\(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x^m)^r)}{x} \, dx\) [146]

Optimal result

Integrand size = 28, antiderivative size = 150 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^m}{e}\right )}{3 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )}{m}+\frac {2 b n r \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2}-\frac {2 b^2 n^2 r \operatorname {PolyLog}\left (4,-\frac {f x^m}{e}\right )}{m^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(741\) vs. \(2(150)=300\).

Time = 0.38 (sec) , antiderivative size = 741, normalized size of antiderivative = 4.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/3*(a*b*m*n*r*Log[x]^3) + (b^2*m*n^2*r*Log[x]^4)/4 - (b^2*m*n*r*Log[x]^3 
*Log[c*x^n])/3 - a*b*n*r*Log[x]^2*Log[1 + e/(f*x^m)] + (2*b^2*n^2*r*Log[x] 
^3*Log[1 + e/(f*x^m)])/3 - b^2*n*r*Log[x]^2*Log[c*x^n]*Log[1 + e/(f*x^m)] 
- a^2*r*Log[x]*Log[e + f*x^m] + 2*a*b*n*r*Log[x]^2*Log[e + f*x^m] - b^2*n^ 
2*r*Log[x]^3*Log[e + f*x^m] + (a^2*r*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - 
 (2*a*b*n*r*Log[x]*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m + (b^2*n^2*r*Log[x] 
^2*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - 2*a*b*r*Log[x]*Log[c*x^n]*Log[e + 
 f*x^m] + 2*b^2*n*r*Log[x]^2*Log[c*x^n]*Log[e + f*x^m] + (2*a*b*r*Log[-((f 
*x^m)/e)]*Log[c*x^n]*Log[e + f*x^m])/m - (2*b^2*n*r*Log[x]*Log[-((f*x^m)/e 
)]*Log[c*x^n]*Log[e + f*x^m])/m - b^2*r*Log[x]*Log[c*x^n]^2*Log[e + f*x^m] 
 + (b^2*r*Log[-((f*x^m)/e)]*Log[c*x^n]^2*Log[e + f*x^m])/m + a^2*Log[x]*Lo 
g[d*(e + f*x^m)^r] - a*b*n*Log[x]^2*Log[d*(e + f*x^m)^r] + (b^2*n^2*Log[x] 
^3*Log[d*(e + f*x^m)^r])/3 + 2*a*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^m)^r] 
- b^2*n*Log[x]^2*Log[c*x^n]*Log[d*(e + f*x^m)^r] + b^2*Log[x]*Log[c*x^n]^2 
*Log[d*(e + f*x^m)^r] + (b*n*r*Log[x]*(-(b*n*Log[x]) + 2*(a + b*Log[c*x^n] 
))*PolyLog[2, -(e/(f*x^m))])/m + (r*(a - b*n*Log[x] + b*Log[c*x^n])^2*Poly 
Log[2, 1 + (f*x^m)/e])/m + (2*a*b*n*r*PolyLog[3, -(e/(f*x^m))])/m^2 + (2*b 
^2*n*r*Log[c*x^n]*PolyLog[3, -(e/(f*x^m))])/m^2 + (2*b^2*n^2*r*PolyLog[4, 
-(e/(f*x^m))])/m^3
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2822, 2775, 2821, 2830, 7143}

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.

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

\(\Big \downarrow \) 2822

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac {f m r \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )^3}{f x^m+e}dx}{3 b n}\)

\(\Big \downarrow \) 2775

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2775

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) 
+ (e_.)*(x_)^(r_)), x_Symbol] :> Simp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c* 
x^n])^p/(e*r)), x] - Simp[b*f^m*n*(p/(e*r))   Int[Log[1 + e*(x^r/d)]*((a + 
b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] & 
& EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2822

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

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \,x^{m}\right )^{r}\right )}{x}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (145) = 290\).

