\(\int \frac {a+b \log (c x^n)}{d+e x} \, dx\) [34]

Optimal result

Integrand size = 18, antiderivative size = 39 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \]

[Out]

(a+b*ln(c*x^n))*ln(1+e*x/d)/e+b*n*polylog(2,-e*x/d)/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2354, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx=\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \]

[In]

Int[(a + b*Log[c*x^n])/(d + e*x),x]

[Out]

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

Rule 2354

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

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.40 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.87

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}-\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \ln \left (e x +d \right )}{e}\)	\(151\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*log(c*x^n))/(d + e*x),x)

[Out]

int((a + b*log(c*x^n))/(d + e*x), x)