\(\int \frac {(a+b \log (c x^n))^2}{x (d+e x)} \, dx\) [96]

Optimal result

Integrand size = 23, antiderivative size = 79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx=-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d} \]

[Out]

-ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d+2*b*n*(a+b*ln(c*x^n))*polylog(2,-d/e/x)/d+2*b^2*n^2*polylog(3,-d/e/x)/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2379, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b d n}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {d+e x}{d}\right )}{d}-\frac {2 b n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{d} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.42 (sec) , antiderivative size = 528, normalized size of antiderivative = 6.68

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{d}+\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (x \right )}{d}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d}+\frac {b^{2} \ln \left (x \right )^{3} n^{2}}{3 d}-\frac {2 b^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d}-\frac {2 b^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d}+\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d}+\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 (-\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (x^{n}\right ) \ln \left (x \right )}{d}-n \left (\frac {\ln \left (x \right )^{2}}{2 d}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d}\right )\right )+\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 (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )}{4}\)	\(528\)

[In]

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

[Out]

-b^2*ln(x^n)^2/d*ln(e*x+d)+b^2*ln(x^n)^2/d*ln(x)-b^2*n/d*ln(x^n)*ln(x)^2+1/3*b^2/d*ln(x)^3*n^2-2*b^2/d*ln(x)*l
n(e*x+d)*ln(-e*x/d)*n^2-2*b^2/d*ln(x)*dilog(-e*x/d)*n^2+2*b^2*n/d*ln(x^n)*ln(e*x+d)*ln(-e*x/d)+2*b^2*n/d*ln(x^
n)*dilog(-e*x/d)+b^2/d*n^2*ln(e*x+d)*ln(x)^2-b^2/d*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2/d*n^2*ln(x)*polylog(2,-e*x/d)
+2*b^2/d*n^2*polylog(3,-e*x/d)+(-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)*b*(-ln(x^n)/d*ln(e*x+d)+ln(x^n)/d*ln(x
)-n*(1/2/d*ln(x)^2-1/d*ln(e*x+d)*ln(-e*x/d)-1/d*dilog(-e*x/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/d*ln(e*x+d)+1/d*ln(x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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