3.139 \(\int \frac{(a+b \log (c x^n))^3 \log (d (e+f x^m)^r)}{x} \, dx\)

Optimal. Leaf size=185 \[ -\frac{6 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^3}+\frac{3 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}+\frac{6 b^3 n^3 r \text{PolyLog}\left (5,-\frac{f x^m}{e}\right )}{m^4}+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[Out]

((a + b*Log[c*x^n])^4*Log[d*(e + f*x^m)^r])/(4*b*n) - (r*(a + b*Log[c*x^n])^4*Log[1 + (f*x^m)/e])/(4*b*n) - (r
*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^m)/e)])/m + (3*b*n*r*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^m)/e)])/m^
2 - (6*b^2*n^2*r*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^m)/e)])/m^3 + (6*b^3*n^3*r*PolyLog[5, -((f*x^m)/e)])/m^4

________________________________________________________________________________________

Rubi [A]  time = 0.298733, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ -\frac{6 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^3}+\frac{3 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}+\frac{6 b^3 n^3 r \text{PolyLog}\left (5,-\frac{f x^m}{e}\right )}{m^4}+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])^4*Log[d*(e + f*x^m)^r])/(4*b*n) - (r*(a + b*Log[c*x^n])^4*Log[1 + (f*x^m)/e])/(4*b*n) - (r
*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^m)/e)])/m + (3*b*n*r*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^m)/e)])/m^
2 - (6*b^2*n^2*r*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^m)/e)])/m^3 + (6*b^3*n^3*r*PolyLog[5, -((f*x^m)/e)])/m^4

Rule 2375

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[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 6589

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*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{(f m r) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^4}{e+f x^m} \, dx}{4 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^m}{e}\right )}{4 b n}+r \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^m}{e}\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{(3 b n r) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^m}{e}\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{3 b n r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{\left (6 b^2 n^2 r\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m^2}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^m}{e}\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{3 b n r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{6 b^2 n^2 r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x^m}{e}\right )}{m^3}+\frac{\left (6 b^3 n^3 r\right ) \int \frac{\text{Li}_4\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m^3}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^m}{e}\right )}{4 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{3 b n r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{6 b^2 n^2 r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x^m}{e}\right )}{m^3}+\frac{6 b^3 n^3 r \text{Li}_5\left (-\frac{f x^m}{e}\right )}{m^4}\\ \end{align*}

