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]

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

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Rubi [A]  time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.45, size = 224, normalized size = 2.91 \[ \frac {1}{3} \, b^{3} n^{3} x^{3} \log \relax (x)^{3} + \frac {1}{3} \, b^{3} x^{3} \log \relax (c)^{3} - \frac {1}{3} \, {\left (b^{3} n - 3 \, a b^{2}\right )} x^{3} \log \relax (c)^{2} + \frac {1}{9} \, {\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b\right )} x^{3} \log \relax (c) - \frac {1}{27} \, {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n - 9 \, a^{3}\right )} x^{3} + \frac {1}{3} \, {\left (3 \, b^{3} n^{2} x^{3} \log \relax (c) - {\left (b^{3} n^{3} - 3 \, a b^{2} n^{2}\right )} x^{3}\right )} \log \relax (x)^{2} + \frac {1}{9} \, {\left (9 \, b^{3} n x^{3} \log \relax (c)^{2} - 6 \, {\left (b^{3} n^{2} - 3 \, a b^{2} n\right )} x^{3} \log \relax (c) + {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n\right )} x^{3}\right )} \log \relax (x) \]

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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giac [B]  time = 0.30, size = 256, normalized size = 3.32 \[ \frac {1}{3} \, b^{3} n^{3} x^{3} \log \relax (x)^{3} - \frac {1}{3} \, b^{3} n^{3} x^{3} \log \relax (x)^{2} + b^{3} n^{2} x^{3} \log \relax (c) \log \relax (x)^{2} + \frac {2}{9} \, b^{3} n^{3} x^{3} \log \relax (x) - \frac {2}{3} \, b^{3} n^{2} x^{3} \log \relax (c) \log \relax (x) + b^{3} n x^{3} \log \relax (c)^{2} \log \relax (x) + a b^{2} n^{2} x^{3} \log \relax (x)^{2} - \frac {2}{27} \, b^{3} n^{3} x^{3} + \frac {2}{9} \, b^{3} n^{2} x^{3} \log \relax (c) - \frac {1}{3} \, b^{3} n x^{3} \log \relax (c)^{2} + \frac {1}{3} \, b^{3} x^{3} \log \relax (c)^{3} - \frac {2}{3} \, a b^{2} n^{2} x^{3} \log \relax (x) + 2 \, a b^{2} n x^{3} \log \relax (c) \log \relax (x) + \frac {2}{9} \, a b^{2} n^{2} x^{3} - \frac {2}{3} \, a b^{2} n x^{3} \log \relax (c) + a b^{2} x^{3} \log \relax (c)^{2} + a^{2} b n x^{3} \log \relax (x) - \frac {1}{3} \, a^{2} b n x^{3} + a^{2} b x^{3} \log \relax (c) + \frac {1}{3} \, a^{3} x^{3} \]

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

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maple [C]  time = 0.31, size = 2650, normalized size = 34.42 \[ \text {Expression too large to display} \]

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

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maxima [A]  time = 0.68, size = 134, normalized size = 1.74 \[ \frac {1}{3} \, b^{3} x^{3} \log \left (c x^{n}\right )^{3} + a b^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac {1}{3} \, a^{2} b n x^{3} + a^{2} b x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a^{3} x^{3} + \frac {2}{9} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} a b^{2} - \frac {1}{27} \, {\left (9 \, n x^{3} \log \left (c x^{n}\right )^{2} + 2 \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} n\right )} b^{3} \]

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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mupad [B]  time = 3.37, size = 108, normalized size = 1.40 \[ x^3\,\left (\frac {a^3}{3}-\frac {a^2\,b\,n}{3}+\frac {2\,a\,b^2\,n^2}{9}-\frac {2\,b^3\,n^3}{27}\right )+\frac {x^3\,\ln \left (c\,x^n\right )\,\left (3\,a^2\,b-2\,a\,b^2\,n+\frac {2\,b^3\,n^2}{3}\right )}{3}+x^3\,{\ln \left (c\,x^n\right )}^2\,\left (a\,b^2-\frac {b^3\,n}{3}\right )+\frac {b^3\,x^3\,{\ln \left (c\,x^n\right )}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*log(c*x^n))^3,x)

[Out]

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

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sympy [B]  time = 2.89, size = 311, normalized size = 4.04 \[ \frac {a^{3} x^{3}}{3} + a^{2} b n x^{3} \log {\relax (x )} - \frac {a^{2} b n x^{3}}{3} + a^{2} b x^{3} \log {\relax (c )} + a b^{2} n^{2} x^{3} \log {\relax (x )}^{2} - \frac {2 a b^{2} n^{2} x^{3} \log {\relax (x )}}{3} + \frac {2 a b^{2} n^{2} x^{3}}{9} + 2 a b^{2} n x^{3} \log {\relax (c )} \log {\relax (x )} - \frac {2 a b^{2} n x^{3} \log {\relax (c )}}{3} + a b^{2} x^{3} \log {\relax (c )}^{2} + \frac {b^{3} n^{3} x^{3} \log {\relax (x )}^{3}}{3} - \frac {b^{3} n^{3} x^{3} \log {\relax (x )}^{2}}{3} + \frac {2 b^{3} n^{3} x^{3} \log {\relax (x )}}{9} - \frac {2 b^{3} n^{3} x^{3}}{27} + b^{3} n^{2} x^{3} \log {\relax (c )} \log {\relax (x )}^{2} - \frac {2 b^{3} n^{2} x^{3} \log {\relax (c )} \log {\relax (x )}}{3} + \frac {2 b^{3} n^{2} x^{3} \log {\relax (c )}}{9} + b^{3} n x^{3} \log {\relax (c )}^{2} \log {\relax (x )} - \frac {b^{3} n x^{3} \log {\relax (c )}^{2}}{3} + \frac {b^{3} x^{3} \log {\relax (c )}^{3}}{3} \]

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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