3.150 \(\int (d x)^m (a+b \log (c x^n))^3 \, dx\)

Optimal. Leaf size=116 \[ \frac {6 b^2 n^2 (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^3}+\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)^2}-\frac {6 b^3 n^3 (d x)^{m+1}}{d (m+1)^4} \]

[Out]

-6*b^3*n^3*(d*x)^(1+m)/d/(1+m)^4+6*b^2*n^2*(d*x)^(1+m)*(a+b*ln(c*x^n))/d/(1+m)^3-3*b*n*(d*x)^(1+m)*(a+b*ln(c*x
^n))^2/d/(1+m)^2+(d*x)^(1+m)*(a+b*ln(c*x^n))^3/d/(1+m)

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Rubi [A]  time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2305, 2304} \[ \frac {6 b^2 n^2 (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^3}+\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)^2}-\frac {6 b^3 n^3 (d x)^{m+1}}{d (m+1)^4} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])^3,x]

[Out]

(-6*b^3*n^3*(d*x)^(1 + m))/(d*(1 + m)^4) + (6*b^2*n^2*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^3) - (3*b*n
*(d*x)^(1 + m)*(a + b*Log[c*x^n])^2)/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])^3)/(d*(1 + m))

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 (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^3}{d (1+m)}-\frac {(3 b n) \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{1+m}\\ &=-\frac {3 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^3}{d (1+m)}+\frac {\left (6 b^2 n^2\right ) \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2}\\ &=-\frac {6 b^3 n^3 (d x)^{1+m}}{d (1+m)^4}+\frac {6 b^2 n^2 (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^3}-\frac {3 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^3}{d (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 76, normalized size = 0.66 \[ \frac {x (d x)^m \left (\left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left ((m+1)^2 \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n-(m+1) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{(m+1)^3}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])^3,x]

[Out]

(x*(d*x)^m*((a + b*Log[c*x^n])^3 - (3*b*n*((1 + m)^2*(a + b*Log[c*x^n])^2 + 2*b*n*(b*n - (1 + m)*(a + b*Log[c*
x^n]))))/(1 + m)^3))/(1 + m)

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fricas [B]  time = 0.45, size = 574, normalized size = 4.95 \[ \frac {{\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{3} x \log \relax (x)^{3} + {\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} x \log \relax (c)^{3} + 3 \, {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} - {\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x \log \relax (c)^{2} + 3 \, {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 2 \, {\left (b^{3} m + b^{3}\right )} n^{2} - 2 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \relax (c) + 3 \, {\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{2} x \log \relax (c) - {\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{3} - {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n^{2}\right )} x\right )} \log \relax (x)^{2} + {\left (a^{3} m^{3} - 6 \, b^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 6 \, {\left (a b^{2} m + a b^{2}\right )} n^{2} - 3 \, {\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x + 3 \, {\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n x \log \relax (c)^{2} - 2 \, {\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{2} - {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \relax (c) + {\left (2 \, {\left (b^{3} m + b^{3}\right )} n^{3} - 2 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n^{2} + {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b\right )} n\right )} x\right )} \log \relax (x)\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3)*n^3*x*log(x)^3 + (b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3)*x*log(c)^3 + 3*(
a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2 - (b^3*m^2 + 2*b^3*m + b^3)*n)*x*log(c)^2 + 3*(a^2*b*m^3 + 3*a^2*b
*m^2 + 3*a^2*b*m + a^2*b + 2*(b^3*m + b^3)*n^2 - 2*(a*b^2*m^2 + 2*a*b^2*m + a*b^2)*n)*x*log(c) + 3*((b^3*m^3 +
3*b^3*m^2 + 3*b^3*m + b^3)*n^2*x*log(c) - ((b^3*m^2 + 2*b^3*m + b^3)*n^3 - (a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2
*m + a*b^2)*n^2)*x)*log(x)^2 + (a^3*m^3 - 6*b^3*n^3 + 3*a^3*m^2 + 3*a^3*m + a^3 + 6*(a*b^2*m + a*b^2)*n^2 - 3*
(a^2*b*m^2 + 2*a^2*b*m + a^2*b)*n)*x + 3*((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3)*n*x*log(c)^2 - 2*((b^3*m^2 + 2
*b^3*m + b^3)*n^2 - (a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2)*n)*x*log(c) + (2*(b^3*m + b^3)*n^3 - 2*(a*b^
2*m^2 + 2*a*b^2*m + a*b^2)*n^2 + (a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b)*n)*x)*log(x))*e^(m*log(d) + m*l
og(x))/(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)

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giac [B]  time = 0.85, size = 1133, normalized size = 9.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.78, size = 9684, normalized size = 83.48 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [B]  time = 0.70, size = 247, normalized size = 2.13 \[ -\frac {3 \, a^{2} b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} b^{3} \log \left (c x^{n}\right )^{3}}{d {\left (m + 1\right )}} - 6 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} a b^{2} - 3 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )^{2}}{{\left (m + 1\right )}^{2}} - \frac {2 \, {\left (\frac {d^{m + 1} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m + 1} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} n}{d {\left (m + 1\right )}}\right )} b^{3} + \frac {3 \, \left (d x\right )^{m + 1} a b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {3 \, \left (d x\right )^{m + 1} a^{2} b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*log(c*x^n))^3,x)