Mathematica [B]  time = 0.618469, size = 1395, normalized size = 7.54 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*b*m*n*r*Log[x]^3)/2 + (3*a*b^2*m*n^2*r*Log[x]^4)/4 - (3*b^3*m*n^3*r*Log[x]^5)/10 - a*b^2*m*n*r*Log[x]^3*
Log[c*x^n] + (3*b^3*m*n^2*r*Log[x]^4*Log[c*x^n])/4 - (b^3*m*n*r*Log[x]^3*Log[c*x^n]^2)/2 - (3*a^2*b*n*r*Log[x]
^2*Log[1 + e/(f*x^m)])/2 + 2*a*b^2*n^2*r*Log[x]^3*Log[1 + e/(f*x^m)] - (3*b^3*n^3*r*Log[x]^4*Log[1 + e/(f*x^m)
])/4 - 3*a*b^2*n*r*Log[x]^2*Log[c*x^n]*Log[1 + e/(f*x^m)] + 2*b^3*n^2*r*Log[x]^3*Log[c*x^n]*Log[1 + e/(f*x^m)]
- (3*b^3*n*r*Log[x]^2*Log[c*x^n]^2*Log[1 + e/(f*x^m)])/2 - a^3*r*Log[x]*Log[e + f*x^m] + 3*a^2*b*n*r*Log[x]^2
*Log[e + f*x^m] - 3*a*b^2*n^2*r*Log[x]^3*Log[e + f*x^m] + b^3*n^3*r*Log[x]^4*Log[e + f*x^m] + (a^3*r*Log[-((f*
x^m)/e)]*Log[e + f*x^m])/m - (3*a^2*b*n*r*Log[x]*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m + (3*a*b^2*n^2*r*Log[x]^2
*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - (b^3*n^3*r*Log[x]^3*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - 3*a^2*b*r*Log
[x]*Log[c*x^n]*Log[e + f*x^m] + 6*a*b^2*n*r*Log[x]^2*Log[c*x^n]*Log[e + f*x^m] - 3*b^3*n^2*r*Log[x]^3*Log[c*x^
n]*Log[e + f*x^m] + (3*a^2*b*r*Log[-((f*x^m)/e)]*Log[c*x^n]*Log[e + f*x^m])/m - (6*a*b^2*n*r*Log[x]*Log[-((f*x
^m)/e)]*Log[c*x^n]*Log[e + f*x^m])/m + (3*b^3*n^2*r*Log[x]^2*Log[-((f*x^m)/e)]*Log[c*x^n]*Log[e + f*x^m])/m -
3*a*b^2*r*Log[x]*Log[c*x^n]^2*Log[e + f*x^m] + 3*b^3*n*r*Log[x]^2*Log[c*x^n]^2*Log[e + f*x^m] + (3*a*b^2*r*Log
[-((f*x^m)/e)]*Log[c*x^n]^2*Log[e + f*x^m])/m - (3*b^3*n*r*Log[x]*Log[-((f*x^m)/e)]*Log[c*x^n]^2*Log[e + f*x^m
])/m - b^3*r*Log[x]*Log[c*x^n]^3*Log[e + f*x^m] + (b^3*r*Log[-((f*x^m)/e)]*Log[c*x^n]^3*Log[e + f*x^m])/m + a^
3*Log[x]*Log[d*(e + f*x^m)^r] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f*x^m)^r])/2 + a*b^2*n^2*Log[x]^3*Log[d*(e + f*
x^m)^r] - (b^3*n^3*Log[x]^4*Log[d*(e + f*x^m)^r])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^m)^r] - 3*a*b^2
*n*Log[x]^2*Log[c*x^n]*Log[d*(e + f*x^m)^r] + b^3*n^2*Log[x]^3*Log[c*x^n]*Log[d*(e + f*x^m)^r] + 3*a*b^2*Log[x
]*Log[c*x^n]^2*Log[d*(e + f*x^m)^r] - (3*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[d*(e + f*x^m)^r])/2 + b^3*Log[x]*Log[
c*x^n]^3*Log[d*(e + f*x^m)^r] + (b*n*r*Log[x]*(b^2*n^2*Log[x]^2 - 3*b*n*Log[x]*(a + b*Log[c*x^n]) + 3*(a + b*L
og[c*x^n])^2)*PolyLog[2, -(e/(f*x^m))])/m + (r*(a - b*n*Log[x] + b*Log[c*x^n])^3*PolyLog[2, 1 + (f*x^m)/e])/m
+ (3*a^2*b*n*r*PolyLog[3, -(e/(f*x^m))])/m^2 + (6*a*b^2*n*r*Log[c*x^n]*PolyLog[3, -(e/(f*x^m))])/m^2 + (3*b^3*
n*r*Log[c*x^n]^2*PolyLog[3, -(e/(f*x^m))])/m^2 + (6*a*b^2*n^2*r*PolyLog[4, -(e/(f*x^m))])/m^3 + (6*b^3*n^2*r*L
og[c*x^n]*PolyLog[4, -(e/(f*x^m))])/m^3 + (6*b^3*n^3*r*PolyLog[5, -(e/(f*x^m))])/m^4

________________________________________________________________________________________

Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( e+f{x}^{m} \right ) ^{r} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(e+f*x^m)^r)/x,x)

[Out]

