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

Optimal. Leaf size=102 \[ -\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^2 r^2}+\frac {b n \log \left (d+e x^r\right )}{d^2 r^2} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2349, 2345, 2391, 2335, 260} \[ \frac {b n \text {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^2 r^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r}+\frac {b n \log \left (d+e x^r\right )}{d^2 r^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2335

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

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 2349

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

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

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Mathematica [A]  time = 0.30, size = 132, normalized size = 1.29 \[ \frac {\frac {d r \left (a+b \log \left (c x^n\right )\right )}{d+e x^r}-a r \log \left (d-d x^r\right )+b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b n \left (\text {Li}_2\left (\frac {e x^r}{d}+1\right )+\left (\log \left (-\frac {e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac {1}{2} r^2 \log ^2(x)\right )+b n \log \left (d-d x^r\right )}{d^2 r^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 214, normalized size = 2.10 \[ \frac {b d n r^{2} \log \relax (x)^{2} + 2 \, b d r \log \relax (c) + 2 \, a d r + {\left (b e n r^{2} \log \relax (x)^{2} + 2 \, {\left (b e r^{2} \log \relax (c) - b e n r + a e r^{2}\right )} \log \relax (x)\right )} x^{r} - 2 \, {\left (b e n x^{r} + b d n\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 2 \, {\left (b d r \log \relax (c) - b d n + a d r + {\left (b e r \log \relax (c) - b e n + a e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b d r^{2} \log \relax (c) + a d r^{2}\right )} \log \relax (x) - 2 \, {\left (b e n r x^{r} \log \relax (x) + b d n r \log \relax (x)\right )} \log \left (\frac {e x^{r} + d}{d}\right )}{2 \, {\left (d^{2} e r^{2} x^{r} + d^{3} r^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d*n*r^2*log(x)^2 + 2*b*d*r*log(c) + 2*a*d*r + (b*e*n*r^2*log(x)^2 + 2*(b*e*r^2*log(c) - b*e*n*r + a*e*r
^2)*log(x))*x^r - 2*(b*e*n*x^r + b*d*n)*dilog(-(e*x^r + d)/d + 1) - 2*(b*d*r*log(c) - b*d*n + a*d*r + (b*e*r*l
og(c) - b*e*n + a*e*r)*x^r)*log(e*x^r + d) + 2*(b*d*r^2*log(c) + a*d*r^2)*log(x) - 2*(b*e*n*r*x^r*log(x) + b*d
*n*r*log(x))*log((e*x^r + d)/d))/(d^2*e*r^2*x^r + d^3*r^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.06, size = 715, normalized size = 7.01 \[ \frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e \,x^{r}+d \right ) 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 (x^{r}\right )}{2 d^{2} 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^{2} 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 )}{2 \left (e \,x^{r}+d \right ) d r}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e \,x^{r}+d \right ) 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^{2} 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^{2} 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^{2} 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^{2} r}-\frac {b n \dilog \left (\frac {e \,x^{r}+d}{d}\right )}{d^{2} r^{2}}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e \,x^{r}+d \right )}{2 d^{2} r}+\frac {b n \ln \relax (x )^{2}}{2 d^{2}}-\frac {b n \ln \relax (x )}{\left (e \,x^{r}+d \right ) d r}-\frac {b n \ln \relax (x ) \ln \left (\frac {e \,x^{r}+d}{d}\right )}{d^{2} r}+\frac {a \ln \left (x^{r}\right )}{d^{2} r}-\frac {a \ln \left (e \,x^{r}+d \right )}{d^{2} r}+\frac {a}{\left (e \,x^{r}+d \right ) d r}-\frac {b n \ln \relax (x ) \ln \left (x^{r}\right )}{d^{2} r}+\frac {b n \ln \relax (x ) \ln \left (e \,x^{r}+d \right )}{d^{2} r}-\frac {b e n \,x^{r} \ln \relax (x )}{\left (e \,x^{r}+d \right ) d^{2} r}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 \left (e \,x^{r}+d \right ) d r}+\frac {b \ln \relax (c ) \ln \left (x^{r}\right )}{d^{2} r}-\frac {b \ln \relax (c ) \ln \left (e \,x^{r}+d \right )}{d^{2} r}+\frac {b \ln \relax (c )}{\left (e \,x^{r}+d \right ) d r}+\frac {b \ln \left (x^{n}\right ) \ln \left (x^{r}\right )}{d^{2} r}-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{r}+d \right )}{d^{2} r}+\frac {b \ln \left (x^{n}\right )}{\left (e \,x^{r}+d \right ) d r}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{r}\right )}{2 d^{2} r}+\frac {b n \ln \left (e \,x^{r}+d \right )}{d^{2} 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)^2,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {1}{d e r x^{r} + d^{2} r} + \frac {\log \relax (x)}{d^{2}} - \frac {\log \left (\frac {e x^{r} + d}{e}\right )}{d^{2} r}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{2} x x^{2 \, r} + 2 \, d e x x^{r} + d^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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