\(\int \frac {(a+b \log (c x^n))^3}{x (d+e x)^2} \, dx\) [122]

Optimal result

Integrand size = 23, antiderivative size = 217 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2389, 2379, 2421, 2430, 6724, 2355, 2354} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\frac {6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}+\frac {3 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d^2} \]

[In]

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

[Out]

-((e*x*(a + b*Log[c*x^n])^3)/(d^2*(d + e*x))) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d^2 + (3*b*n*(a + b*Lo
g[c*x^n])^2*Log[1 + (e*x)/d])/d^2 + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))])/d^2 + (6*b^2*n^2*(a +
b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^2 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])/d^2 - (6*b^3
*n^3*PolyLog[3, -((e*x)/d)])/d^2 + (6*b^3*n^3*PolyLog[4, -(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 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 {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{d} \\ & = -\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\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^2}+\frac {(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^2} \\ & = -\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^2}+\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^2}-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2}-\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^2} \\ & = -\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^2}+\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^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2}+\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^2}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^2}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {d}{e x}\right )}{x} \, dx}{d^2} \\ & = -\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^2}+\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^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2}+\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^2}-\frac {6 b^3 n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {d}{e x}\right )}{d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\frac {4 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 (d+e x) \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 (d+e x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (-2 e x \log (x)+(d+e x) \log ^2(x)+2 (d+e x) \log (d+e x)-2 (d+e x) \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )+4 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log (x) \left ((d+e x) \log ^2(x)+6 (d+e x) \log \left (1+\frac {e x}{d}\right )-3 \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )\right )-6 (d+e x) (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )+b^3 n^3 \left ((d+e x) \log ^4(x)-4 \left (\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-6 (d+e x) \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-4 (d+e x) \left (\log ^3(x) \log \left (1+\frac {e x}{d}\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )\right )}{4 d^2 (d+e x)} \]

[In]

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

[Out]

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

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 = 1373, normalized size of antiderivative = 6.33

method	result	size
risch	\(\text {Expression too large to display}\)	\(1373\)

[In]

int((a+b*ln(c*x^n))^3/x/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]

[In]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]

[In]

integrate((a+b*log(c*x^n))^3/x/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]

[In]

integrate((a+b*log(c*x^n))^3/x/(e*x+d)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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