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

Optimal. Leaf size=94 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d r^2}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )}{d r^3}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d r} \]

[Out]

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

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Rubi [A]  time = 0.135188, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2345, 2374, 6589} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d r^2}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )}{d r^3}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d r} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d r}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d r^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{x} \, dx}{d r^2}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d r^2}+\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{d x^{-r}}{e}\right )}{d r^3}\\ \end{align*}

Mathematica [B]  time = 0.293643, size = 270, normalized size = 2.87 \[ -\frac{-2 a b n r \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+2 b^2 n r \left (n \log (x)-\log \left (c x^n\right )\right ) \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+b^2 n^2 \left (-2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )-2 r \log (x) \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )+r^2 \log ^2(x) \log \left (\frac{d x^{-r}}{e}+1\right )\right )+a^2 r^2 \log \left (d-d x^r\right )-2 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b^2 r^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 \log \left (d-d x^r\right )}{d r^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.236, size = 3012, normalized size = 32. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*c*x^n)^6-1/4/r/d*ln(x^r)*Pi^2*b^2*csgn(I*c*x^n)^6+b^2/r/d*ln(x^r)*ln(x^n)^
2+I/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+I/r/d*ln(d+e*x^r)*Pi*a*b*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)+I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+I/r/d*ln(d+e*x^r)
*n*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(d+e*x^r)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I/r/d*l
n(d+e*x^r)*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I/r/d*ln(x^r)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)-I/r/d*ln(x^r)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I/r/d*ln(x^r)*ln(x^n)*b^2*Pi*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)-I/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(x^r)*Pi*a*b*csg
n(I*c*x^n)^2*csgn(I*c)+I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^3+I/r/d*ln(x^r)*ln(x^n)*b^2*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)^2-I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*
Pi*csgn(I*c*x^n)^2*csgn(I*c)-I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r/d*ln(d+e*x^r)*ln(x^n
)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r/d*ln(d+e*x^r)*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-I/r/d*ln(d+e*x^r)*Pi*a
*b*csgn(I*x^n)*csgn(I*c*x^n)^2+2/r/d*ln(d+e*x^r)*n*ln(x)*b^2*ln(c)-I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^
2*csgn(I*c)+b^2/d*ln(x)^2*ln(x^n)*n-b^2/r/d*ln(d+e*x^r)*ln(x)^2*n^2+b^2/r/d*ln(x^r)*ln(x)^2*n^2+b^2/r*n^2/d*ln
(x)^2*ln(1+e*x^r/d)-2*b^2/r^2*n/d*polylog(2,-e*x^r/d)*ln(x^n)+2/r/d*ln(x^r)*ln(c)*a*b-2*b/r/d*ln(d+e*x^r)*ln(x
^n)*a+2*b^2/r^3*n^2/d*polylog(3,-e*x^r/d)-1/r/d*ln(d+e*x^r)*ln(c)^2*b^2+1/r/d*ln(x^r)*ln(c)^2*b^2+1/4/r/d*ln(d
+e*x^r)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+1/2/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^
3*csgn(I*c)^2+I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I*c*x^n)^3-1/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c
*x^n)^4*csgn(I*c)-I/r/d*ln(x^r)*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^3+I/r/d*ln(d+e*x^r)*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^
3+I/r/d*ln(d+e*x^r)*Pi*a*b*csgn(I*c*x^n)^3-2*b^2/r*n/d*ln(x)*ln(1+e*x^r/d)*ln(x^n)-1/4/r/d*ln(x^r)*Pi^2*b^2*cs
gn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/2/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+I
/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+1/2/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)
-I/r/d*ln(x^r)*Pi*a*b*csgn(I*c*x^n)^3-1/2/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-2/r
*n/d*ln(x)*ln(1+e*x^r/d)*b^2*ln(c)-2/r/d*ln(x^r)*n*ln(x)*b^2*ln(c)-I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I
*c*x^n)^2*csgn(I*c)-I/r/d*ln(d+e*x^r)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^3-I/r/d*ln(d+e*x^r)*ln(x^n)*b^2*Pi*csgn(I*c
*x^n)^2*csgn(I*c)-1/2*I*n/d*ln(x)^2*b^2*Pi*csgn(I*c*x^n)^3-1/r/d*ln(d+e*x^r)*a^2+1/r/d*ln(x^r)*a^2-2/r^2*n/d*p
olylog(2,-e*x^r/d)*b^2*ln(c)-2/r/d*ln(d+e*x^r)*ln(c)*a*b-2/r/d*ln(d+e*x^r)*ln(x^n)*b^2*ln(c)+2/r/d*ln(x^r)*ln(
x^n)*b^2*ln(c)-I/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+I/r/d*ln(x^r)*Pi*a*b*csgn(I*x^n)*csgn(
I*c*x^n)^2-b^2/r/d*ln(d+e*x^r)*ln(x^n)^2+1/4/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2+2*b/r/d*ln(d
+e*x^r)*n*ln(x)*a-2*b/r/d*ln(x^r)*n*ln(x)*a-2*b/r*n/d*ln(x)*ln(1+e*x^r/d)*a+1/2*I*n/d*ln(x)^2*b^2*Pi*csgn(I*x^
n)*csgn(I*c*x^n)^2+1/2*I*n/d*ln(x)^2*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b/r/d*ln(x^r)*ln(x^n)*a-2*b/r^2*n/d*po
lylog(2,-e*x^r/d)*a+1/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-I/r/d*ln(x^r)*ln(c)*Pi*b^
2*csgn(I*c*x^n)^3+1/4/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-1/4/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x
^n)^2*csgn(I*c*x^n)^4+1/2/r/d*ln(x^r)*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)-1/2/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*c
*x^n)^5*csgn(I*c)-1/4/r/d*ln(x^r)*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2+1/2/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x^n)*cs
gn(I*c*x^n)^5-1/2/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+2*b^2/r/d*ln(d+e*x^r)*ln(x)*ln(x^n)*n-2
*b^2/r/d*ln(x^r)*ln(x)*ln(x^n)*n-I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(x^r
)*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*c*x^n)^3+I/r/d*ln(x^r)*ln
(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(x^r)*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)-1/2*I*n/d*ln(x)^2*
b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2/3*b^2*n^2/d*ln(x)^3+n/d*ln(x)^2*b^2*ln(c)+b*n/d*ln(x)^2*a+I/r*n/d
*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I/r/d*ln(d+e*x^r)*n*ln(x)*b^2*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)+I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)

