Integrand size = 18, antiderivative size = 116 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {6 b^3 n^3 (d x)^{1+m}}{d (1+m)^4}+\frac {6 b^2 n^2 (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^3}-\frac {3 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^3}{d (1+m)} \]
-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)
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\frac {x (d x)^m \left (\left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left ((1+m)^2 \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n-(1+m) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{(1+m)^3}\right )}{1+m} \]
(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)
Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2742, 2742, 2741}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2dx}{m+1}\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n \left (\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac {2 b n \int (d x)^m \left (a+b \log \left (c x^n\right )\right )dx}{m+1}\right )}{m+1}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n \left (\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac {2 b n \left (\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1}}{d (m+1)^2}\right )}{m+1}\right )}{m+1}\) |
((d*x)^(1 + m)*(a + b*Log[c*x^n])^3)/(d*(1 + m)) - (3*b*n*(((d*x)^(1 + m)* (a + b*Log[c*x^n])^2)/(d*(1 + m)) - (2*b*n*(-((b*n*(d*x)^(1 + m))/(d*(1 + m)^2)) + ((d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m))))/(1 + m)))/(1 + m )
3.2.50.3.1 Defintions of rubi rules used
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(577\) vs. \(2(116)=232\).
Time = 0.78 (sec) , antiderivative size = 578, normalized size of antiderivative = 4.98
method | result | size |
parallelrisch | \(-\frac {-x \left (d x \right )^{m} a^{3}+6 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a \,b^{2} m^{2} n +12 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a \,b^{2} m n -x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{3} b^{3} m^{3}-3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{3} b^{3} m^{2}-3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{3} b^{3} m +3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{3} n -6 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) b^{3} n^{2}-3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} a \,b^{2}-6 x \left (d x \right )^{m} a \,b^{2} n^{2}-3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a^{2} b +3 x \left (d x \right )^{m} a^{2} b n -x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{3} b^{3}-x \left (d x \right )^{m} a^{3} m^{3}+6 x \left (d x \right )^{m} b^{3} n^{3}-3 x \left (d x \right )^{m} a^{3} m^{2}-3 x \left (d x \right )^{m} a^{3} m -9 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} a \,b^{2} m -9 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a^{2} b \,m^{2}+3 x \left (d x \right )^{m} a^{2} b \,m^{2} n -6 x \left (d x \right )^{m} a \,b^{2} m \,n^{2}-9 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a^{2} b m +6 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a \,b^{2} n +6 x \left (d x \right )^{m} a^{2} b m n -3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} a \,b^{2} m^{3}+3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{3} m^{2} n -9 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} a \,b^{2} m^{2}+6 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right )^{2} b^{3} m n -3 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) a^{2} b \,m^{3}-6 x \left (d x \right )^{m} \ln \left (c \,x^{n}\right ) b^{3} m \,n^{2}}{\left (m^{3}+3 m^{2}+3 m +1\right ) \left (1+m \right )}\) | \(578\) |
risch | \(\text {Expression too large to display}\) | \(9552\) |
-(-x*(d*x)^m*a^3+6*x*(d*x)^m*ln(c*x^n)*a*b^2*m^2*n+12*x*(d*x)^m*ln(c*x^n)* a*b^2*m*n-x*(d*x)^m*ln(c*x^n)^3*b^3*m^3-3*x*(d*x)^m*ln(c*x^n)^3*b^3*m^2-3* x*(d*x)^m*ln(c*x^n)^3*b^3*m+3*x*(d*x)^m*ln(c*x^n)^2*b^3*n-6*x*(d*x)^m*ln(c *x^n)*b^3*n^2-3*x*(d*x)^m*ln(c*x^n)^2*a*b^2-6*x*(d*x)^m*a*b^2*n^2-3*x*(d*x )^m*ln(c*x^n)*a^2*b+3*x*(d*x)^m*a^2*b*n-x*(d*x)^m*ln(c*x^n)^3*b^3-x*(d*x)^ m*a^3*m^3+6*x*(d*x)^m*b^3*n^3-3*x*(d*x)^m*a^3*m^2-3*x*(d*x)^m*a^3*m-9*x*(d *x)^m*ln(c*x^n)^2*a*b^2*m-9*x*(d*x)^m*ln(c*x^n)*a^2*b*m^2+3*x*(d*x)^m*a^2* b*m^2*n-6*x*(d*x)^m*a*b^2*m*n^2-9*x*(d*x)^m*ln(c*x^n)*a^2*b*m+6*x*(d*x)^m* ln(c*x^n)*a*b^2*n+6*x*(d*x)^m*a^2*b*m*n-3*x*(d*x)^m*ln(c*x^n)^2*a*b^2*m^3+ 3*x*(d*x)^m*ln(c*x^n)^2*b^3*m^2*n-9*x*(d*x)^m*ln(c*x^n)^2*a*b^2*m^2+6*x*(d *x)^m*ln(c*x^n)^2*b^3*m*n-3*x*(d*x)^m*ln(c*x^n)*a^2*b*m^3-6*x*(d*x)^m*ln(c *x^n)*b^3*m*n^2)/(m^3+3*m^2+3*m+1)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (116) = 232\).
