3.424 \(\int \frac{a+b \log (c x^n)}{x (d+e x^r)} \, dx\)

Optimal. Leaf size=54 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d r^2}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]

[Out]

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

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Rubi [A]  time = 0.0753664, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2345, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d r^2}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2345

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

Rule 2391

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*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{(b n) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d r}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{b n \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d r^2}\\ \end{align*}

Mathematica [A]  time = 0.102542, size = 108, normalized size = 2. \[ \frac{2 b n \text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )-2 r \log \left (d-d x^r\right ) \left (a+b \log \left (c x^n\right )\right )+2 b n r \log (x) \left (\log \left (d-d x^r\right )-\log \left (d+e x^r\right )\right )+2 b n \log \left (-\frac{e x^r}{d}\right ) \log \left (d+e x^r\right )+b n r^2 \log ^2(x)}{2 d r^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.059, size = 451, normalized size = 8.4 \begin{align*}{\frac{b\ln \left ( d+e{x}^{r} \right ) n\ln \left ( x \right ) }{dr}}-{\frac{b\ln \left ( d+e{x}^{r} \right ) \ln \left ({x}^{n} \right ) }{dr}}-{\frac{b\ln \left ({x}^{r} \right ) n\ln \left ( x \right ) }{dr}}+{\frac{b\ln \left ({x}^{r} \right ) \ln \left ({x}^{n} \right ) }{dr}}+{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}}{2\,d}}-{\frac{\ln \left ( x \right ) bn}{dr}\ln \left ( 1+{\frac{e{x}^{r}}{d}} \right ) }-{\frac{bn}{{r}^{2}d}{\it polylog} \left ( 2,-{\frac{e{x}^{r}}{d}} \right ) }+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( d+e{x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ({x}^{r} \right ) }{dr}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ({x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( d+e{x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ({x}^{r} \right ) }{dr}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ({x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( d+e{x}^{r} \right ) }{dr}}-{\frac{b\ln \left ( c \right ) \ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{b\ln \left ( c \right ) \ln \left ({x}^{r} \right ) }{dr}}-{\frac{a\ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{a\ln \left ({x}^{r} \right ) }{dr}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{\log \left (x\right )}{d} - \frac{\log \left (\frac{e x^{r} + d}{e}\right )}{d r}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x x^{r} + d x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.26406, size = 235, normalized size = 4.35 \begin{align*} \frac{b n r^{2} \log \left (x\right )^{2} - 2 \, b n r \log \left (x\right ) \log \left (\frac{e x^{r} + d}{d}\right ) - 2 \, b n{\rm Li}_2\left (-\frac{e x^{r} + d}{d} + 1\right ) - 2 \,{\left (b r \log \left (c\right ) + a r\right )} \log \left (e x^{r} + d\right ) + 2 \,{\left (b r^{2} \log \left (c\right ) + a r^{2}\right )} \log \left (x\right )}{2 \, d r^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*n*r^2*log(x)^2 - 2*b*n*r*log(x)*log((e*x^r + d)/d) - 2*b*n*dilog(-(e*x^r + d)/d + 1) - 2*(b*r*log(c) +
a*r)*log(e*x^r + d) + 2*(b*r^2*log(c) + a*r^2)*log(x))/(d*r^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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