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

[Out]

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

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Fricas [C]  time = 1.57036, size = 544, normalized size = 5.79 \begin{align*} \frac{b^{2} n^{2} r^{3} \log \left (x\right )^{3} + 6 \, b^{2} n^{2}{\rm polylog}\left (3, -\frac{e x^{r}}{d}\right ) + 3 \,{\left (b^{2} n r^{3} \log \left (c\right ) + a b n r^{3}\right )} \log \left (x\right )^{2} - 6 \,{\left (b^{2} n^{2} r \log \left (x\right ) + b^{2} n r \log \left (c\right ) + a b n r\right )}{\rm Li}_2\left (-\frac{e x^{r} + d}{d} + 1\right ) - 3 \,{\left (b^{2} r^{2} \log \left (c\right )^{2} + 2 \, a b r^{2} \log \left (c\right ) + a^{2} r^{2}\right )} \log \left (e x^{r} + d\right ) + 3 \,{\left (b^{2} r^{3} \log \left (c\right )^{2} + 2 \, a b r^{3} \log \left (c\right ) + a^{2} r^{3}\right )} \log \left (x\right ) - 3 \,{\left (b^{2} n^{2} r^{2} \log \left (x\right )^{2} + 2 \,{\left (b^{2} n r^{2} \log \left (c\right ) + a b n r^{2}\right )} \log \left (x\right )\right )} \log \left (\frac{e x^{r} + d}{d}\right )}{3 \, d r^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*n^2*r^3*log(x)^3 + 6*b^2*n^2*polylog(3, -e*x^r/d) + 3*(b^2*n*r^3*log(c) + a*b*n*r^3)*log(x)^2 - 6*(b^
2*n^2*r*log(x) + b^2*n*r*log(c) + a*b*n*r)*dilog(-(e*x^r + d)/d + 1) - 3*(b^2*r^2*log(c)^2 + 2*a*b*r^2*log(c)
+ a^2*r^2)*log(e*x^r + d) + 3*(b^2*r^3*log(c)^2 + 2*a*b*r^3*log(c) + a^2*r^3)*log(x) - 3*(b^2*n^2*r^2*log(x)^2
 + 2*(b^2*n*r^2*log(c) + a*b*n*r^2)*log(x))*log((e*x^r + d)/d))/(d*r^3)

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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))**2/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{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\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))^2/x/(d+e*x^r),x, algorithm="giac")

[Out]

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

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