int((a+b*ln(c*x^n))^3*ln(d*(e+f*x^m)^r)/x,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2
+ 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b
^3*n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x))*log(
x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)*log(x))*log((f*x^m + e)^r) - integrate(-1/4*
(4*b^3*e*log(c)^3*log(d) + 12*a*b^2*e*log(c)^2*log(d) + 12*a^2*b*e*log(c)*log(d) + 4*a^3*e*log(d) + 4*(b^3*e*l
og(d) - (b^3*f*m*r*log(x) - b^3*f*log(d))*x^m)*log(x^n)^3 + 6*(2*b^3*e*log(c)*log(d) + 2*a*b^2*e*log(d) + (b^3
*f*m*n*r*log(x)^2 + 2*b^3*f*log(c)*log(d) + 2*a*b^2*f*log(d) - 2*(b^3*f*m*r*log(c) + a*b^2*f*m*r)*log(x))*x^m)
*log(x^n)^2 + (b^3*f*m*n^3*r*log(x)^4 + 4*b^3*f*log(c)^3*log(d) + 12*a*b^2*f*log(c)^2*log(d) + 12*a^2*b*f*log(
c)*log(d) + 4*a^3*f*log(d) - 4*(b^3*f*m*n^2*r*log(c) + a*b^2*f*m*n^2*r)*log(x)^3 + 6*(b^3*f*m*n*r*log(c)^2 + 2
*a*b^2*f*m*n*r*log(c) + a^2*b*f*m*n*r)*log(x)^2 - 4*(b^3*f*m*r*log(c)^3 + 3*a*b^2*f*m*r*log(c)^2 + 3*a^2*b*f*m
*r*log(c) + a^3*f*m*r)*log(x))*x^m + 4*(3*b^3*e*log(c)^2*log(d) + 6*a*b^2*e*log(c)*log(d) + 3*a^2*b*e*log(d) -
(b^3*f*m*n^2*r*log(x)^3 - 3*b^3*f*log(c)^2*log(d) - 6*a*b^2*f*log(c)*log(d) - 3*a^2*b*f*log(d) - 3*(b^3*f*m*n
*r*log(c) + a*b^2*f*m*n*r)*log(x)^2 + 3*(b^3*f*m*r*log(c)^2 + 2*a*b^2*f*m*r*log(c) + a^2*b*f*m*r)*log(x))*x^m)
*log(x^n))/(f*x*x^m + e*x), x)

________________________________________________________________________________________

Fricas [C]  time = 1.16021, size = 1824, normalized size = 9.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^3*m^4*n^3*log(d)*log(x)^4 + 24*b^3*n^3*r*polylog(5, -f*x^m/e) + 4*(b^3*m^4*n^2*log(c) + a*b^2*m^4*n^2)*
log(d)*log(x)^3 + 6*(b^3*m^4*n*log(c)^2 + 2*a*b^2*m^4*n*log(c) + a^2*b*m^4*n)*log(d)*log(x)^2 + 4*(b^3*m^4*log
(c)^3 + 3*a*b^2*m^4*log(c)^2 + 3*a^2*b*m^4*log(c) + a^3*m^4)*log(d)*log(x) - 4*(b^3*m^3*n^3*r*log(x)^3 + b^3*m
^3*r*log(c)^3 + 3*a*b^2*m^3*r*log(c)^2 + 3*a^2*b*m^3*r*log(c) + a^3*m^3*r + 3*(b^3*m^3*n^2*r*log(c) + a*b^2*m^
3*n^2*r)*log(x)^2 + 3*(b^3*m^3*n*r*log(c)^2 + 2*a*b^2*m^3*n*r*log(c) + a^2*b*m^3*n*r)*log(x))*dilog(-(f*x^m +
e)/e + 1) + (b^3*m^4*n^3*r*log(x)^4 + 4*(b^3*m^4*n^2*r*log(c) + a*b^2*m^4*n^2*r)*log(x)^3 + 6*(b^3*m^4*n*r*log
(c)^2 + 2*a*b^2*m^4*n*r*log(c) + a^2*b*m^4*n*r)*log(x)^2 + 4*(b^3*m^4*r*log(c)^3 + 3*a*b^2*m^4*r*log(c)^2 + 3*
a^2*b*m^4*r*log(c) + a^3*m^4*r)*log(x))*log(f*x^m + e) - (b^3*m^4*n^3*r*log(x)^4 + 4*(b^3*m^4*n^2*r*log(c) + a
*b^2*m^4*n^2*r)*log(x)^3 + 6*(b^3*m^4*n*r*log(c)^2 + 2*a*b^2*m^4*n*r*log(c) + a^2*b*m^4*n*r)*log(x)^2 + 4*(b^3
*m^4*r*log(c)^3 + 3*a*b^2*m^4*r*log(c)^2 + 3*a^2*b*m^4*r*log(c) + a^3*m^4*r)*log(x))*log((f*x^m + e)/e) - 24*(
b^3*m*n^3*r*log(x) + b^3*m*n^2*r*log(c) + a*b^2*m*n^2*r)*polylog(4, -f*x^m/e) + 12*(b^3*m^2*n^3*r*log(x)^2 + b
^3*m^2*n*r*log(c)^2 + 2*a*b^2*m^2*n*r*log(c) + a^2*b*m^2*n*r + 2*(b^3*m^2*n^2*r*log(c) + a*b^2*m^2*n^2*r)*log(
x))*polylog(3, -f*x^m/e))/m^4

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{m} + e\right )}^{r} d\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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