\(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x)^m)}{x^4} \, dx\) [90]

Optimal result

Integrand size = 26, antiderivative size = 420 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=-\frac {19 b^2 f m n^2}{108 e x^2}+\frac {26 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac {8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}-\frac {2 b f^3 m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}-\frac {f^3 m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {2 b^2 f^3 m n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{9 e^3}+\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{3 e^3}+\frac {2 b^2 f^3 m n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right )}{3 e^3} \] Output:

-19/108*b^2*f*m*n^2/e/x^2+26/27*b^2*f^2*m*n^2/e^2/x+2/27*b^2*f^3*m*n^2*ln( 
x)/e^3-5/18*b*f*m*n*(a+b*ln(c*x^n))/e/x^2+8/9*b*f^2*m*n*(a+b*ln(c*x^n))/e^ 
2/x-2/9*b*f^3*m*n*ln(1+e/f/x)*(a+b*ln(c*x^n))/e^3-1/6*f*m*(a+b*ln(c*x^n))^ 
2/e/x^2+1/3*f^2*m*(a+b*ln(c*x^n))^2/e^2/x-1/3*f^3*m*ln(1+e/f/x)*(a+b*ln(c* 
x^n))^2/e^3-2/27*b^2*f^3*m*n^2*ln(f*x+e)/e^3-2/27*b^2*n^2*ln(d*(f*x+e)^m)/ 
x^3-2/9*b*n*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)/x^3-1/3*(a+b*ln(c*x^n))^2*ln(d 
*(f*x+e)^m)/x^3+2/9*b^2*f^3*m*n^2*polylog(2,-e/f/x)/e^3+2/3*b*f^3*m*n*(a+b 
*ln(c*x^n))*polylog(2,-e/f/x)/e^3+2/3*b^2*f^3*m*n^2*polylog(3,-e/f/x)/e^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(420)=840\).

Time = 0.50 (sec) , antiderivative size = 909, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/x^4,x]
 

Output:

-1/108*(18*a^2*e^2*f*m*x + 30*a*b*e^2*f*m*n*x + 19*b^2*e^2*f*m*n^2*x - 36* 
a^2*e*f^2*m*x^2 - 96*a*b*e*f^2*m*n*x^2 - 104*b^2*e*f^2*m*n^2*x^2 - 36*a^2* 
f^3*m*x^3*Log[x] - 24*a*b*f^3*m*n*x^3*Log[x] - 8*b^2*f^3*m*n^2*x^3*Log[x] 
+ 36*a*b*f^3*m*n*x^3*Log[x]^2 + 12*b^2*f^3*m*n^2*x^3*Log[x]^2 - 12*b^2*f^3 
*m*n^2*x^3*Log[x]^3 + 36*a*b*e^2*f*m*x*Log[c*x^n] + 30*b^2*e^2*f*m*n*x*Log 
[c*x^n] - 72*a*b*e*f^2*m*x^2*Log[c*x^n] - 96*b^2*e*f^2*m*n*x^2*Log[c*x^n] 
- 72*a*b*f^3*m*x^3*Log[x]*Log[c*x^n] - 24*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n 
] + 36*b^2*f^3*m*n*x^3*Log[x]^2*Log[c*x^n] + 18*b^2*e^2*f*m*x*Log[c*x^n]^2 
 - 36*b^2*e*f^2*m*x^2*Log[c*x^n]^2 - 36*b^2*f^3*m*x^3*Log[x]*Log[c*x^n]^2 
+ 36*a^2*f^3*m*x^3*Log[e + f*x] + 24*a*b*f^3*m*n*x^3*Log[e + f*x] + 8*b^2* 
f^3*m*n^2*x^3*Log[e + f*x] - 72*a*b*f^3*m*n*x^3*Log[x]*Log[e + f*x] - 24*b 
^2*f^3*m*n^2*x^3*Log[x]*Log[e + f*x] + 36*b^2*f^3*m*n^2*x^3*Log[x]^2*Log[e 
 + f*x] + 72*a*b*f^3*m*x^3*Log[c*x^n]*Log[e + f*x] + 24*b^2*f^3*m*n*x^3*Lo 
g[c*x^n]*Log[e + f*x] - 72*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n]*Log[e + f*x] 
+ 36*b^2*f^3*m*x^3*Log[c*x^n]^2*Log[e + f*x] + 36*a^2*e^3*Log[d*(e + f*x)^ 
m] + 24*a*b*e^3*n*Log[d*(e + f*x)^m] + 8*b^2*e^3*n^2*Log[d*(e + f*x)^m] + 
72*a*b*e^3*Log[c*x^n]*Log[d*(e + f*x)^m] + 24*b^2*e^3*n*Log[c*x^n]*Log[d*( 
e + f*x)^m] + 36*b^2*e^3*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 72*a*b*f^3*m*n* 
x^3*Log[x]*Log[1 + (f*x)/e] + 24*b^2*f^3*m*n^2*x^3*Log[x]*Log[1 + (f*x)/e] 
 - 36*b^2*f^3*m*n^2*x^3*Log[x]^2*Log[1 + (f*x)/e] + 72*b^2*f^3*m*n*x^3*...
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 403, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -f m \int \left (-\frac {2 b^2 n^2}{27 x^3 (e+f x)}-\frac {2 b \left (a+b \log \left (c x^n\right )\right ) n}{9 x^3 (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3 (e+f x)}\right )dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -f m \left (-\frac {2 b f^2 n \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^2 \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}+\frac {2 b f^2 n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}-\frac {8 b f n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {5 b n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}-\frac {2 b^2 f^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x}\right )}{9 e^3}-\frac {2 b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x}\right )}{3 e^3}-\frac {2 b^2 f^2 n^2 \log (x)}{27 e^3}+\frac {2 b^2 f^2 n^2 \log (e+f x)}{27 e^3}-\frac {26 b^2 f n^2}{27 e^2 x}+\frac {19 b^2 n^2}{108 e x^2}\right )-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}\)

