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

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

[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)

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Rubi [A]  time = 0.229273, antiderivative size = 102, normalized size of antiderivative = 1., 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 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 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]

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 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]

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*}

Mathematica [A]  time = 0.313321, size = 132, normalized size = 1.29 \[ \frac{b n \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+\frac{d r \left (a+b \log \left (c x^n\right )\right )}{d+e x^r}-a r \log \left (d-d x^r\right )+b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b n \log \left (d-d x^r\right )}{d^2 r^2} \]

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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Maple [C]  time = 0.066, size = 715, normalized size = 7. \begin{align*} \text{result too large to display} \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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{1}{d e r x^{r} + d^{2} r} + \frac{\log \left (x\right )}{d^{2}} - \frac{\log \left (\frac{e x^{r} + d}{e}\right )}{d^{2} r}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{2} x x^{2 \, r} + 2 \, d e x x^{r} + d^{2} 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)^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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Fricas [B]  time = 1.38566, size = 527, normalized size = 5.17 \begin{align*} \frac{b d n r^{2} \log \left (x\right )^{2} + 2 \, b d r \log \left (c\right ) + 2 \, a d r +{\left (b e n r^{2} \log \left (x\right )^{2} + 2 \,{\left (b e r^{2} \log \left (c\right ) - b e n r + a e r^{2}\right )} \log \left (x\right )\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 \left (c\right ) - b d n + a d r +{\left (b e r \log \left (c\right ) - b e n + a e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \,{\left (b d r^{2} \log \left (c\right ) + a d r^{2}\right )} \log \left (x\right ) - 2 \,{\left (b e n r x^{r} \log \left (x\right ) + b d n r \log \left (x\right )\right )} \log \left (\frac{e x^{r} + d}{d}\right )}{2 \,{\left (d^{2} e r^{2} x^{r} + d^{3} r^{2}\right )}} \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)^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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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)**2,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 )}^{2} 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)^2,x, algorithm="giac")

[Out]

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