\(\int \frac {(a+b \log (c x^n))^3}{x (d+e x)} \, dx\) [121]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d} \]

[Out]

-ln(1+d/e/x)*(a+b*ln(c*x^n))^3/d+3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-d/e/x)/d+6*b^2*n^2*(a+b*ln(c*x^n))*polylog
(3,-d/e/x)/d+6*b^3*n^3*polylog(4,-d/e/x)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2379, 2421, 2430, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\frac {6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d} \]

[In]

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

[Out]

-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))])/d + (6*b^2*
n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])/d + (6*b^3*n^3*PolyLog[4, -(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 2430

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

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 )^3}{d}+\frac {(3 b n) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d} \\ & = -\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d}-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \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 )^3}{d}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {d}{e x}\right )}{d}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\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 )^3}{d}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {d}{e x}\right )}{d}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {d}{e x}\right )}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(113)=226\).

Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\frac {4 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )-4 b^2 n^2 \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \left (\log (x)-3 \log \left (1+\frac {e x}{d}\right )\right )-6 \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )+b^3 n^3 \left (\log ^4(x)-4 \log ^3(x) \log \left (1+\frac {e x}{d}\right )-12 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+24 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-24 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )}{4 d} \]

[In]

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

[Out]

(4*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n])^3 - 4*(a - b*n*Log[x] + b*Log[c*x^n])^3*Log[d + e*x] + 6*b*n*(a - b*
n*Log[x] + b*Log[c*x^n])^2*(Log[x]^2 - 2*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)])) - 4*b^2*n^2*(-a +
b*n*Log[x] - b*Log[c*x^n])*(Log[x]^2*(Log[x] - 3*Log[1 + (e*x)/d]) - 6*Log[x]*PolyLog[2, -((e*x)/d)] + 6*Poly
Log[3, -((e*x)/d)]) + b^3*n^3*(Log[x]^4 - 4*Log[x]^3*Log[1 + (e*x)/d] - 12*Log[x]^2*PolyLog[2, -((e*x)/d)] + 2
4*Log[x]*PolyLog[3, -((e*x)/d)] - 24*PolyLog[4, -((e*x)/d)]))/(4*d)

Maple [C] (warning: unable to verify)

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

Time = 0.55 (sec) , antiderivative size = 967, normalized size of antiderivative = 8.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(967\)

[In]

int((a+b*ln(c*x^n))^3/x/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

-b^3*ln(x^n)^3/d*ln(e*x+d)+b^3*ln(x^n)^3/d*ln(x)-3/2*b^3*n/d*ln(x^n)^2*ln(x)^2+b^3/d*n^2*ln(x^n)*ln(x)^3-1/4*b
^3/d*ln(x)^4*n^3+3*b^3/d*ln(x)^2*ln(e*x+d)*ln(-e*x/d)*n^3+3*b^3/d*ln(x)^2*dilog(-e*x/d)*n^3-6*b^3/d*ln(x)*ln(x
^n)*ln(e*x+d)*ln(-e*x/d)*n^2-6*b^3/d*ln(x)*ln(x^n)*dilog(-e*x/d)*n^2+3*b^3*n/d*ln(x^n)^2*ln(e*x+d)*ln(-e*x/d)+
3*b^3*n/d*ln(x^n)^2*dilog(-e*x/d)-2*b^3/d*n^3*ln(e*x+d)*ln(x)^3+2*b^3/d*n^3*ln(x)^3*ln(1+e*x/d)+3*b^3/d*n^3*ln
(x)^2*polylog(2,-e*x/d)-6*b^3/d*n^3*polylog(4,-e*x/d)+3*b^3/d*n^2*ln(x)^2*ln(x^n)*ln(e*x+d)-3*b^3/d*n^2*ln(x)^
2*ln(x^n)*ln(1+e*x/d)-6*b^3/d*n^2*ln(x)*ln(x^n)*polylog(2,-e*x/d)+6*b^3/d*n^2*ln(x^n)*polylog(3,-e*x/d)+1/8*(-
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)^3*(-1/d*ln(e*x+d)+1/d*ln(x))+3/2*(-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^2*(-ln(x^n)^2/d*ln(e*x+d)+ln(x^n)^2/d*ln(x)-2*n*(1/2/d*ln(x^n)*ln(x)^2-1/6/d*ln(x)^3*n-1/d*((ln(x^n)-n*ln
(x))*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))+n*(1/2*ln(e*x+d)*ln(x)^2-1/2*ln(x)^2*ln(1+e*x/d)-ln(x)*polylog(2,-e*
x/d)+polylog(3,-e*x/d)))))+3/4*(-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*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)))

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )} x} \,d x } \]

[In]

integrate((a+b*log(c*x^n))^3/x/(e*x+d),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )}\, dx \]

[In]

integrate((a+b*ln(c*x**n))**3/x/(e*x+d),x)

[Out]

Integral((a + b*log(c*x**n))**3/(x*(d + e*x)), x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )} x} \,d x } \]

[In]

integrate((a+b*log(c*x^n))^3/x/(e*x+d),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )} x} \,d x } \]

[In]

integrate((a+b*log(c*x^n))^3/x/(e*x+d),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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