\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x)^m)}{x^2} \, dx\) [94]

Optimal result

Integrand size = 26, antiderivative size = 411 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^2} \, dx=\frac {6 b^3 f m n^3 \log (x)}{e}-\frac {6 b^2 f m n^2 \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {3 b f m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {f m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac {6 b^3 f m n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+\frac {6 b^3 f m n^3 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{e}+\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{e}+\frac {6 b^3 f m n^3 \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right )}{e}+\frac {6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right )}{e}+\frac {6 b^3 f m n^3 \operatorname {PolyLog}\left (4,-\frac {e}{f x}\right )}{e} \] Output:

6*b^3*f*m*n^3*ln(x)/e-6*b^2*f*m*n^2*ln(1+e/f/x)*(a+b*ln(c*x^n))/e-3*b*f*m* 
n*ln(1+e/f/x)*(a+b*ln(c*x^n))^2/e-f*m*ln(1+e/f/x)*(a+b*ln(c*x^n))^3/e-6*b^ 
3*f*m*n^3*ln(f*x+e)/e-6*b^3*n^3*ln(d*(f*x+e)^m)/x-6*b^2*n^2*(a+b*ln(c*x^n) 
)*ln(d*(f*x+e)^m)/x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)/x-(a+b*ln(c*x^ 
n))^3*ln(d*(f*x+e)^m)/x+6*b^3*f*m*n^3*polylog(2,-e/f/x)/e+6*b^2*f*m*n^2*(a 
+b*ln(c*x^n))*polylog(2,-e/f/x)/e+3*b*f*m*n*(a+b*ln(c*x^n))^2*polylog(2,-e 
/f/x)/e+6*b^3*f*m*n^3*polylog(3,-e/f/x)/e+6*b^2*f*m*n^2*(a+b*ln(c*x^n))*po 
lylog(3,-e/f/x)/e+6*b^3*f*m*n^3*polylog(4,-e/f/x)/e
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1347\) vs. \(2(411)=822\).

Time = 0.75 (sec) , antiderivative size = 1347, normalized size of antiderivative = 3.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(-4*a^3*f*m*x*Log[x] - 12*a^2*b*f*m*n*x*Log[x] - 24*a*b^2*f*m*n^2*x*L 
og[x] - 24*b^3*f*m*n^3*x*Log[x] + 6*a^2*b*f*m*n*x*Log[x]^2 + 12*a*b^2*f*m* 
n^2*x*Log[x]^2 + 12*b^3*f*m*n^3*x*Log[x]^2 - 4*a*b^2*f*m*n^2*x*Log[x]^3 - 
4*b^3*f*m*n^3*x*Log[x]^3 + b^3*f*m*n^3*x*Log[x]^4 - 12*a^2*b*f*m*x*Log[x]* 
Log[c*x^n] - 24*a*b^2*f*m*n*x*Log[x]*Log[c*x^n] - 24*b^3*f*m*n^2*x*Log[x]* 
Log[c*x^n] + 12*a*b^2*f*m*n*x*Log[x]^2*Log[c*x^n] + 12*b^3*f*m*n^2*x*Log[x 
]^2*Log[c*x^n] - 4*b^3*f*m*n^2*x*Log[x]^3*Log[c*x^n] - 12*a*b^2*f*m*x*Log[ 
x]*Log[c*x^n]^2 - 12*b^3*f*m*n*x*Log[x]*Log[c*x^n]^2 + 6*b^3*f*m*n*x*Log[x 
]^2*Log[c*x^n]^2 - 4*b^3*f*m*x*Log[x]*Log[c*x^n]^3 + 4*a^3*f*m*x*Log[e + f 
*x] + 12*a^2*b*f*m*n*x*Log[e + f*x] + 24*a*b^2*f*m*n^2*x*Log[e + f*x] + 24 
*b^3*f*m*n^3*x*Log[e + f*x] - 12*a^2*b*f*m*n*x*Log[x]*Log[e + f*x] - 24*a* 
b^2*f*m*n^2*x*Log[x]*Log[e + f*x] - 24*b^3*f*m*n^3*x*Log[x]*Log[e + f*x] + 
 12*a*b^2*f*m*n^2*x*Log[x]^2*Log[e + f*x] + 12*b^3*f*m*n^3*x*Log[x]^2*Log[ 
e + f*x] - 4*b^3*f*m*n^3*x*Log[x]^3*Log[e + f*x] + 12*a^2*b*f*m*x*Log[c*x^ 
n]*Log[e + f*x] + 24*a*b^2*f*m*n*x*Log[c*x^n]*Log[e + f*x] + 24*b^3*f*m*n^ 
2*x*Log[c*x^n]*Log[e + f*x] - 24*a*b^2*f*m*n*x*Log[x]*Log[c*x^n]*Log[e + f 
*x] - 24*b^3*f*m*n^2*x*Log[x]*Log[c*x^n]*Log[e + f*x] + 12*b^3*f*m*n^2*x*L 
og[x]^2*Log[c*x^n]*Log[e + f*x] + 12*a*b^2*f*m*x*Log[c*x^n]^2*Log[e + f*x] 
 + 12*b^3*f*m*n*x*Log[c*x^n]^2*Log[e + f*x] - 12*b^3*f*m*n*x*Log[x]*Log[c* 
x^n]^2*Log[e + f*x] + 4*b^3*f*m*x*Log[c*x^n]^3*Log[e + f*x] + 4*a^3*e*L...
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2825, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -f m \int \left (-\frac {6 b^3 n^3}{x (e+f x)}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{x (e+f x)}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{x (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (e+f x)}\right )dx-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-f m \left (-\frac {6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {6 b^2 n^2 \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {3 b n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {\log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{e}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right )}{e}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {e}{f x}\right )}{e}+\frac {6 b^3 n^3 \log (e+f x)}{e}-\frac {6 b^3 n^3 \log (x)}{e}\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac {6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}\)

