\(\int \frac {(a+b \log (c x^n))^2}{x (d+e x^r)} \, dx\) [430]

Optimal result

Integrand size = 25, antiderivative size = 94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d r^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2379, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d r}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d r^3} \]

[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 2379

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 6724

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*} \text {integral}& = -\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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(94)=188\).

Time = 0.19 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=-\frac {a^2 r^2 \log \left (d-d x^r\right )-2 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b^2 r^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2 \log \left (d-d x^r\right )-2 a b n r \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+2 b^2 n r \left (n \log (x)-\log \left (c x^n\right )\right ) \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+b^2 n^2 \left (r^2 \log ^2(x) \log \left (1+\frac {d x^{-r}}{e}\right )-2 r \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )-2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )\right )}{d r^3} \]

[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))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.80 (sec) , antiderivative size = 580, normalized size of antiderivative = 6.17

method	result	size
risch	\(-\frac {b^{2} \ln \left (d +e \,x^{r}\right ) \ln \left (x \right )^{2} n^{2}}{r d}+\frac {2 b^{2} \ln \left (d +e \,x^{r}\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n}{r d}-\frac {b^{2} \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )^{2}}{r d}+\frac {b^{2} \ln \left (x^{r}\right ) \ln \left (x \right )^{2} n^{2}}{r d}-\frac {2 b^{2} \ln \left (x^{r}\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n}{r d}+\frac {b^{2} \ln \left (x^{r}\right ) \ln \left (x^{n}\right )^{2}}{r d}-\frac {2 b^{2} \ln \left (x \right )^{3} n^{2}}{3 d}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e \,x^{r}}{d}\right )}{r d}+\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e \,x^{r}}{d}\right )}{r^{3} d}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d}-\frac {2 b^{2} n \ln \left (x \right ) \ln \left (1+\frac {e \,x^{r}}{d}\right ) \ln \left (x^{n}\right )}{r d}-\frac {2 b^{2} n \,\operatorname {Li}_{2}\left (-\frac {e \,x^{r}}{d}\right ) \ln \left (x^{n}\right )}{r^{2} d}+\frac {\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \left (-\frac {\ln \left (d +e \,x^{r}\right )}{d}+\frac {\ln \left (x^{r}\right )}{d}\right )-\frac {n \left (-\frac {r^{2} \ln \left (x \right )^{2}}{2}+r \ln \left (x \right ) \ln \left (1+\frac {e \,x^{r}}{d}\right )+\operatorname {Li}_{2}\left (-\frac {e \,x^{r}}{d}\right )\right )}{r d}\right )}{r}+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (-\frac {\ln \left (d +e \,x^{r}\right )}{r d}+\frac {\ln \left (x^{r}\right )}{r d}\right )}{4}\)	\(580\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (93) = 186\).

Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\frac {b^{2} n^{2} r^{3} \log \left (x\right )^{3} + 6 \, b^{2} n^{2} {\rm polylog}\left (3, -\frac {e x^{r}}{d}\right ) + 3 \, {\left (b^{2} n r^{3} \log \left (c\right ) + a b n r^{3}\right )} \log \left (x\right )^{2} - 6 \, {\left (b^{2} n^{2} r \log \left (x\right ) + b^{2} n r \log \left (c\right ) + a b n r\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 3 \, {\left (b^{2} r^{2} \log \left (c\right )^{2} + 2 \, a b r^{2} \log \left (c\right ) + a^{2} r^{2}\right )} \log \left (e x^{r} + d\right ) + 3 \, {\left (b^{2} r^{3} \log \left (c\right )^{2} + 2 \, a b r^{3} \log \left (c\right ) + a^{2} r^{3}\right )} \log \left (x\right ) - 3 \, {\left (b^{2} n^{2} r^{2} \log \left (x\right )^{2} + 2 \, {\left (b^{2} n r^{2} \log \left (c\right ) + a b n r^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {e x^{r} + d}{d}\right )}{3 \, d r^{3}} \]

[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)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x \left (d + e x^{r}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{r} + d\right )} x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{r} + d\right )} x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,\left (d+e\,x^r\right )} \,d x \]

[In]

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

[Out]

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