[Out]

int((d*x)^m*(a + b*log(c*x^n))^3, x)

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sympy [A]  time = 43.51, size = 2778, normalized size = 23.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*d**m*m**3*x*x**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a**3*d**m*m**2*x*x**m/(m**4 + 4*m**3 +
6*m**2 + 4*m + 1) + 3*a**3*d**m*m*x*x**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + a**3*d**m*x*x**m/(m**4 + 4*m**3
+ 6*m**2 + 4*m + 1) + 3*a**2*b*d**m*m**3*n*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a**2*b*d**m*m
**3*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2*b*d**m*m**2*n*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m
**2 + 4*m + 1) - 3*a**2*b*d**m*m**2*n*x*x**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2*b*d**m*m**2*x*x**m*lo
g(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2*b*d**m*m*n*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) -
6*a**2*b*d**m*m*n*x*x**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2*b*d**m*m*x*x**m*log(c)/(m**4 + 4*m**3 +
6*m**2 + 4*m + 1) + 3*a**2*b*d**m*n*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*a**2*b*d**m*n*x*x**m/
(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a**2*b*d**m*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2
*d**m*m**3*n**2*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*d**m*m**3*n*x*x**m*log(c)*log(x
)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2*d**m*m**3*x*x**m*log(c)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1)
+ 9*a*b**2*d**m*m**2*n**2*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*a*b**2*d**m*m**2*n**2*x*x**m
*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 18*a*b**2*d**m*m**2*n*x*x**m*log(c)*log(x)/(m**4 + 4*m**3 + 6*m**
2 + 4*m + 1) - 6*a*b**2*d**m*m**2*n*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a*b**2*d**m*m**2*x*x*
*m*log(c)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a*b**2*d**m*m*n**2*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m**
2 + 4*m + 1) - 12*a*b**2*d**m*m*n**2*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*d**m*m*n**2*x
*x**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 18*a*b**2*d**m*m*n*x*x**m*log(c)*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4
*m + 1) - 12*a*b**2*d**m*m*n*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a*b**2*d**m*m*x*x**m*log(c)*
*2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2*d**m*n**2*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1)
- 6*a*b**2*d**m*n**2*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*d**m*n**2*x*x**m/(m**4 + 4*m
**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*d**m*n*x*x**m*log(c)*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*a*b**2*d
**m*n*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2*d**m*x*x**m*log(c)**2/(m**4 + 4*m**3 + 6*m**
2 + 4*m + 1) + b**3*d**m*m**3*n**3*x*x**m*log(x)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d**m*m**3*n**2
*x*x**m*log(c)*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d**m*m**3*n*x*x**m*log(c)**2*log(x)/(m**4
+ 4*m**3 + 6*m**2 + 4*m + 1) + b**3*d**m*m**3*x*x**m*log(c)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d*
*m*m**2*n**3*x*x**m*log(x)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b**3*d**m*m**2*n**3*x*x**m*log(x)**2/(m**
4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*b**3*d**m*m**2*n**2*x*x**m*log(c)*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m +
1) - 6*b**3*d**m*m**2*n**2*x*x**m*log(c)*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*b**3*d**m*m**2*n*x*x**
m*log(c)**2*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b**3*d**m*m**2*n*x*x**m*log(c)**2/(m**4 + 4*m**3 + 6
*m**2 + 4*m + 1) + 3*b**3*d**m*m**2*x*x**m*log(c)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d**m*m*n**3*x
*x**m*log(x)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*b**3*d**m*m*n**3*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m*
*2 + 4*m + 1) + 6*b**3*d**m*m*n**3*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*b**3*d**m*m*n**2*x*x**
m*log(c)*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 12*b**3*d**m*m*n**2*x*x**m*log(c)*log(x)/(m**4 + 4*m**
3 + 6*m**2 + 4*m + 1) + 6*b**3*d**m*m*n**2*x*x**m*log(c)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*b**3*d**m*m*n*
x*x**m*log(c)**2*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*b**3*d**m*m*n*x*x**m*log(c)**2/(m**4 + 4*m**3 +
6*m**2 + 4*m + 1) + 3*b**3*d**m*m*x*x**m*log(c)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + b**3*d**m*n**3*x*x**m
*log(x)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b**3*d**m*n**3*x*x**m*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*
m + 1) + 6*b**3*d**m*n**3*x*x**m*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*b**3*d**m*n**3*x*x**m/(m**4 + 4
*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d**m*n**2*x*x**m*log(c)*log(x)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*b*
*3*d**m*n**2*x*x**m*log(c)*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*b**3*d**m*n**2*x*x**m*log(c)/(m**4 +
4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*d**m*n*x*x**m*log(c)**2*log(x)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b**3
*d**m*n*x*x**m*log(c)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + b**3*d**m*x*x**m*log(c)**3/(m**4 + 4*m**3 + 6*m*
*2 + 4*m + 1), Ne(m, -1)), (Piecewise(((a**3*log(c*x**n) + 3*a**2*b*log(c*x**n)**2/2 + a*b**2*log(c*x**n)**3 +
b**3*log(c*x**n)**4/4)/n, Ne(n, 0)), ((a**3 + 3*a**2*b*log(c) + 3*a*b**2*log(c)**2 + b**3*log(c)**3)*log(x),
True))/d, True))

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