Integrand size = 23, antiderivative size = 83 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {g p x^n}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \] Output:
-g*p*x^n/n+g*(d+e*x^n)*ln(c*(d+e*x^n)^p)/e/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
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-e g p x^n+\left (d g+e g x^n+e f \log \left (-\frac {e x^n}{d}\right )\right ) \log \left (c \left (d+e x^n\right )^p\right )+e f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{e n} \] Input:
Integrate[((f + g*x^n)*Log[c*(d + e*x^n)^p])/x,x]
Output:
(-(e*g*p*x^n) + (d*g + e*g*x^n + e*f*Log[-((e*x^n)/d)])*Log[c*(d + e*x^n)^ p] + e*f*p*PolyLog[2, 1 + (e*x^n)/d])/(e*n)
Time = 0.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
| \(\displaystyle \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {\int x^{-n} \left (g x^n+f\right ) \log \left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\int \left (f \log \left (c \left (e x^n+d\right )^p\right ) x^{-n}+g \log \left (c \left (e x^n+d\right )^p\right )\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e}+f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )-g p x^n}{n}\) |
Input:
Int[((f + g*x^n)*Log[c*(d + e*x^n)^p])/x,x]
Output:
(-(g*p*x^n) + (g*(d + e*x^n)*Log[c*(d + e*x^n)^p])/e + f*Log[-((e*x^n)/d)] *Log[c*(d + e*x^n)^p] + f*p*PolyLog[2, 1 + (e*x^n)/d])/n
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]
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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69
| method | result | size |
| risch | \(\frac {\left (f \ln \left (x \right ) n +g \,x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (f \ln \left (x \right )+\frac {g \,x^{n}}{n}\right )-\frac {g p \,x^{n}}{n}+\frac {p g d \ln \left (d +e \,x^{n}\right )}{e n}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(223\) |
Input:
int((f+g*x^n)*ln(c*(d+e*x^n)^p)/x,x,method=_RETURNVERBOSE)
Output:
(f*ln(x)*n+g*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)+g*x^n/n)-g*p*x^n/n+1/e*p/n*g*d*ln(d+e*x^n)-p/n*f*dilo g((d+e*x^n)/d)-p*f*ln(x)*ln((d+e*x^n)/d)
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.20 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {e f n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - e f n \log \left (c\right ) \log \left (x\right ) + e f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e g p - e g \log \left (c\right )\right )} x^{n} - {\left (e f n p \log \left (x\right ) + e g p x^{n} + d g p\right )} \log \left (e x^{n} + d\right )}{e n} \] Input:
integrate((f+g*x^n)*log(c*(d+e*x^n)^p)/x,x, algorithm="fricas")
Output:
-(e*f*n*p*log(x)*log((e*x^n + d)/d) - e*f*n*log(c)*log(x) + e*f*p*dilog(-( e*x^n + d)/d + 1) + (e*g*p - e*g*log(c))*x^n - (e*f*n*p*log(x) + e*g*p*x^n + d*g*p)*log(e*x^n + d))/(e*n)
\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \] Input:
integrate((f+g*x**n)*ln(c*(d+e*x**n)**p)/x,x)
Output:
Integral((f + g*x**n)*log(c*(d + e*x**n)**p)/x, x)
\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate((f+g*x^n)*log(c*(d+e*x^n)^p)/x,x, algorithm="maxima")
Output:
-1/2*(e*f*n^2*p*log(x)^2 + 2*(e*g*p - e*g*log(c))*x^n - 2*(e*f*n*log(x) + e*g*x^n)*log((e*x^n + d)^p) - 2*(d*g*n*p + e*f*n*log(c))*log(x))/(e*n) + i ntegrate((d*e*f*n*p*log(x) - d^2*g*p)/(e^2*x*x^n + d*e*x), x)
\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate((f+g*x^n)*log(c*(d+e*x^n)^p)/x,x, algorithm="giac")
Output:
integrate((g*x^n + f)*log((e*x^n + d)^p*c)/x, x)
Timed out. \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^n\right )}{x} \,d x \] Input:
int((log(c*(d + e*x^n)^p)*(f + g*x^n))/x,x)
Output:
int((log(c*(d + e*x^n)^p)*(f + g*x^n))/x, x)
\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {2 x^{n} \mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) e g p -2 x^{n} e g \,p^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}{x^{n} e x +d x}d x \right ) d e f n p +{\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{2} e f +2 \,\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) d g p}{2 e n p} \] Input:
int((f+g*x^n)*log(c*(d+e*x^n)^p)/x,x)
Output:
(2*x**n*log((x**n*e + d)**p*c)*e*g*p - 2*x**n*e*g*p**2 + 2*int(log((x**n*e + d)**p*c)/(x**n*e*x + d*x),x)*d*e*f*n*p + log((x**n*e + d)**p*c)**2*e*f + 2*log((x**n*e + d)**p*c)*d*g*p)/(2*e*n*p)