\(\int \frac {(f+g x^{3 n}) \log (c (d+e x^n)^p)}{x} \, dx\) [360]

Optimal result

Integrand size = 25, antiderivative size = 144 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {d^2 g p x^n}{3 e^2 n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \] Output:

-1/3*d^2*g*p*x^n/e^2/n+1/6*d*g*p*x^(2*n)/e/n-1/9*g*p*x^(3*n)/n+1/3*d^3*g*p 
*ln(d+e*x^n)/e^3/n+1/3*g*x^(3*n)*ln(c*(d+e*x^n)^p)/n+f*ln(-e*x^n/d)*ln(c*( 
d+e*x^n)^p)/n+f*p*polylog(2,1+e*x^n/d)/n
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.82 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-\frac {g p \left (e x^n \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right )-6 d^3 \log \left (d+e x^n\right )\right )}{e^3}+6 g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )+18 f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )\right )}{18 n} \] Input:

Integrate[((f + g*x^(3*n))*Log[c*(d + e*x^n)^p])/x,x]
 

Output:

(-((g*p*(e*x^n*(6*d^2 - 3*d*e*x^n + 2*e^2*x^(2*n)) - 6*d^3*Log[d + e*x^n]) 
)/e^3) + 6*g*x^(3*n)*Log[c*(d + e*x^n)^p] + 18*f*(Log[-((e*x^n)/d)]*Log[c* 
(d + e*x^n)^p] + p*PolyLog[2, 1 + (e*x^n)/d]))/(18*n)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 2863, 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.

\(\displaystyle \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\)

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2863

\(\displaystyle \frac {\int \left (f \log \left (c \left (e x^n+d\right )^p\right ) x^{-n}+g \log \left (c \left (e x^n+d\right )^p\right ) x^{2 n}\right )dx^n}{n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {1}{3} g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3}-\frac {d^2 g p x^n}{3 e^2}+f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )+\frac {d g p x^{2 n}}{6 e}-\frac {1}{9} g p x^{3 n}}{n}\)

Input:

Int[((f + g*x^(3*n))*Log[c*(d + e*x^n)^p])/x,x]
 

Output:

(-1/3*(d^2*g*p*x^n)/e^2 + (d*g*p*x^(2*n))/(6*e) - (g*p*x^(3*n))/9 + (d^3*g 
*p*Log[d + e*x^n])/(3*e^3) + (g*x^(3*n)*Log[c*(d + e*x^n)^p])/3 + f*Log[-( 
(e*x^n)/d)]*Log[c*(d + e*x^n)^p] + f*p*PolyLog[2, 1 + (e*x^n)/d])/n
 

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

Maple [C] (warning: unable to verify)

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

Time = 6.83 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.85

method	result	size
risch	\(\frac {\left (g \,x^{3 n}+3 f \ln \left (x \right ) n \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{3 n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (f \ln \left (x \right )+\frac {g \,x^{3 n}}{3 n}\right )-\frac {g p \,x^{3 n}}{9 n}+\frac {d g p \,x^{2 n}}{6 e n}-\frac {d^{2} g p \,x^{n}}{3 e^{2} n}+\frac {d^{3} g p \ln \left (d +e \,x^{n}\right )}{3 e^{3} n}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\)	\(267\)

Input:

int((f+g*x^(3*n))*ln(c*(d+e*x^n)^p)/x,x,method=_RETURNVERBOSE)
 

Output:

1/3*(g*(x^n)^3+3*f*ln(x)*n)/n*ln((d+e*x^n)^p)+(1/2*I*Pi*csgn(I*(d+e*x^n)^p 
)*csgn(I*c*(d+e*x^n)^p)^2-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^ 
p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*csgn(I*c*(d+e*x^n)^ 
p)^2*csgn(I*c)+ln(c))*(f*ln(x)+1/3*g/n*(x^n)^3)-1/9*p/n*g*(x^n)^3+1/6/e*p/ 
n*g*d*(x^n)^2-1/3*d^2*g*p*x^n/e^2/n+1/3*d^3*g*p*ln(d+e*x^n)/e^3/n-p/n*f*di 
log((d+e*x^n)/d)-p*f*ln(x)*ln((d+e*x^n)/d)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.03 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {18 \, e^{3} f n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 18 \, e^{3} f n \log \left (c\right ) \log \left (x\right ) - 3 \, d e^{2} g p x^{2 \, n} + 6 \, d^{2} e g p x^{n} + 18 \, e^{3} f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + 2 \, {\left (e^{3} g p - 3 \, e^{3} g \log \left (c\right )\right )} x^{3 \, n} - 6 \, {\left (3 \, e^{3} f n p \log \left (x\right ) + e^{3} g p x^{3 \, n} + d^{3} g p\right )} \log \left (e x^{n} + d\right )}{18 \, e^{3} n} \] Input:

integrate((f+g*x^(3*n))*log(c*(d+e*x^n)^p)/x,x, algorithm="fricas")
 

Output:

-1/18*(18*e^3*f*n*p*log(x)*log((e*x^n + d)/d) - 18*e^3*f*n*log(c)*log(x) - 
 3*d*e^2*g*p*x^(2*n) + 6*d^2*e*g*p*x^n + 18*e^3*f*p*dilog(-(e*x^n + d)/d + 
 1) + 2*(e^3*g*p - 3*e^3*g*log(c))*x^(3*n) - 6*(3*e^3*f*n*p*log(x) + e^3*g 
*p*x^(3*n) + d^3*g*p)*log(e*x^n + d))/(e^3*n)
 

Sympy [F]

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

integrate((f+g*x**(3*n))*ln(c*(d+e*x**n)**p)/x,x)
 

Output:

Integral((f + g*x**(3*n))*log(c*(d + e*x**n)**p)/x, x)
 

Maxima [F]

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

integrate((f+g*x^(3*n))*log(c*(d+e*x^n)^p)/x,x, algorithm="maxima")
 

Output:

-1/18*(9*e^3*f*n^2*p*log(x)^2 - 3*d*e^2*g*p*x^(2*n) + 6*d^2*e*g*p*x^n + 2* 
(e^3*g*p - 3*e^3*g*log(c))*x^(3*n) - 6*(3*e^3*f*n*log(x) + e^3*g*x^(3*n))* 
log((e*x^n + d)^p) - 6*(d^3*g*n*p + 3*e^3*f*n*log(c))*log(x))/(e^3*n) + in 
tegrate(1/3*(3*d*e^3*f*n*p*log(x) - d^4*g*p)/(e^4*x*x^n + d*e^3*x), x)
 

Giac [F]

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

integrate((f+g*x^(3*n))*log(c*(d+e*x^n)^p)/x,x, algorithm="giac")
 

Output:

integrate((g*x^(3*n) + f)*log((e*x^n + d)^p*c)/x, x)
 

Mupad [F(-1)]

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

int((log(c*(d + e*x^n)^p)*(f + g*x^(3*n)))/x,x)
 

Output:

int((log(c*(d + e*x^n)^p)*(f + g*x^(3*n)))/x, x)
 

Reduce [F]

\[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {6 x^{3 n} \mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) e^{3} g p -2 x^{3 n} e^{3} g \,p^{2}+3 x^{2 n} d \,e^{2} g \,p^{2}-6 x^{n} d^{2} e g \,p^{2}+18 \left (\int \frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}{x^{n} e x +d x}d x \right ) d \,e^{3} f n p +9 {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{2} e^{3} f +6 \,\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) d^{3} g p}{18 e^{3} n p} \] Input:

int((f+g*x^(3*n))*log(c*(d+e*x^n)^p)/x,x)
 

Output:

(6*x**(3*n)*log((x**n*e + d)**p*c)*e**3*g*p - 2*x**(3*n)*e**3*g*p**2 + 3*x 
**(2*n)*d*e**2*g*p**2 - 6*x**n*d**2*e*g*p**2 + 18*int(log((x**n*e + d)**p* 
c)/(x**n*e*x + d*x),x)*d*e**3*f*n*p + 9*log((x**n*e + d)**p*c)**2*e**3*f + 
 6*log((x**n*e + d)**p*c)*d**3*g*p)/(18*e**3*n*p)