{3} \log \left (c\right ) -{\left (b^{3} n^{3} - 3 \, a b^{2} n^{2}\right )} x^{3}\right )} \log \left (x\right )^{2} + \frac{1}{9} \,{\left (9 \, b^{3} n x^{3} \log \left (c\right )^{2} - 6 \,{\left (b^{3} n^{2} - 3 \, a b^{2} n\right )} x^{3} \log \left (c\right ) +{\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n\right )} x^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

1/3*b^3*n^3*x^3*log(x)^3 + 1/3*b^3*x^3*log(c)^3 - 1/3*(b^3*n - 3*a*b^2)*x^3*log(c)^2 + 1/9*(2*b^3*n^2 - 6*a*b^
2*n + 9*a^2*b)*x^3*log(c) - 1/27*(2*b^3*n^3 - 6*a*b^2*n^2 + 9*a^2*b*n - 9*a^3)*x^3 + 1/3*(3*b^3*n^2*x^3*log(c)
- (b^3*n^3 - 3*a*b^2*n^2)*x^3)*log(x)^2 + 1/9*(9*b^3*n*x^3*log(c)^2 - 6*(b^3*n^2 - 3*a*b^2*n)*x^3*log(c) + (2
*b^3*n^3 - 6*a*b^2*n^2 + 9*a^2*b*n)*x^3)*log(x)

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Sympy [B]  time = 4.03597, size = 311, normalized size = 4.04 \begin{align*} \frac{a^{3} x^{3}}{3} + a^{2} b n x^{3} \log{\left (x \right )} - \frac{a^{2} b n x^{3}}{3} + a^{2} b x^{3} \log{\left (c \right )} + a b^{2} n^{2} x^{3} \log{\left (x \right )}^{2} - \frac{2 a b^{2} n^{2} x^{3} \log{\left (x \right )}}{3} + \frac{2 a b^{2} n^{2} x^{3}}{9} + 2 a b^{2} n x^{3} \log{\left (c \right )} \log{\left (x \right )} - \frac{2 a b^{2} n x^{3} \log{\left (c \right )}}{3} + a b^{2} x^{3} \log{\left (c \right )}^{2} + \frac{b^{3} n^{3} x^{3} \log{\left (x \right )}^{3}}{3} - \frac{b^{3} n^{3} x^{3} \log{\left (x \right )}^{2}}{3} + \frac{2 b^{3} n^{3} x^{3} \log{\left (x \right )}}{9} - \frac{2 b^{3} n^{3} x^{3}}{27} + b^{3} n^{2} x^{3} \log{\left (c \right )} \log{\left (x \right )}^{2} - \frac{2 b^{3} n^{2} x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} + \frac{2 b^{3} n^{2} x^{3} \log{\left (c \right )}}{9} + b^{3} n x^{3} \log{\left (c \right )}^{2} \log{\left (x \right )} - \frac{b^{3} n x^{3} \log{\left (c \right )}^{2}}{3} + \frac{b^{3} x^{3} \log{\left (c \right )}^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))**3,x)

[Out]

a**3*x**3/3 + a**2*b*n*x**3*log(x) - a**2*b*n*x**3/3 + a**2*b*x**3*log(c) + a*b**2*n**2*x**3*log(x)**2 - 2*a*b
**2*n**2*x**3*log(x)/3 + 2*a*b**2*n**2*x**3/9 + 2*a*b**2*n*x**3*log(c)*log(x) - 2*a*b**2*n*x**3*log(c)/3 + a*b
**2*x**3*log(c)**2 + b**3*n**3*x**3*log(x)**3/3 - b**3*n**3*x**3*log(x)**2/3 + 2*b**3*n**3*x**3*log(x)/9 - 2*b
**3*n**3*x**3/27 + b**3*n**2*x**3*log(c)*log(x)**2 - 2*b**3*n**2*x**3*log(c)*log(x)/3 + 2*b**3*n**2*x**3*log(c
)/9 + b**3*n*x**3*log(c)**2*log(x) - b**3*n*x**3*log(c)**2/3 + b**3*x**3*log(c)**3/3

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Giac [B]  time = 1.29104, size = 346, normalized size = 4.49 \begin{align*} \frac{1}{3} \, b^{3} n^{3} x^{3} \log \left (x\right )^{3} - \frac{1}{3} \, b^{3} n^{3} x^{3} \log \left (x\right )^{2} + b^{3} n^{2} x^{3} \log \left (c\right ) \log \left (x\right )^{2} + \frac{2}{9} \, b^{3} n^{3} x^{3} \log \left (x\right ) - \frac{2}{3} \, b^{3} n^{2} x^{3} \log \left (c\right ) \log \left (x\right ) + b^{3} n x^{3} \log \left (c\right )^{2} \log \left (x\right ) + a b^{2} n^{2} x^{3} \log \left (x\right )^{2} - \frac{2}{27} \, b^{3} n^{3} x^{3} + \frac{2}{9} \, b^{3} n^{2} x^{3} \log \left (c\right ) - \frac{1}{3} \, b^{3} n x^{3} \log \left (c\right )^{2} + \frac{1}{3} \, b^{3} x^{3} \log \left (c\right )^{3} - \frac{2}{3} \, a b^{2} n^{2} x^{3} \log \left (x\right ) + 2 \, a b^{2} n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac{2}{9} \, a b^{2} n^{2} x^{3} - \frac{2}{3} \, a b^{2} n x^{3} \log \left (c\right ) + a b^{2} x^{3} \log \left (c\right )^{2} + a^{2} b n x^{3} \log \left (x\right ) - \frac{1}{3} \, a^{2} b n x^{3} + a^{2} b x^{3} \log \left (c\right ) + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

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