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

Optimal. Leaf size=54 \[ \frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]

[Out]

-(a+b*ln(c*x^n))*ln(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 = 0.08, antiderivative size = 54, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.11, size = 108, normalized size = 2.00 \[ \frac {-2 r \log \left (d-d x^r\right ) \left (a+b \log \left (c x^n\right )\right )+2 b n \text {Li}_2\left (\frac {e x^r}{d}+1\right )+2 b n r \log (x) \left (\log \left (d-d x^r\right )-\log \left (d+e x^r\right )\right )+2 b n \log \left (-\frac {e x^r}{d}\right ) \log \left (d+e x^r\right )+b n r^2 \log ^2(x)}{2 d r^2} \]

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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fricas [A]  time = 0.44, size = 93, normalized size = 1.72 \[ \frac {b n r^{2} \log \relax (x)^{2} - 2 \, b n r \log \relax (x) \log \left (\frac {e x^{r} + d}{d}\right ) - 2 \, b n {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 2 \, {\left (b r \log \relax (c) + a r\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b r^{2} \log \relax (c) + a r^{2}\right )} \log \relax (x)}{2 \, d r^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x}\,{d x} \]

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)

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maple [C]  time = 0.06, size = 451, normalized size = 8.35 \[ \frac {b n \ln \relax (x )^{2}}{2 d}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x^{r}\right )}{2 d r}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e \,x^{r}+d \right )}{2 d r}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{r}\right )}{2 d r}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{r}+d \right )}{2 d r}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{r}\right )}{2 d r}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e \,x^{r}+d \right )}{2 d r}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{r}\right )}{2 d r}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e \,x^{r}+d \right )}{2 d r}-\frac {b n \ln \relax (x ) \ln \left (x^{r}\right )}{d r}-\frac {b n \ln \relax (x ) \ln \left (\frac {e \,x^{r}}{d}+1\right )}{d r}+\frac {b n \ln \relax (x ) \ln \left (e \,x^{r}+d \right )}{d r}+\frac {b \ln \relax (c ) \ln \left (x^{r}\right )}{d r}-\frac {b \ln \relax (c ) \ln \left (e \,x^{r}+d \right )}{d r}+\frac {b \ln \left (x^{n}\right ) \ln \left (x^{r}\right )}{d r}-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{r}+d \right )}{d r}+\frac {a \ln \left (x^{r}\right )}{d r}-\frac {a \ln \left (e \,x^{r}+d \right )}{d r}-\frac {b n \polylog \left (2, -\frac {e \,x^{r}}{d}\right )}{d \,r^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {\log \relax (x)}{d} - \frac {\log \left (\frac {e x^{r} + d}{e}\right )}{d r}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x x^{r} + d x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+e\,x^r\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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