Time = 0.33 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.95 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\frac {{\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{3} x \log \left (x\right )^{3} + {\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} x \log \left (c\right )^{3} + 3 \, {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} - {\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x \log \left (c\right )^{2} + 3 \, {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 2 \, {\left (b^{3} m + b^{3}\right )} n^{2} - 2 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \left (c\right ) + 3 \, {\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{2} x \log \left (c\right ) - {\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{3} - {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n^{2}\right )} x\right )} \log \left (x\right )^{2} + {\left (a^{3} m^{3} - 6 \, b^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 6 \, {\left (a b^{2} m + a b^{2}\right )} n^{2} - 3 \, {\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x + 3 \, {\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n x \log \left (c\right )^{2} - 2 \, {\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{2} - {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \left (c\right ) + {\left (2 \, {\left (b^{3} m + b^{3}\right )} n^{3} - 2 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n^{2} + {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b\right )} n\right )} x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \]
((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*log(x))/( m^4 + 4*m^3 + 6*m^2 + 4*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1273 vs. \(2 (107) = 214\).
Time = 8.99 (sec) , antiderivative size = 1273, normalized size of antiderivative = 10.97 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*m**3*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a** 3*m**2*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a**3*m*x*(d*x)**m /(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + a**3*x*(d*x)**m/(m**4 + 4*m**3 + 6*m **2 + 4*m + 1) + 3*a**2*b*m**3*x*(d*x)**m*log(c*x**n)/(m**4 + 4*m**3 + 6*m **2 + 4*m + 1) - 3*a**2*b*m**2*n*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2*b*m**2*x*(d*x)**m*log(c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*a**2*b*m*n*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a**2 *b*m*x*(d*x)**m*log(c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*a**2*b* n*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a**2*b*x*(d*x)**m*log( c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2*m**3*x*(d*x)**m*log( c*x**n)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*a*b**2*m**2*n*x*(d*x)**m *log(c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a*b**2*m**2*x*(d*x)**m *log(c*x**n)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*m*n**2*x*(d* x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 12*a*b**2*m*n*x*(d*x)**m*log(c* x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 9*a*b**2*m*x*(d*x)**m*log(c*x** n)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*a*b**2*n**2*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*a*b**2*n*x*(d*x)**m*log(c*x**n)/(m**4 + 4 *m**3 + 6*m**2 + 4*m + 1) + 3*a*b**2*x*(d*x)**m*log(c*x**n)**2/(m**4 + 4*m **3 + 6*m**2 + 4*m + 1) + b**3*m**3*x*(d*x)**m*log(c*x**n)**3/(m**4 + 4*m* *3 + 6*m**2 + 4*m + 1) - 3*b**3*m**2*n*x*(d*x)**m*log(c*x**n)**2/(m**4 ...
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (116) = 232\).
Time = 0.21 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.13 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {3 \, a^{2} b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} b^{3} \log \left (c x^{n}\right )^{3}}{d {\left (m + 1\right )}} - 6 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} a b^{2} - 3 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )^{2}}{{\left (m + 1\right )}^{2}} - \frac {2 \, {\left (\frac {d^{m + 1} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m + 1} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} n}{d {\left (m + 1\right )}}\right )} b^{3} + \frac {3 \, \left (d x\right )^{m + 1} a b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {3 \, \left (d x\right )^{m + 1} a^{2} b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]
-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))
Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (116) = 232\).
Time = 0.41 (sec) , antiderivative size = 1133, normalized size of antiderivative = 9.77 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Too large to display} \]
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 ...
Timed out. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]