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

Optimal. Leaf size=77 \[ \frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \]

[Out]

(-3*b^3*n^3*x^2)/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n]))/4 - (3*b*n*x^2*(a + b*Log[c*x^n])^2)/4 + (x^2*(a + b*L
og[c*x^n])^3)/2

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Rubi [A]  time = 0.038703, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2305, 2304} \[ \frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Log[c*x^n])^3,x]

[Out]

(-3*b^3*n^3*x^2)/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n]))/4 - (3*b*n*x^2*(a + b*Log[c*x^n])^2)/4 + (x^2*(a + b*L
og[c*x^n])^3)/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 x \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{1}{2} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{2} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{3}{8} b^3 n^3 x^2+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0295424, size = 60, normalized size = 0.78 \[ \frac{1}{8} x^2 \left (4 \left (a+b \log \left (c x^n\right )\right )^3-3 b n \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n])^3,x]

[Out]

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

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Maple [C]  time = 0.334, 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*(a+b*ln(c*x^n))^3,x)

[Out]

1/2*x^2*b^3*ln(x^n)^3+3/4*b^2*x^2*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)-I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)-b*n+2*a)*ln(x^n)^2+3/8*b*x^2*(4*I*ln(c)*P
i*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+4*ln(c)^2*b^2-Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-4*a*b*n+2*b^2*n^2+4*a^2-2
*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-2*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c
*x^n)^3*csgn(I*c)^2+2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*
csgn(I*c)^2-4*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+2*I*Pi*b^2*n*csgn(I*c*x^n)^3+4*I*ln(c)*Pi*b^2*csg
n(I*c*x^n)^2*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-4*I*Pi*a*b*
csgn(I*c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a
*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-Pi^2*b^2*csgn(I*c*x^n)^6+8*ln(c)*a*b-4*ln(c)*b^2*n+2*I*Pi*b^2*n*csgn(I*
x^n)*csgn(I*c*x^n)*csgn(I*c)+2*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-Pi^2*
b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4)*ln(x^n)+1/16*x^2*(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)*csgn(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*csg
n(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)+12*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^3+12*I*Pi*a*b^2*n*c
sgn(I*c*x^n)^3+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*c
sgn(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*csgn(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*cs
gn(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*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^2*c
sgn(I*c)-12*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-12*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b^
3*n^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*I*Pi*a*b^2*n*cs
gn(I*c*x^n)^2*csgn(I*c)-12*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c))

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Maxima [A]  time = 1.12363, size = 182, normalized size = 2.36 \begin{align*} \frac{1}{2} \, b^{3} x^{2} \log \left (c x^{n}\right )^{3} + \frac{3}{2} \, a b^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac{3}{4} \, a^{2} b n x^{2} + \frac{3}{2} \, a^{2} b x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a^{3} x^{2} + \frac{3}{4} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} a b^{2} - \frac{3}{8} \,{\left (2 \, n x^{2} \log \left (c x^{n}\right )^{2} +{\left (n^{2} x^{2} - 2 \, n x^{2} \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*(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.814465, size = 505, normalized size = 6.56 \begin{align*} \frac{1}{2} \, b^{3} n^{3} x^{2} \log \left (x\right )^{3} + \frac{1}{2} \, b^{3} x^{2} \log \left (c\right )^{3} - \frac{3}{4} \,{\left (b^{3} n - 2 \, a b^{2}\right )} x^{2} \log \left (c\right )^{2} + \frac{3}{4} \,{\left (b^{3} n^{2} - 2 \, a b^{2} n + 2 \, a^{2} b\right )} x^{2} \log \left (c\right ) - \frac{1}{8} \,{\left (3 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 6 \, a^{2} b n - 4 \, a^{3}\right )} x^{2} + \frac{3}{4} \,{\left (2 \, b^{3} n^{2} x^{2} \log \left (c\right ) -{\left (b^{3} n^{3} - 2 \, a b^{2} n^{2}\right )} x^{2}\right )} \log \left (x\right )^{2} + \frac{3}{4} \,{\left (2 \, b^{3} n x^{2} \log \left (c\right )^{2} - 2 \,{\left (b^{3} n^{2} - 2 \, a b^{2} n\right )} x^{2} \log \left (c\right ) +{\left (b^{3} n^{3} - 2 \, a b^{2} n^{2} + 2 \, a^{2} b n\right )} x^{2}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*n^3*x^2*log(x)^3 + 1/2*b^3*x^2*log(c)^3 - 3/4*(b^3*n - 2*a*b^2)*x^2*log(c)^2 + 3/4*(b^3*n^2 - 2*a*b^2*
n + 2*a^2*b)*x^2*log(c) - 1/8*(3*b^3*n^3 - 6*a*b^2*n^2 + 6*a^2*b*n - 4*a^3)*x^2 + 3/4*(2*b^3*n^2*x^2*log(c) -
(b^3*n^3 - 2*a*b^2*n^2)*x^2)*log(x)^2 + 3/4*(2*b^3*n*x^2*log(c)^2 - 2*(b^3*n^2 - 2*a*b^2*n)*x^2*log(c) + (b^3*
n^3 - 2*a*b^2*n^2 + 2*a^2*b*n)*x^2)*log(x)

