3.426 \(\int \frac{a+b \log (c x^n)}{x (d+e x^r)^3} \, dx\)

Optimal. Leaf size=169 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}+\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac{b n}{2 d^2 r^2 \left (d+e x^r\right )}+\frac{3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac{b n \log (x)}{2 d^3 r} \]

[Out]

-(b*n)/(2*d^2*r^2*(d + e*x^r)) - (b*n*Log[x])/(2*d^3*r) + (a + b*Log[c*x^n])/(2*d*r*(d + e*x^r)^2) - (e*x^r*(a
+ b*Log[c*x^n]))/(d^3*r*(d + e*x^r)) - ((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^3*r) + (3*b*n*Log[d + e*x^r
])/(2*d^3*r^2) + (b*n*PolyLog[2, -(d/(e*x^r))])/(d^3*r^2)

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Rubi [A]  time = 0.409281, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2349, 2345, 2391, 2335, 260, 2338, 266, 44} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}+\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac{b n}{2 d^2 r^2 \left (d+e x^r\right )}+\frac{3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac{b n \log (x)}{2 d^3 r} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x*(d + e*x^r)^3),x]

[Out]

-(b*n)/(2*d^2*r^2*(d + e*x^r)) - (b*n*Log[x])/(2*d^3*r) + (a + b*Log[c*x^n])/(2*d*r*(d + e*x^r)^2) - (e*x^r*(a
+ b*Log[c*x^n]))/(d^3*r*(d + e*x^r)) - ((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^3*r) + (3*b*n*Log[d + e*x^r
])/(2*d^3*r^2) + (b*n*PolyLog[2, -(d/(e*x^r))])/(d^3*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]

Rule 2338

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, 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 )^3} \, dx}{d}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac{\int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2}-\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^2}-\frac{(b n) \int \frac{1}{x \left (d+e x^r\right )^2} \, dx}{2 d r}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 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^3 r}-\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)^2} \, dx,x,x^r\right )}{2 d r^2}+\frac{(b n) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}+\frac{(b e n) \int \frac{x^{-1+r}}{d+e x^r} \, dx}{d^3 r}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 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^3 r}+\frac{b n \log \left (d+e x^r\right )}{d^3 r^2}+\frac{b n \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{(b n) \operatorname{Subst}\left (\int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx,x,x^r\right )}{2 d r^2}\\ &=-\frac{b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac{b n \log (x)}{2 d^3 r}+\frac{a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 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^3 r}+\frac{3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac{b n \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^2}\\ \end{align*}

Mathematica [A]  time = 0.234533, size = 170, normalized size = 1.01 \[ \frac{2 b n \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+\frac{d^2 r \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2}+\frac{d \left (2 a r+2 b r \log \left (c x^n\right )-b n\right )}{d+e x^r}-2 a r \log \left (d-d x^r\right )+2 b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+3 b n \log \left (d-d x^r\right )}{2 d^3 r^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*x^n])/(x*(d + e*x^r)^3),x]

[Out]

((d^2*r*(a + b*Log[c*x^n]))/(d + e*x^r)^2 + (d*(-(b*n) + 2*a*r + 2*b*r*Log[c*x^n]))/(d + e*x^r) + 3*b*n*Log[d
- d*x^r] - 2*a*r*Log[d - d*x^r] + 2*b*r*(n*Log[x] - Log[c*x^n])*Log[d - d*x^r] + 2*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]))/(2*d^3*r^2)

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Maple [C]  time = 0.25, size = 1012, normalized size = 6. \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)^3,x)

[Out]

1/2*b*n/d^3*ln(x)^2-b/r/d^2/(d+e*x^r)*n*ln(x)-1/2*b/r/d/(d+e*x^r)^2*n*ln(x)+b/r/d^3*ln(d+e*x^r)*n*ln(x)-1/r*a/
d^3*ln(d+e*x^r)+1/r*a/d^2/(d+e*x^r)+1/2/r*a/d/(d+e*x^r)^2+1/r*a/d^3*ln(x^r)-1/4*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)/d/(d+e*x^r)^2+1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^3*ln(d+e*x^r)+1/2*I/r*b*Pi*csg
n(I*c*x^n)^3/d^3*ln(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*ln(x^r)+1/4*I/r*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2/d/(d+e*x^r)^2-1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*ln(d+e*x^r)-1/2*I/r*b*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)/d^3*ln(x^r)-1/r*b*ln(c)/d^3*ln(d+e*x^r)+1/r*b*ln(c)/d^2/(d+e*x^r)+1/2/r*b*ln(c)/d/(d+e*
x^r)^2+1/r*b*ln(c)/d^3*ln(x^r)+1/2*b/r/d/(d+e*x^r)^2*ln(x^n)+b/r/d^3*ln(x^r)*ln(x^n)-b/r^2*n/d^3*dilog((d+e*x^
r)/d)-b/r/d^3*ln(d+e*x^r)*ln(x^n)+b/r/d^2/(d+e*x^r)*ln(x^n)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^2/(d+e*x^r)-1/4*I/r
*b*Pi*csgn(I*c*x^n)^3/d/(d+e*x^r)^2-b/r/d^3*ln(x^r)*n*ln(x)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^3*ln(x^r)-1/2*I/r*b
*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^2/(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^3*ln(x^r)+1/2*I
/r*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^2/(d+e*x^r)-1/2*I/r*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^3*ln(d+e*x^r)-b/r*n*e
/d^2*ln(x)*x^r/(d+e*x^r)^2-1/2*b/r*n*e^2/d^3*ln(x)*(x^r)^2/(d+e*x^r)^2+1/4*I/r*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/
d/(d+e*x^r)^2+1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^2/(d+e*x^r)-b/r*n*e/d^3*ln(x)*x^r/(d+e*x^r)-1/2*b*n/d
^2/r^2/(d+e*x^r)+3/2*b*n*ln(d+e*x^r)/d^3/r^2-b/r*n/d^3*ln(x)*ln((d+e*x^r)/d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{2 \, e x^{r} + 3 \, d}{d^{2} e^{2} r x^{2 \, r} + 2 \, d^{3} e r x^{r} + d^{4} r} + \frac{2 \, \log \left (x\right )}{d^{3}} - \frac{2 \, \log \left (\frac{e x^{r} + d}{e}\right )}{d^{3} r}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{3} x x^{3 \, r} + 3 \, d e^{2} x x^{2 \, r} + 3 \, d^{2} e x x^{r} + d^{3} 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)^3,x, algorithm="maxima")

