\(\int (d x)^m (a+b \log (c x^n))^3 \, dx\) [150]

Optimal result

Integrand size = 18, antiderivative size = 116 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {6 b^3 n^3 (d x)^{1+m}}{d (1+m)^4}+\frac {6 b^2 n^2 (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^3}-\frac {3 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^3}{d (1+m)} \]

[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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2342, 2341} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\frac {6 b^2 n^2 (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^3}+\frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac {3 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)^2}-\frac {6 b^3 n^3 (d x)^{m+1}}{d (m+1)^4} \]

[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 2341

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 2342

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\frac {x (d x)^m \left (\left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left ((1+m)^2 \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n-(1+m) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{(1+m)^3}\right )}{1+m} \]

[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)

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(9552\)

[In]

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

[Out]

-(-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)

Fricas [B] (verification not implemented)

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

[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)

Sympy [B] (verification not implemented)

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 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*m**2*x*(d*x)**m*log(c*x**
n)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*b**3*m*n**2*x*(d*x)**m*log(c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m
+ 1) - 6*b**3*m*n*x*(d*x)**m*log(c*x**n)**2/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*b**3*m*x*(d*x)**m*log(c*x**
n)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 6*b**3*n**3*x*(d*x)**m/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*b**3*
n**2*x*(d*x)**m*log(c*x**n)/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b**3*n*x*(d*x)**m*log(c*x**n)**2/(m**4 + 4*
m**3 + 6*m**2 + 4*m + 1) + b**3*x*(d*x)**m*log(c*x**n)**3/(m**4 + 4*m**3 + 6*m**2 + 4*m + 1), Ne(m, -1)), (Pie
cewise(((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))

Maxima [B] (verification not implemented)

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 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {3 \, a^{2} b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} b^{3} \log \left (c x^{n}\right )^{3}}{d {\left (m + 1\right )}} - 6 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} a b^{2} - 3 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )^{2}}{{\left (m + 1\right )}^{2}} - \frac {2 \, {\left (\frac {d^{m + 1} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m + 1} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} n}{d {\left (m + 1\right )}}\right )} b^{3} + \frac {3 \, \left (d x\right )^{m + 1} a b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {3 \, \left (d x\right )^{m + 1} a^{2} b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]

[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))

Giac [B] (verification not implemented)

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 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Too large to display} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]

[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)