\(\int \frac {\log (c x^n) (a x^m+b \log ^2(c x^n))^3}{x} \, dx\) [9]

Optimal result

Integrand size = 28, antiderivative size = 272 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx=-\frac {360 a b^2 n^5 x^m}{m^6}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \] Output:

-360*a*b^2*n^5*x^m/m^6-9/8*a^2*b*n^3*x^(2*m)/m^4-1/9*a^3*n*x^(3*m)/m^2+360 
*a*b^2*n^4*x^m*ln(c*x^n)/m^5+9/4*a^2*b*n^2*x^(2*m)*ln(c*x^n)/m^3+1/3*a^3*x 
^(3*m)*ln(c*x^n)/m-180*a*b^2*n^3*x^m*ln(c*x^n)^2/m^4-9/4*a^2*b*n*x^(2*m)*l 
n(c*x^n)^2/m^2+60*a*b^2*n^2*x^m*ln(c*x^n)^3/m^3+3/2*a^2*b*x^(2*m)*ln(c*x^n 
)^3/m-15*a*b^2*n*x^m*ln(c*x^n)^4/m^2+3*a*b^2*x^m*ln(c*x^n)^5/m+1/8*b^3*ln( 
c*x^n)^8/n
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx=-\frac {a n x^m \left (25920 b^2 n^4+81 a b m^2 n^2 x^m+8 a^2 m^4 x^{2 m}\right )}{72 m^6}+\frac {a x^m \left (4320 b^2 n^4+27 a b m^2 n^2 x^m+4 a^2 m^4 x^{2 m}\right ) \log \left (c x^n\right )}{12 m^5}-\frac {9 a b n x^m \left (80 b n^2+a m^2 x^m\right ) \log ^2\left (c x^n\right )}{4 m^4}+\frac {3 a b x^m \left (40 b n^2+a m^2 x^m\right ) \log ^3\left (c x^n\right )}{2 m^3}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n} \] Input:

Integrate[(Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^3)/x,x]
 

Output:

-1/72*(a*n*x^m*(25920*b^2*n^4 + 81*a*b*m^2*n^2*x^m + 8*a^2*m^4*x^(2*m)))/m 
^6 + (a*x^m*(4320*b^2*n^4 + 27*a*b*m^2*n^2*x^m + 4*a^2*m^4*x^(2*m))*Log[c* 
x^n])/(12*m^5) - (9*a*b*n*x^m*(80*b*n^2 + a*m^2*x^m)*Log[c*x^n]^2)/(4*m^4) 
 + (3*a*b*x^m*(40*b*n^2 + a*m^2*x^m)*Log[c*x^n]^3)/(2*m^3) - (15*a*b^2*n*x 
^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^8)/( 
8*n)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3019, 2009}

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.

\(\displaystyle \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx\)

\(\Big \downarrow \) 3019

\(\displaystyle \int \left (a^3 x^{3 m-1} \log \left (c x^n\right )+3 a^2 b x^{2 m-1} \log ^3\left (c x^n\right )+3 a b^2 x^{m-1} \log ^5\left (c x^n\right )+\frac {b^3 \log ^7\left (c x^n\right )}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac {a^3 n x^{3 m}}{9 m^2}+\frac {9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac {9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac {3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac {9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac {360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac {180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac {60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac {15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac {3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac {360 a b^2 n^5 x^m}{m^6}+\frac {b^3 \log ^8\left (c x^n\right )}{8 n}\)

Input:

Int[(Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^3)/x,x]
 

Output:

(-360*a*b^2*n^5*x^m)/m^6 - (9*a^2*b*n^3*x^(2*m))/(8*m^4) - (a^3*n*x^(3*m)) 
/(9*m^2) + (360*a*b^2*n^4*x^m*Log[c*x^n])/m^5 + (9*a^2*b*n^2*x^(2*m)*Log[c 
*x^n])/(4*m^3) + (a^3*x^(3*m)*Log[c*x^n])/(3*m) - (180*a*b^2*n^3*x^m*Log[c 
*x^n]^2)/m^4 - (9*a^2*b*n*x^(2*m)*Log[c*x^n]^2)/(4*m^2) + (60*a*b^2*n^2*x^ 
m*Log[c*x^n]^3)/m^3 + (3*a^2*b*x^(2*m)*Log[c*x^n]^3)/(2*m) - (15*a*b^2*n*x 
^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^8)/( 
8*n)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3019

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)* 
(x_)^(m_.))^(p_.))/(x_), x_Symbol] :> Int[ExpandIntegrand[Log[c*x^n]^r/x, ( 
a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && 
EqQ[r, q - 1] && IGtQ[p, 0]
 

