\(\int \frac {\log ^2(f x^p) (a+b \log (c (d+e x^m)^n))}{x} \, dx\) [622]

Optimal result

Integrand size = 28, antiderivative size = 132 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {2 b n p^2 \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(456\) vs. \(2(132)=264\).

Time = 0.28 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.45 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {1}{4} b m n p^2 \log ^4(x)-\frac {1}{3} b m n p \log ^3(x) \log \left (f x^p\right )+\frac {a \log ^3\left (f x^p\right )}{3 p}+\frac {2}{3} b n p^2 \log ^3(x) \log \left (1+\frac {d x^{-m}}{e}\right )-b n p \log ^2(x) \log \left (f x^p\right ) \log \left (1+\frac {d x^{-m}}{e}\right )-b n p^2 \log ^3(x) \log \left (d+e x^m\right )+\frac {b n p^2 \log ^2(x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+2 b n p \log ^2(x) \log \left (f x^p\right ) \log \left (d+e x^m\right )-\frac {2 b n p \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (f x^p\right ) \log \left (d+e x^m\right )}{m}-b n \log (x) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log \left (-\frac {e x^m}{d}\right ) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )}{m}+\frac {1}{3} b p^2 \log ^3(x) \log \left (c \left (d+e x^m\right )^n\right )-b p \log ^2(x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log ^2\left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )-\frac {b n p \log (x) \left (p \log (x)-2 \log \left (f x^p\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{m}+\frac {b n \left (-p \log (x)+\log \left (f x^p\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )}{m^2}+\frac {2 b n p^2 \operatorname {PolyLog}\left (4,-\frac {d x^{-m}}{e}\right )}{m^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 149, 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 = {2931, 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.

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

\(\Big \downarrow \) 2931

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

\(\Big \downarrow \) 2775

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 2931

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

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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (127) = 254\).

Time = 0.13 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.13 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=-\frac {6 \, b n p^{2} {\rm polylog}\left (4, -\frac {e x^{m}}{d}\right ) - 3 \, {\left (b m^{3} \log \left (c\right ) + a m^{3}\right )} \log \left (f\right )^{2} \log \left (x\right ) - 3 \, {\left (b m^{3} p \log \left (c\right ) + a m^{3} p\right )} \log \left (f\right ) \log \left (x\right )^{2} - {\left (b m^{3} p^{2} \log \left (c\right ) + a m^{3} p^{2}\right )} \log \left (x\right )^{3} + 3 \, {\left (b m^{2} n p^{2} \log \left (x\right )^{2} + 2 \, b m^{2} n p \log \left (f\right ) \log \left (x\right ) + b m^{2} n \log \left (f\right )^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left (b m^{3} n p^{2} \log \left (x\right )^{3} + 3 \, b m^{3} n p \log \left (f\right ) \log \left (x\right )^{2} + 3 \, b m^{3} n \log \left (f\right )^{2} \log \left (x\right )\right )} \log \left (e x^{m} + d\right ) + {\left (b m^{3} n p^{2} \log \left (x\right )^{3} + 3 \, b m^{3} n p \log \left (f\right ) \log \left (x\right )^{2} + 3 \, b m^{3} n \log \left (f\right )^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right ) - 6 \, {\left (b m n p^{2} \log \left (x\right ) + b m n p \log \left (f\right )\right )} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{3 \, m^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d + e*x**m)**n))*log(f*x**p)**2/x, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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