Time = 0.10 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\frac {b^{2} m^{3} n^{2} \log \left (d\right ) \log \left (x\right )^{3} - 6 \, b^{2} n^{2} r {\rm polylog}\left (4, -\frac {f x^{m}}{e}\right ) + 3 \, {\left (b^{2} m^{3} n \log \left (c\right ) + a b m^{3} n\right )} \log \left (d\right ) \log \left (x\right )^{2} + 3 \, {\left (b^{2} m^{3} \log \left (c\right )^{2} + 2 \, a b m^{3} \log \left (c\right ) + a^{2} m^{3}\right )} \log \left (d\right ) \log \left (x\right ) - 3 \, {\left (b^{2} m^{2} n^{2} r \log \left (x\right )^{2} + b^{2} m^{2} r \log \left (c\right )^{2} + 2 \, a b m^{2} r \log \left (c\right ) + a^{2} m^{2} r + 2 \, {\left (b^{2} m^{2} n r \log \left (c\right ) + a b m^{2} n r\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) + {\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \, {\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) - {\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \, {\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (\frac {f x^{m} + e}{e}\right ) + 6 \, {\left (b^{2} m n^{2} r \log \left (x\right ) + b^{2} m n r \log \left (c\right ) + a b m n r\right )} {\rm polylog}\left (3, -\frac {f x^{m}}{e}\right )}{3 \, m^{3}} \] Input:

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

Output:

1/3*(b^2*m^3*n^2*log(d)*log(x)^3 - 6*b^2*n^2*r*polylog(4, -f*x^m/e) + 3*(b 
^2*m^3*n*log(c) + a*b*m^3*n)*log(d)*log(x)^2 + 3*(b^2*m^3*log(c)^2 + 2*a*b 
*m^3*log(c) + a^2*m^3)*log(d)*log(x) - 3*(b^2*m^2*n^2*r*log(x)^2 + b^2*m^2 
*r*log(c)^2 + 2*a*b*m^2*r*log(c) + a^2*m^2*r + 2*(b^2*m^2*n*r*log(c) + a*b 
*m^2*n*r)*log(x))*dilog(-(f*x^m + e)/e + 1) + (b^2*m^3*n^2*r*log(x)^3 + 3* 
(b^2*m^3*n*r*log(c) + a*b*m^3*n*r)*log(x)^2 + 3*(b^2*m^3*r*log(c)^2 + 2*a* 
b*m^3*r*log(c) + a^2*m^3*r)*log(x))*log(f*x^m + e) - (b^2*m^3*n^2*r*log(x) 
^3 + 3*(b^2*m^3*n*r*log(c) + a*b*m^3*n*r)*log(x)^2 + 3*(b^2*m^3*r*log(c)^2 
 + 2*a*b*m^3*r*log(c) + a^2*m^3*r)*log(x))*log((f*x^m + e)/e) + 6*(b^2*m*n 
^2*r*log(x) + b^2*m*n*r*log(c) + a*b*m*n*r)*polylog(3, -f*x^m/e))/m^3
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

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

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

Output:

1/3*(b^2*n^2*log(x)^3 + 3*b^2*log(x)*log(x^n)^2 - 3*(b^2*n*log(c) + a*b*n) 
*log(x)^2 - 3*(b^2*n*log(x)^2 - 2*(b^2*log(c) + a*b)*log(x))*log(x^n) + 3* 
(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*log(x))*log((f*x^m + e)^r) - integrate 
(-1/3*(3*b^2*e*log(c)^2*log(d) + 6*a*b*e*log(c)*log(d) + 3*a^2*e*log(d) + 
3*(b^2*e*log(d) - (b^2*f*m*r*log(x) - b^2*f*log(d))*x^m)*log(x^n)^2 - (b^2 
*f*m*n^2*r*log(x)^3 - 3*b^2*f*log(c)^2*log(d) - 6*a*b*f*log(c)*log(d) - 3* 
a^2*f*log(d) - 3*(b^2*f*m*n*r*log(c) + a*b*f*m*n*r)*log(x)^2 + 3*(b^2*f*m* 
r*log(c)^2 + 2*a*b*f*m*r*log(c) + a^2*f*m*r)*log(x))*x^m + 3*(2*b^2*e*log( 
c)*log(d) + 2*a*b*e*log(d) + (b^2*f*m*n*r*log(x)^2 + 2*b^2*f*log(c)*log(d) 
 + 2*a*b*f*log(d) - 2*(b^2*f*m*r*log(c) + a*b*f*m*r)*log(x))*x^m)*log(x^n) 
)/(f*x*x^m + e*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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