Input:

Int[((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/x^4,x]
 

Output:

(-2*b^2*n^2*Log[d*(e + f*x)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d 
*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(3*x^3) 
 - f*m*((19*b^2*n^2)/(108*e*x^2) - (26*b^2*f*n^2)/(27*e^2*x) - (2*b^2*f^2* 
n^2*Log[x])/(27*e^3) + (5*b*n*(a + b*Log[c*x^n]))/(18*e*x^2) - (8*b*f*n*(a 
 + b*Log[c*x^n]))/(9*e^2*x) + (2*b*f^2*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n 
]))/(9*e^3) + (a + b*Log[c*x^n])^2/(6*e*x^2) - (f*(a + b*Log[c*x^n])^2)/(3 
*e^2*x) + (f^2*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(3*e^3) + (2*b^2*f^2 
*n^2*Log[e + f*x])/(27*e^3) - (2*b^2*f^2*n^2*PolyLog[2, -(e/(f*x))])/(9*e^ 
3) - (2*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/(3*e^3) - (2*b^ 
2*f^2*n^2*PolyLog[3, -(e/(f*x))])/(3*e^3))
 

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 = 66.92 (sec) , antiderivative size = 6242, normalized size of antiderivative = 14.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(6242\)

Input:

int((a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)/x^4,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(f*x+e)^m)/x^4,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**2*ln(d*(f*x+e)**m)/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(f*x+e)^m)/x^4,x, algorithm="maxima")
 

Output:

-1/54*(9*(2*b^2*f^3*m*x^3*log(f*x + e) - 2*b^2*f^3*m*x^3*log(x) - 2*b^2*e* 
f^2*m*x^2 + b^2*e^2*f*m*x + 2*b^2*e^3*log(d))*log(x^n)^2 + 2*(9*b^2*e^3*lo 
g(x^n)^2 + 9*a^2*e^3 + 6*(e^3*n + 3*e^3*log(c))*a*b + (2*e^3*n^2 + 6*e^3*n 
*log(c) + 9*e^3*log(c)^2)*b^2 + 6*(3*a*b*e^3 + (e^3*n + 3*e^3*log(c))*b^2) 
*log(x^n))*log((f*x + e)^m))/(e^3*x^3) + integrate(1/27*(27*b^2*e^4*log(c) 
^2*log(d) + 54*a*b*e^4*log(c)*log(d) + 27*a^2*e^4*log(d) + (9*(e^3*f*m + 3 
*e^3*f*log(d))*a^2 + 6*(e^3*f*m*n + 3*(e^3*f*m + 3*e^3*f*log(d))*log(c))*a 
*b + (2*e^3*f*m*n^2 + 6*e^3*f*m*n*log(c) + 9*(e^3*f*m + 3*e^3*f*log(d))*lo 
g(c)^2)*b^2)*x - 3*(6*b^2*e*f^3*m*n*x^3 + 3*b^2*e^2*f^2*m*n*x^2 - 18*a*b*e 
^4*log(d) - 6*(e^4*n*log(d) + 3*e^4*log(c)*log(d))*b^2 - (6*(e^3*f*m + 3*e 
^3*f*log(d))*a*b + (5*e^3*f*m*n + 6*e^3*f*n*log(d) + 6*(e^3*f*m + 3*e^3*f* 
log(d))*log(c))*b^2)*x - 6*(b^2*f^4*m*n*x^4 + b^2*e*f^3*m*n*x^3)*log(f*x + 
 e) + 6*(b^2*f^4*m*n*x^4 + b^2*e*f^3*m*n*x^3)*log(x))*log(x^n))/(e^3*f*x^5 
 + e^4*x^4), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(f*x+e)^m)/x^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2*log((f*x + e)^m*d)/x^4, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4} \,d x \] Input:

int((log(d*(e + f*x)^m)*(a + b*log(c*x^n))^2)/x^4,x)
 

Output:

int((log(d*(e + f*x)^m)*(a + b*log(c*x^n))^2)/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx=\frac {-36 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) a b \,f^{3} n \,x^{3}+12 \,\mathrm {log}\left (x \right ) b^{2} f^{3} m \,n^{2} x^{3}-6 b^{2} e^{2} f m \,n^{2} x +12 b^{2} e \,f^{2} m \,n^{2} x^{2}-108 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{f \,x^{5}+e \,x^{4}}d x \right ) a b \,e^{4} m \,x^{3}-36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{f \,x^{5}+e \,x^{4}}d x \right ) b^{2} e^{4} m n \,x^{3}-54 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{3}-54 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) a^{2} f^{3} x^{3}-12 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) b^{2} e^{3} n^{2}-18 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{3} m -8 b^{2} e^{3} m \,n^{2}-108 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right ) a b \,e^{3}-36 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right ) b^{2} e^{3} n -36 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) a b \,e^{3} n -12 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) b^{2} f^{3} n^{2} x^{3}-36 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{3} m -24 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{3} m n +54 \,\mathrm {log}\left (x \right ) a^{2} f^{3} m \,x^{3}-27 a^{2} e^{2} f m x +54 a^{2} e \,f^{2} m \,x^{2}-12 a b \,e^{3} m n -54 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{f \,x^{5}+e \,x^{4}}d x \right ) b^{2} e^{4} m \,x^{3}+36 \,\mathrm {log}\left (x \right ) a b \,f^{3} m n \,x^{3}-18 a b \,e^{2} f m n x +36 a b e \,f^{2} m n \,x^{2}-54 \,\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) a^{2} e^{3}}{162 e^{3} x^{3}} \] Input:

int((a+b*log(c*x^n))^2*log(d*(f*x+e)^m)/x^4,x)
 

Output:

( - 54*int(log(x**n*c)**2/(e*x**4 + f*x**5),x)*b**2*e**4*m*x**3 - 108*int( 
log(x**n*c)/(e*x**4 + f*x**5),x)*a*b*e**4*m*x**3 - 36*int(log(x**n*c)/(e*x 
**4 + f*x**5),x)*b**2*e**4*m*n*x**3 - 54*log((e + f*x)**m*d)*log(x**n*c)** 
2*b**2*e**3 - 108*log((e + f*x)**m*d)*log(x**n*c)*a*b*e**3 - 36*log((e + f 
*x)**m*d)*log(x**n*c)*b**2*e**3*n - 54*log((e + f*x)**m*d)*a**2*e**3 - 54* 
log((e + f*x)**m*d)*a**2*f**3*x**3 - 36*log((e + f*x)**m*d)*a*b*e**3*n - 3 
6*log((e + f*x)**m*d)*a*b*f**3*n*x**3 - 12*log((e + f*x)**m*d)*b**2*e**3*n 
**2 - 12*log((e + f*x)**m*d)*b**2*f**3*n**2*x**3 - 18*log(x**n*c)**2*b**2* 
e**3*m - 36*log(x**n*c)*a*b*e**3*m - 24*log(x**n*c)*b**2*e**3*m*n + 54*log 
(x)*a**2*f**3*m*x**3 + 36*log(x)*a*b*f**3*m*n*x**3 + 12*log(x)*b**2*f**3*m 
*n**2*x**3 - 27*a**2*e**2*f*m*x + 54*a**2*e*f**2*m*x**2 - 12*a*b*e**3*m*n 
- 18*a*b*e**2*f*m*n*x + 36*a*b*e*f**2*m*n*x**2 - 8*b**2*e**3*m*n**2 - 6*b* 
*2*e**2*f*m*n**2*x + 12*b**2*e*f**2*m*n**2*x**2)/(162*e**3*x**3)