[Out]

1/2*a*((2*e*x^r + 3*d)/(d^2*e^2*r*x^(2*r) + 2*d^3*e*r*x^r + d^4*r) + 2*log(x)/d^3 - 2*log((e*x^r + d)/e)/(d^3*
r)) + b*integrate((log(c) + log(x^n))/(e^3*x*x^(3*r) + 3*d*e^2*x*x^(2*r) + 3*d^2*e*x*x^r + d^3*x), x)

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Fricas [B]  time = 1.60719, size = 959, normalized size = 5.67 \begin{align*} \frac{b d^{2} n r^{2} \log \left (x\right )^{2} + 3 \, b d^{2} r \log \left (c\right ) - b d^{2} n + 3 \, a d^{2} r +{\left (b e^{2} n r^{2} \log \left (x\right )^{2} +{\left (2 \, b e^{2} r^{2} \log \left (c\right ) - 3 \, b e^{2} n r + 2 \, a e^{2} r^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} +{\left (2 \, b d e n r^{2} \log \left (x\right )^{2} + 2 \, b d e r \log \left (c\right ) - b d e n + 2 \, a d e r + 4 \,{\left (b d e r^{2} \log \left (c\right ) - b d e n r + a d e r^{2}\right )} \log \left (x\right )\right )} x^{r} - 2 \,{\left (b e^{2} n x^{2 \, r} + 2 \, b d e n x^{r} + b d^{2} n\right )}{\rm Li}_2\left (-\frac{e x^{r} + d}{d} + 1\right ) -{\left (2 \, b d^{2} r \log \left (c\right ) - 3 \, b d^{2} n + 2 \, a d^{2} r +{\left (2 \, b e^{2} r \log \left (c\right ) - 3 \, b e^{2} n + 2 \, a e^{2} r\right )} x^{2 \, r} + 2 \,{\left (2 \, b d e r \log \left (c\right ) - 3 \, b d e n + 2 \, a d e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \,{\left (b d^{2} r^{2} \log \left (c\right ) + a d^{2} r^{2}\right )} \log \left (x\right ) - 2 \,{\left (b e^{2} n r x^{2 \, r} \log \left (x\right ) + 2 \, b d e n r x^{r} \log \left (x\right ) + b d^{2} n r \log \left (x\right )\right )} \log \left (\frac{e x^{r} + d}{d}\right )}{2 \,{\left (d^{3} e^{2} r^{2} x^{2 \, r} + 2 \, d^{4} e r^{2} x^{r} + d^{5} 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)^3,x, algorithm="fricas")

[Out]

1/2*(b*d^2*n*r^2*log(x)^2 + 3*b*d^2*r*log(c) - b*d^2*n + 3*a*d^2*r + (b*e^2*n*r^2*log(x)^2 + (2*b*e^2*r^2*log(
c) - 3*b*e^2*n*r + 2*a*e^2*r^2)*log(x))*x^(2*r) + (2*b*d*e*n*r^2*log(x)^2 + 2*b*d*e*r*log(c) - b*d*e*n + 2*a*d
*e*r + 4*(b*d*e*r^2*log(c) - b*d*e*n*r + a*d*e*r^2)*log(x))*x^r - 2*(b*e^2*n*x^(2*r) + 2*b*d*e*n*x^r + b*d^2*n
)*dilog(-(e*x^r + d)/d + 1) - (2*b*d^2*r*log(c) - 3*b*d^2*n + 2*a*d^2*r + (2*b*e^2*r*log(c) - 3*b*e^2*n + 2*a*
e^2*r)*x^(2*r) + 2*(2*b*d*e*r*log(c) - 3*b*d*e*n + 2*a*d*e*r)*x^r)*log(e*x^r + d) + 2*(b*d^2*r^2*log(c) + a*d^
2*r^2)*log(x) - 2*(b*e^2*n*r*x^(2*r)*log(x) + 2*b*d*e*n*r*x^r*log(x) + b*d^2*n*r*log(x))*log((e*x^r + d)/d))/(
d^3*e^2*r^2*x^(2*r) + 2*d^4*e*r^2*x^r + d^5*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)**3,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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

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