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Sympy [B]  time = 2.49179, size = 337, normalized size = 4.38 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b n x^{2} \log{\left (x \right )}}{2} - \frac{3 a^{2} b n x^{2}}{4} + \frac{3 a^{2} b x^{2} \log{\left (c \right )}}{2} + \frac{3 a b^{2} n^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{3 a b^{2} n^{2} x^{2} \log{\left (x \right )}}{2} + \frac{3 a b^{2} n^{2} x^{2}}{4} + 3 a b^{2} n x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{3 a b^{2} n x^{2} \log{\left (c \right )}}{2} + \frac{3 a b^{2} x^{2} \log{\left (c \right )}^{2}}{2} + \frac{b^{3} n^{3} x^{2} \log{\left (x \right )}^{3}}{2} - \frac{3 b^{3} n^{3} x^{2} \log{\left (x \right )}^{2}}{4} + \frac{3 b^{3} n^{3} x^{2} \log{\left (x \right )}}{4} - \frac{3 b^{3} n^{3} x^{2}}{8} + \frac{3 b^{3} n^{2} x^{2} \log{\left (c \right )} \log{\left (x \right )}^{2}}{2} - \frac{3 b^{3} n^{2} x^{2} \log{\left (c \right )} \log{\left (x \right )}}{2} + \frac{3 b^{3} n^{2} x^{2} \log{\left (c \right )}}{4} + \frac{3 b^{3} n x^{2} \log{\left (c \right )}^{2} \log{\left (x \right )}}{2} - \frac{3 b^{3} n x^{2} \log{\left (c \right )}^{2}}{4} + \frac{b^{3} x^{2} \log{\left (c \right )}^{3}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 3*a**2*b*n*x**2*log(x)/2 - 3*a**2*b*n*x**2/4 + 3*a**2*b*x**2*log(c)/2 + 3*a*b**2*n**2*x**2*log(x
)**2/2 - 3*a*b**2*n**2*x**2*log(x)/2 + 3*a*b**2*n**2*x**2/4 + 3*a*b**2*n*x**2*log(c)*log(x) - 3*a*b**2*n*x**2*
log(c)/2 + 3*a*b**2*x**2*log(c)**2/2 + b**3*n**3*x**2*log(x)**3/2 - 3*b**3*n**3*x**2*log(x)**2/4 + 3*b**3*n**3
*x**2*log(x)/4 - 3*b**3*n**3*x**2/8 + 3*b**3*n**2*x**2*log(c)*log(x)**2/2 - 3*b**3*n**2*x**2*log(c)*log(x)/2 +
3*b**3*n**2*x**2*log(c)/4 + 3*b**3*n*x**2*log(c)**2*log(x)/2 - 3*b**3*n*x**2*log(c)**2/4 + b**3*x**2*log(c)**
3/2

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Giac [B]  time = 1.18497, size = 354, normalized size = 4.6 \begin{align*} \frac{1}{2} \, b^{3} n^{3} x^{2} \log \left (x\right )^{3} - \frac{3}{4} \, b^{3} n^{3} x^{2} \log \left (x\right )^{2} + \frac{3}{2} \, b^{3} n^{2} x^{2} \log \left (c\right ) \log \left (x\right )^{2} + \frac{3}{4} \, b^{3} n^{3} x^{2} \log \left (x\right ) - \frac{3}{2} \, b^{3} n^{2} x^{2} \log \left (c\right ) \log \left (x\right ) + \frac{3}{2} \, b^{3} n x^{2} \log \left (c\right )^{2} \log \left (x\right ) + \frac{3}{2} \, a b^{2} n^{2} x^{2} \log \left (x\right )^{2} - \frac{3}{8} \, b^{3} n^{3} x^{2} + \frac{3}{4} \, b^{3} n^{2} x^{2} \log \left (c\right ) - \frac{3}{4} \, b^{3} n x^{2} \log \left (c\right )^{2} + \frac{1}{2} \, b^{3} x^{2} \log \left (c\right )^{3} - \frac{3}{2} \, a b^{2} n^{2} x^{2} \log \left (x\right ) + 3 \, a b^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac{3}{4} \, a b^{2} n^{2} x^{2} - \frac{3}{2} \, a b^{2} n x^{2} \log \left (c\right ) + \frac{3}{2} \, a b^{2} x^{2} \log \left (c\right )^{2} + \frac{3}{2} \, a^{2} b n x^{2} \log \left (x\right ) - \frac{3}{4} \, a^{2} b n x^{2} + \frac{3}{2} \, a^{2} b x^{2} \log \left (c\right ) + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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