\(\int \frac {a+b \log (c (d+e x)^n)}{(f+\frac {g}{x}) x} \, dx\) [306]

Optimal result

Integrand size = 27, antiderivative size = 63 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \]

[Out]

(a+b*ln(c*(e*x+d)^n))*ln(-e*(f*x+g)/(d*f-e*g))/f+b*n*polylog(2,f*(e*x+d)/(d*f-e*g))/f

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2459, 2441, 2440, 2438} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \]

[In]

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

[Out]

((a + b*Log[c*(d + e*x)^n])*Log[-((e*(g + f*x))/(d*f - e*g))])/f + (b*n*PolyLog[2, (f*(d + e*x))/(d*f - e*g)])
/f

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2459

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \log \left (c (d+e x)^n\right )}{g+f x} \, dx \\ & = \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}-\frac {(b e n) \int \frac {\log \left (\frac {e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx}{f} \\ & = \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}-\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1+\frac {f x}{-d f+e g}\right )}{x} \, dx,x,d+e x\right )}{f} \\ & = \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}+\frac {b n \text {Li}_2\left (\frac {f (d+e x)}{d f-e g}\right )}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (g+f x)}{-d f+e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \]

[In]

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

[Out]

((a + b*Log[c*(d + e*x)^n])*Log[(e*(g + f*x))/(-(d*f) + e*g)])/f + (b*n*PolyLog[2, (f*(d + e*x))/(d*f - e*g)])
/f

Maple [C] (warning: unable to verify)

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

Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.44

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (f x +g \right )}{f}-\frac {b n \operatorname {dilog}\left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}-\frac {b n \ln \left (f x +g \right ) \ln \left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \ln \left (f x +g \right )}{f}\)	\(217\)

[In]

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

[Out]

b*ln((e*x+d)^n)*ln(f*x+g)/f-b/f*n*dilog(((f*x+g)*e+d*f-e*g)/(d*f-e*g))-b/f*n*ln(f*x+g)*ln(((f*x+g)*e+d*f-e*g)/
(d*f-e*g))+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^
n)^2+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+b*ln(c)+a)*ln(f*x+g)/
f

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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