Input:

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

Output:

(-6*b^3*n^3*Log[d*(e + f*x)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e 
 + f*x)^m])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/x - ((a + 
b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/x - f*m*((-6*b^3*n^3*Log[x])/e + (6*b^ 
2*n^2*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/e + (3*b*n*Log[1 + e/(f*x)]*(a 
+ b*Log[c*x^n])^2)/e + (Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^3)/e + (6*b^3* 
n^3*Log[e + f*x])/e - (6*b^3*n^3*PolyLog[2, -(e/(f*x))])/e - (6*b^2*n^2*(a 
 + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/e - (3*b*n*(a + b*Log[c*x^n])^2*P 
olyLog[2, -(e/(f*x))])/e - (6*b^3*n^3*PolyLog[3, -(e/(f*x))])/e - (6*b^2*n 
^2*(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x))])/e - (6*b^3*n^3*PolyLog[4, -( 
e/(f*x))])/e)
 

Defintions of rubi rules used

rule 2009

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

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 181.47 (sec) , antiderivative size = 16533, normalized size of antiderivative = 40.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(16533\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a 
^3)*log((f*x + e)^m*d)/x^2, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-((b^3*f*m*x*log(f*x + e) - b^3*f*m*x*log(x) + b^3*e*log(d))*log(x^n)^3 + 
(b^3*e*log(x^n)^3 + 3*(e*n + e*log(c))*a^2*b + 3*(2*e*n^2 + 2*e*n*log(c) + 
 e*log(c)^2)*a*b^2 + (6*e*n^3 + 6*e*n^2*log(c) + 3*e*n*log(c)^2 + e*log(c) 
^3)*b^3 + a^3*e + 3*((e*n + e*log(c))*b^3 + a*b^2*e)*log(x^n)^2 + 3*(2*(e* 
n + e*log(c))*a*b^2 + (2*e*n^2 + 2*e*n*log(c) + e*log(c)^2)*b^3 + a^2*b*e) 
*log(x^n))*log((f*x + e)^m))/(e*x) + integrate((b^3*e^2*log(c)^3*log(d) + 
3*a*b^2*e^2*log(c)^2*log(d) + 3*a^2*b*e^2*log(c)*log(d) + a^3*e^2*log(d) + 
 3*(a*b^2*e^2*log(d) + (e^2*n*log(d) + e^2*log(c)*log(d))*b^3 + ((e*f*m + 
e*f*log(d))*a*b^2 + (e*f*m*n + e*f*n*log(d) + (e*f*m + e*f*log(d))*log(c)) 
*b^3)*x + (b^3*f^2*m*n*x^2 + b^3*e*f*m*n*x)*log(f*x + e) - (b^3*f^2*m*n*x^ 
