\(\int \frac {x (a+b \log (c x^n))}{(d+e x)^2} \, dx\) [41]

Optimal result

Integrand size = 19, antiderivative size = 65 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2384, 2354, 2438} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{e^2}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2} \]

[In]

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

[Out]

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

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 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 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 {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \frac {a+b n+b \log \left (c x^n\right )}{d+e x} \, dx}{e} \\ & = -\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^2} \\ & = -\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {\frac {d \left (a+b \log \left (c x^n\right )\right )}{d+e x}-b n (\log (x)-\log (d+e x))+\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.78 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.15

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2}}+\frac {b \ln \left (x^{n}\right ) d}{e^{2} \left (e x +d \right )}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}+\frac {b n \ln \left (e x +d \right )}{e^{2}}-\frac {b n \ln \left (e x \right )}{e^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\ln \left (e x +d \right )}{e^{2}}+\frac {d}{e^{2} \left (e x +d \right )}\right )\)	\(205\)

[In]

int(x*(a+b*ln(c*x^n))/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

b*ln(x^n)/e^2*ln(e*x+d)+b*ln(x^n)/e^2*d/(e*x+d)-b*n/e^2*ln(e*x+d)*ln(-e*x/d)-b*n/e^2*dilog(-e*x/d)+b*n/e^2*ln(
e*x+d)-b*n/e^2*ln(e*x)+(-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)*(1/e^2*ln(e*x+d)+1/e^2*d/(e*x+d))

Fricas [F]

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

[In]

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

[Out]

integral((b*x*log(c*x^n) + a*x)/(e^2*x^2 + 2*d*e*x + d^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

a*(d/(e^3*x + d*e^2) + log(e*x + d)/e^2) + b*integrate((x*log(c) + x*log(x^n))/(e^2*x^2 + 2*d*e*x + d^2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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