\(\int \frac {(a+b \log (c x^n))^2}{x (d+e x)^2} \, dx\) [104]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2} \]

[In]

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

[Out]

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

Rule 2354

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

Rule 2355

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

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 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 2438

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\frac {-3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{b n}+6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 d^2} \]

[In]

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

[Out]

(-3*(a + b*Log[c*x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/(d + e*x) + (a + b*Log[c*x^n])^3/(b*n) + 6*b*n*(a + b*Lo
g[c*x^n])*Log[1 + (e*x)/d] - 3*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 6*b^2*n^2*PolyLog[2, -((e*x)/d)] - 6*b*
n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 6*b^2*n^2*PolyLog[3, -((e*x)/d)])/(3*d^2)

Maple [C] (warning: unable to verify)

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

Time = 0.43 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.52

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{d^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2}}{d \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (x \right )}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{d^{2}}-\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{2}}+\frac {b^{2} \ln \left (x \right )^{3} n^{2}}{3 d^{2}}-\frac {2 b^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}-\frac {2 b^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{2}}+\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{2}}+\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^{2}}+\frac {\ln \left (x^{n}\right )}{d \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}-n \left (\frac {\ln \left (x \right )^{2}}{2 d^{2}}-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {\ln \left (x \right )}{d^{2}}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}\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^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )}{4}\)	\(683\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^3 + 2*d*e*x^2 + d^2*x), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

a^2*(1/(d*e*x + d^2) - log(e*x + d)/d^2 + log(x)/d^2) + 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^2*x^3 + 2*d*e*x^2 + d^2*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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