Maple [A] (warning: unable to verify)

Time = 14.78 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(-\frac {-9 b^{3} \ln \left (c \,x^{n}\right )^{8} m^{6}-216 a \,b^{2} \ln \left (c \,x^{n}\right )^{5} x^{m} m^{5} n -108 a^{2} b \ln \left (c \,x^{n}\right )^{3} x^{2 m} m^{5} n +1080 n^{2} a \,b^{2} \ln \left (c \,x^{n}\right )^{4} x^{m} m^{4}-24 a^{3} \ln \left (c \,x^{n}\right ) x^{3 m} m^{5} n +162 a^{2} b \,n^{2} \ln \left (c \,x^{n}\right )^{2} x^{2 m} m^{4}-4320 n^{3} a \,b^{2} \ln \left (c \,x^{n}\right )^{3} x^{m} m^{3}+8 a^{3} n^{2} x^{3 m} m^{4}-162 a^{2} b \,n^{3} \ln \left (c \,x^{n}\right ) x^{2 m} m^{3}+12960 n^{4} a \,b^{2} \ln \left (c \,x^{n}\right )^{2} x^{m} m^{2}+81 n^{4} a^{2} b \,x^{2 m} m^{2}-25920 a \,b^{2} n^{5} \ln \left (c \,x^{n}\right ) x^{m} m +25920 a \,b^{2} n^{6} x^{m}}{72 m^{6} n}\)	\(271\)
risch	\(\text {Expression too large to display}\)	\(61910\)

Input:

int(ln(c*x^n)*(a*x^m+b*ln(c*x^n)^2)^3/x,x,method=_RETURNVERBOSE)
 

Output:

-1/72*(-9*b^3*ln(c*x^n)^8*m^6-216*a*b^2*ln(c*x^n)^5*x^m*m^5*n-108*a^2*b*ln 
(c*x^n)^3*(x^m)^2*m^5*n+1080*n^2*a*b^2*ln(c*x^n)^4*x^m*m^4-24*a^3*ln(c*x^n 
)*(x^m)^3*m^5*n+162*a^2*b*n^2*ln(c*x^n)^2*(x^m)^2*m^4-4320*n^3*a*b^2*ln(c* 
x^n)^3*x^m*m^3+8*a^3*n^2*(x^m)^3*m^4-162*a^2*b*n^3*ln(c*x^n)*(x^m)^2*m^3+1 
2960*n^4*a*b^2*ln(c*x^n)^2*x^m*m^2+81*n^4*a^2*b*(x^m)^2*m^2-25920*a*b^2*n^ 
5*ln(c*x^n)*x^m*m+25920*a*b^2*n^6*x^m)/m^6/n
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (258) = 516\).

Time = 0.09 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.41 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx =\text {Too large to display} \] Input:

integrate(log(c*x^n)*(a*x^m+b*log(c*x^n)^2)^3/x,x, algorithm="fricas")
 

Output:

1/72*(9*b^3*m^6*n^7*log(x)^8 + 72*b^3*m^6*n^6*log(c)*log(x)^7 + 252*b^3*m^ 
6*n^5*log(c)^2*log(x)^6 + 504*b^3*m^6*n^4*log(c)^3*log(x)^5 + 630*b^3*m^6* 
n^3*log(c)^4*log(x)^4 + 504*b^3*m^6*n^2*log(c)^5*log(x)^3 + 252*b^3*m^6*n* 
log(c)^6*log(x)^2 + 72*b^3*m^6*log(c)^7*log(x) + 8*(3*a^3*m^5*n*log(x) + 3 
*a^3*m^5*log(c) - a^3*m^4*n)*x^(3*m) + 27*(4*a^2*b*m^5*n^3*log(x)^3 + 4*a^ 
2*b*m^5*log(c)^3 - 6*a^2*b*m^4*n*log(c)^2 + 6*a^2*b*m^3*n^2*log(c) - 3*a^2 
*b*m^2*n^3 + 6*(2*a^2*b*m^5*n^2*log(c) - a^2*b*m^4*n^3)*log(x)^2 + 6*(2*a^ 
2*b*m^5*n*log(c)^2 - 2*a^2*b*m^4*n^2*log(c) + a^2*b*m^3*n^3)*log(x))*x^(2* 
m) + 216*(a*b^2*m^5*n^5*log(x)^5 + a*b^2*m^5*log(c)^5 - 5*a*b^2*m^4*n*log( 
c)^4 + 20*a*b^2*m^3*n^2*log(c)^3 - 60*a*b^2*m^2*n^3*log(c)^2 + 120*a*b^2*m 
*n^4*log(c) - 120*a*b^2*n^5 + 5*(a*b^2*m^5*n^4*log(c) - a*b^2*m^4*n^5)*log 
(x)^4 + 10*(a*b^2*m^5*n^3*log(c)^2 - 2*a*b^2*m^4*n^4*log(c) + 2*a*b^2*m^3* 
n^5)*log(x)^3 + 10*(a*b^2*m^5*n^2*log(c)^3 - 3*a*b^2*m^4*n^3*log(c)^2 + 6* 
a*b^2*m^3*n^4*log(c) - 6*a*b^2*m^2*n^5)*log(x)^2 + 5*(a*b^2*m^5*n*log(c)^4 
 - 4*a*b^2*m^4*n^2*log(c)^3 + 12*a*b^2*m^3*n^3*log(c)^2 - 24*a*b^2*m^2*n^4 
*log(c) + 24*a*b^2*m*n^5)*log(x))*x^m)/m^6
 