2 + b^3*e*f*m*n*x)*log(x))*log(x^n)^2 + ((e*f*m + e*f*log(d))*a^3 + 3*(e*f 
*m*n + (e*f*m + e*f*log(d))*log(c))*a^2*b + 3*(2*e*f*m*n^2 + 2*e*f*m*n*log 
(c) + (e*f*m + e*f*log(d))*log(c)^2)*a*b^2 + (6*e*f*m*n^3 + 6*e*f*m*n^2*lo 
g(c) + 3*e*f*m*n*log(c)^2 + (e*f*m + e*f*log(d))*log(c)^3)*b^3)*x + 3*(b^3 
*e^2*log(c)^2*log(d) + 2*a*b^2*e^2*log(c)*log(d) + a^2*b*e^2*log(d) + ((e* 
f*m + e*f*log(d))*a^2*b + 2*(e*f*m*n + (e*f*m + e*f*log(d))*log(c))*a*b^2 
+ (2*e*f*m*n^2 + 2*e*f*m*n*log(c) + (e*f*m + e*f*log(d))*log(c)^2)*b^3)*x) 
*log(x^n))/(e*f*x^3 + e^2*x^2), x)
 

Giac [F]

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

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

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x + e)^m*d)/x^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - int(log(x**n*c)**3/(e*x**2 + f*x**3),x)*b**3*e**2*m*x - 3*int(log(x**n 
*c)**2/(e*x**2 + f*x**3),x)*a*b**2*e**2*m*x - 3*int(log(x**n*c)**2/(e*x**2 
 + f*x**3),x)*b**3*e**2*m*n*x - 3*int(log(x**n*c)/(e*x**2 + f*x**3),x)*a** 
2*b*e**2*m*x - 6*int(log(x**n*c)/(e*x**2 + f*x**3),x)*a*b**2*e**2*m*n*x - 
6*int(log(x**n*c)/(e*x**2 + f*x**3),x)*b**3*e**2*m*n**2*x - log((e + f*x)* 
*m*d)*log(x**n*c)**3*b**3*e - 3*log((e + f*x)**m*d)*log(x**n*c)**2*a*b**2* 
e - 3*log((e + f*x)**m*d)*log(x**n*c)**2*b**3*e*n - 3*log((e + f*x)**m*d)* 
log(x**n*c)*a**2*b*e - 6*log((e + f*x)**m*d)*log(x**n*c)*a*b**2*e*n - 6*lo 
g((e + f*x)**m*d)*log(x**n*c)*b**3*e*n**2 - log((e + f*x)**m*d)*a**3*e - l 
og((e + f*x)**m*d)*a**3*f*x - 3*log((e + f*x)**m*d)*a**2*b*e*n - 3*log((e 
+ f*x)**m*d)*a**2*b*f*n*x - 6*log((e + f*x)**m*d)*a*b**2*e*n**2 - 6*log((e 
 + f*x)**m*d)*a*b**2*f*n**2*x - 6*log((e + f*x)**m*d)*b**3*e*n**3 - 6*log( 
(e + f*x)**m*d)*b**3*f*n**3*x - log(x**n*c)**3*b**3*e*m - 3*log(x**n*c)**2 
*a*b**2*e*m - 6*log(x**n*c)**2*b**3*e*m*n - 3*log(x**n*c)*a**2*b*e*m - 12* 
log(x**n*c)*a*b**2*e*m*n - 18*log(x**n*c)*b**3*e*m*n**2 + log(x)*a**3*f*m* 
x + 3*log(x)*a**2*b*f*m*n*x + 6*log(x)*a*b**2*f*m*n**2*x + 6*log(x)*b**3*f 
*m*n**3*x - 3*a**2*b*e*m*n - 12*a*b**2*e*m*n**2 - 18*b**3*e*m*n**3)/(e*x)