Sympy [A] (verification not implemented)

Time = 30.46 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.51 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx =\text {Too large to display} \] Input:

integrate(ln(c*x**n)*(a*x**m+b*ln(c*x**n)**2)**3/x,x)
                                                                                    
                                                                                    
 

Output:

-a**3*n*Piecewise((Piecewise((x**(3*m)/(3*m), Ne(m, 0)), (log(x), True))/( 
3*m), (m > -oo) & (m < oo) & Ne(m, 0)), (log(x)**2/2, True)) + a**3*Piecew 
ise((x**(3*m)/(3*m), Ne(m, 0)), (log(x), True))*log(c*x**n) + 3*a**2*b*Pie 
cewise((x**(2*m)*log(c*x**n)**3/(2*m) - 3*n*x**(2*m)*log(c*x**n)**2/(4*m** 
2) + 3*n**2*x**(2*m)*log(c*x**n)/(4*m**3) - 3*n**3*x**(2*m)/(8*m**4), Ne(m 
, 0)), (Piecewise((0, (Abs(c*x**n) < 1) & (1/Abs(c*x**n) < 1)), (log(c*x** 
n)**4/(4*n), Abs(c*x**n) < 1), (log(1/(c*x**n))**4/(4*n), 1/Abs(c*x**n) < 
1), (6*meijerg(((), (1, 1, 1, 1, 1)), ((0, 0, 0, 0, 0), ()), c*x**n)/n + 6 
*meijerg(((1, 1, 1, 1, 1), ()), ((), (0, 0, 0, 0, 0)), c*x**n)/n, True)), 
True)) + 3*a*b**2*Piecewise((x**m*log(c*x**n)**5/m - 5*n*x**m*log(c*x**n)* 
*4/m**2 + 20*n**2*x**m*log(c*x**n)**3/m**3 - 60*n**3*x**m*log(c*x**n)**2/m 
**4 + 120*n**4*x**m*log(c*x**n)/m**5 - 120*n**5*x**m/m**6, Ne(m, 0)), (Pie 
cewise((0, (Abs(c*x**n) < 1) & (1/Abs(c*x**n) < 1)), (log(c*x**n)**6/(6*n) 
, Abs(c*x**n) < 1), (log(1/(c*x**n))**6/(6*n), 1/Abs(c*x**n) < 1), (120*me 
ijerg(((), (1, 1, 1, 1, 1, 1, 1)), ((0, 0, 0, 0, 0, 0, 0), ()), c*x**n)/n 
+ 120*meijerg(((1, 1, 1, 1, 1, 1, 1), ()), ((), (0, 0, 0, 0, 0, 0, 0)), c* 
x**n)/n, True)), True)) - b**3*Piecewise((-log(c)**7*log(x), Eq(n, 0)), (- 
log(c*x**n)**8/(8*n), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (258) = 516\).

Time = 0.06 (sec) , antiderivative size = 1115, normalized size of antiderivative = 4.10 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx=\text {Too large to display} \] Input:

integrate(log(c*x^n)*(a*x^m+b*log(c*x^n)^2)^3/x,x, algorithm="maxima")
 

Output:

1/84*(12*b^3*log(c*x^n)^7/n + 252*a*b^2*x^m*log(c*x^n)^4/m + 126*a^2*b*x^( 
2*m)*log(c*x^n)^2/m - 1008*(n*x^m*log(c*x^n)^3/m^2 - 3*(n*x^m*log(c*x^n)^2 
/m^2 - 2*n*(n*x^m*log(c*x^n)/m^2 - n^2*x^m/m^3)/m)*n/m)*a*b^2 - 63*a^2*b*( 
2*n*x^(2*m)*log(c*x^n)/m^2 - n^2*x^(2*m)/m^3) + 28*a^3*x^(3*m)/m)*log(c*x^ 
n) + 1/504*(9*b^3*m^6*n^7*log(x)^8 - 72*b^3*m^6*n^6*log(c)*log(x)^7 + 252* 
b^3*m^6*n^5*log(c)^2*log(x)^6 - 504*b^3*m^6*n^4*log(c)^3*log(x)^5 + 630*b^ 
3*m^6*n^3*log(c)^4*log(x)^4 - 504*b^3*m^6*n^2*log(c)^5*log(x)^3 + 252*b^3* 
m^6*n*log(c)^6*log(x)^2 - 72*b^3*m^6*log(c)^7*log(x) - 72*b^3*m^6*log(x)*l 
og(x^n)^7 - 56*a^3*m^4*n*x^(3*m) + 252*(b^3*m^6*n*log(x)^2 - 2*b^3*m^6*log 
(c)*log(x))*log(x^n)^6 - 504*(b^3*m^6*n^2*log(x)^3 - 3*b^3*m^6*n*log(c)*lo 
g(x)^2 + 3*b^3*m^6*log(c)^2*log(x))*log(x^n)^5 - 189*(2*m^4*n*log(c)^2 - 4 
*m^3*n^2*log(c) + 3*m^2*n^3)*a^2*b*x^(2*m) - 1512*(m^4*n*log(c)^4 - 8*m^3* 
n^2*log(c)^3 + 36*m^2*n^3*log(c)^2 - 96*m*n^4*log(c) + 120*n^5)*a*b^2*x^m 
+ 126*(5*b^3*m^6*n^3*log(x)^4 - 20*b^3*m^6*n^2*log(c)*log(x)^3 + 30*b^3*m^ 
6*n*log(c)^2*log(x)^2 - 20*b^3*m^6*log(c)^3*log(x) - 12*a*b^2*m^4*n*x^m)*l 
og(x^n)^4 - 504*(b^3*m^6*n^4*log(x)^5 - 5*b^3*m^6*n^3*log(c)*log(x)^4 + 10 
*b^3*m^6*n^2*log(c)^2*log(x)^3 - 10*b^3*m^6*n*log(c)^3*log(x)^2 + 5*b^3*m^ 
6*log(c)^4*log(x) + 12*(m^4*n*log(c) - 2*m^3*n^2)*a*b^2*x^m)*log(x^n)^3 + 
126*(2*b^3*m^6*n^5*log(x)^6 - 12*b^3*m^6*n^4*log(c)*log(x)^5 + 30*b^3*m^6* 
n^3*log(c)^2*log(x)^4 - 40*b^3*m^6*n^2*log(c)^3*log(x)^3 + 30*b^3*m^6*n...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (258) = 516\).

Time = 0.14 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.82 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx =\text {Too large to display} \] Input:

integrate(log(c*x^n)*(a*x^m+b*log(c*x^n)^2)^3/x,x, algorithm="giac")
 

Output:

1/8*b^3*n^7*log(x)^8 + b^3*n^6*log(c)*log(x)^7 + 7/2*b^3*n^5*log(c)^2*log( 
x)^6 + 7*b^3*n^4*log(c)^3*log(x)^5 + 35/4*b^3*n^3*log(c)^4*log(x)^4 + 7*b^ 
3*n^2*log(c)^5*log(x)^3 + 3*a*b^2*n^5*x^m*log(x)^5/m + 7/2*b^3*n*log(c)^6* 
log(x)^2 + 15*a*b^2*n^4*x^m*log(c)*log(x)^4/m + b^3*log(c)^7*log(x) + 30*a 
*b^2*n^3*x^m*log(c)^2*log(x)^3/m - 15*a*b^2*n^5*x^m*log(x)^4/m^2 + 30*a*b^ 
2*n^2*x^m*log(c)^3*log(x)^2/m - 60*a*b^2*n^4*x^m*log(c)*log(x)^3/m^2 + 15* 
a*b^2*n*x^m*log(c)^4*log(x)/m - 90*a*b^2*n^3*x^m*log(c)^2*log(x)^2/m^2 + 3 
/2*a^2*b*n^3*x^(2*m)*log(x)^3/m + 60*a*b^2*n^5*x^m*log(x)^3/m^3 + 3*a*b^2* 
x^m*log(c)^5/m - 60*a*b^2*n^2*x^m*log(c)^3*log(x)/m^2 + 9/2*a^2*b*n^2*x^(2 
*m)*log(c)*log(x)^2/m + 180*a*b^2*n^4*x^m*log(c)*log(x)^2/m^3 - 15*a*b^2*n 
*x^m*log(c)^4/m^2 + 9/2*a^2*b*n*x^(2*m)*log(c)^2*log(x)/m + 180*a*b^2*n^3* 
x^m*log(c)^2*log(x)/m^3 - 9/4*a^2*b*n^3*x^(2*m)*log(x)^2/m^2 - 180*a*b^2*n 
^5*x^m*log(x)^2/m^4 + 3/2*a^2*b*x^(2*m)*log(c)^3/m + 60*a*b^2*n^2*x^m*log( 
c)^3/m^3 - 9/2*a^2*b*n^2*x^(2*m)*log(c)*log(x)/m^2 - 360*a*b^2*n^4*x^m*log 
(c)*log(x)/m^4 - 9/4*a^2*b*n*x^(2*m)*log(c)^2/m^2 - 180*a*b^2*n^3*x^m*log( 
c)^2/m^4 + 1/3*a^3*n*x^(3*m)*log(x)/m + 9/4*a^2*b*n^3*x^(2*m)*log(x)/m^3 + 
 360*a*b^2*n^5*x^m*log(x)/m^5 + 1/3*a^3*x^(3*m)*log(c)/m + 9/4*a^2*b*n^2*x 
^(2*m)*log(c)/m^3 + 360*a*b^2*n^4*x^m*log(c)/m^5 - 1/9*a^3*n*x^(3*m)/m^2 - 
 9/8*a^2*b*n^3*x^(2*m)/m^4 - 360*a*b^2*n^5*x^m/m^6
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx=\int \frac {\ln \left (c\,x^n\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )}^3}{x} \,d x \] Input:

int((log(c*x^n)*(a*x^m + b*log(c*x^n)^2)^3)/x,x)
 

Output:

int((log(c*x^n)*(a*x^m + b*log(c*x^n)^2)^3)/x, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.99 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx=\frac {24 x^{3 m} \mathrm {log}\left (x^{n} c \right ) a^{3} m^{5} n -8 x^{3 m} a^{3} m^{4} n^{2}+108 x^{2 m} \mathrm {log}\left (x^{n} c \right )^{3} a^{2} b \,m^{5} n -162 x^{2 m} \mathrm {log}\left (x^{n} c \right )^{2} a^{2} b \,m^{4} n^{2}+162 x^{2 m} \mathrm {log}\left (x^{n} c \right ) a^{2} b \,m^{3} n^{3}-81 x^{2 m} a^{2} b \,m^{2} n^{4}+216 x^{m} \mathrm {log}\left (x^{n} c \right )^{5} a \,b^{2} m^{5} n -1080 x^{m} \mathrm {log}\left (x^{n} c \right )^{4} a \,b^{2} m^{4} n^{2}+4320 x^{m} \mathrm {log}\left (x^{n} c \right )^{3} a \,b^{2} m^{3} n^{3}-12960 x^{m} \mathrm {log}\left (x^{n} c \right )^{2} a \,b^{2} m^{2} n^{4}+25920 x^{m} \mathrm {log}\left (x^{n} c \right ) a \,b^{2} m \,n^{5}-25920 x^{m} a \,b^{2} n^{6}+9 \mathrm {log}\left (x^{n} c \right )^{8} b^{3} m^{6}}{72 m^{6} n} \] Input:

int(log(c*x^n)*(a*x^m+b*log(c*x^n)^2)^3/x,x)
 

Output:

(24*x**(3*m)*log(x**n*c)*a**3*m**5*n - 8*x**(3*m)*a**3*m**4*n**2 + 108*x** 
(2*m)*log(x**n*c)**3*a**2*b*m**5*n - 162*x**(2*m)*log(x**n*c)**2*a**2*b*m* 
*4*n**2 + 162*x**(2*m)*log(x**n*c)*a**2*b*m**3*n**3 - 81*x**(2*m)*a**2*b*m 
**2*n**4 + 216*x**m*log(x**n*c)**5*a*b**2*m**5*n - 1080*x**m*log(x**n*c)** 
4*a*b**2*m**4*n**2 + 4320*x**m*log(x**n*c)**3*a*b**2*m**3*n**3 - 12960*x** 
m*log(x**n*c)**2*a*b**2*m**2*n**4 + 25920*x**m*log(x**n*c)*a*b**2*m*n**5 - 
 25920*x**m*a*b**2*n**6 + 9*log(x**n*c)**8*b**3*m**6)/(72*m**6*n)