3.1.64 \(\int \frac {(a+b \log (c x^n))^4 \log (d (\frac {1}{d}+f x^m))}{x} \, dx\) [64]

Optimal. Leaf size=137 \[ -\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_4\left (-d f x^m\right )}{m^3}+\frac {24 b^3 n^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_5\left (-d f x^m\right )}{m^4}-\frac {24 b^4 n^4 \text {Li}_6\left (-d f x^m\right )}{m^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2421, 2430, 6724} \begin {gather*} \frac {24 b^3 n^3 \text {PolyLog}\left (5,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m^4}-\frac {12 b^2 n^2 \text {PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m^3}+\frac {4 b n \text {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m^2}-\frac {\text {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^4}{m}-\frac {24 b^4 n^4 \text {PolyLog}\left (6,-d f x^m\right )}{m^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^4*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

-(((a + b*Log[c*x^n])^4*PolyLog[2, -(d*f*x^m)])/m) + (4*b*n*(a + b*Log[c*x^n])^3*PolyLog[3, -(d*f*x^m)])/m^2 -
(12*b^2*n^2*(a + b*Log[c*x^n])^2*PolyLog[4, -(d*f*x^m)])/m^3 + (24*b^3*n^3*(a + b*Log[c*x^n])*PolyLog[5, -(d*
f*x^m)])/m^4 - (24*b^4*n^4*PolyLog[6, -(d*f*x^m)])/m^5

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*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {(4 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-d f x^m\right )}{x} \, dx}{m}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {\left (12 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-d f x^m\right )}{x} \, dx}{m^2}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_4\left (-d f x^m\right )}{m^3}+\frac {\left (24 b^3 n^3\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-d f x^m\right )}{x} \, dx}{m^3}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_4\left (-d f x^m\right )}{m^3}+\frac {24 b^3 n^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_5\left (-d f x^m\right )}{m^4}-\frac {\left (24 b^4 n^4\right ) \int \frac {\text {Li}_5\left (-d f x^m\right )}{x} \, dx}{m^4}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_4\left (-d f x^m\right )}{m^3}+\frac {24 b^3 n^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_5\left (-d f x^m\right )}{m^4}-\frac {24 b^4 n^4 \text {Li}_6\left (-d f x^m\right )}{m^5}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1700\) vs. \(2(137)=274\).
time = 0.44, size = 1700, normalized size = 12.41 \begin {gather*} -\frac {2}{3} a^3 b m n \log ^3(x)+\frac {3}{2} a^2 b^2 m n^2 \log ^4(x)-\frac {6}{5} a b^3 m n^3 \log ^5(x)+\frac {1}{3} b^4 m n^4 \log ^6(x)-2 a^2 b^2 m n \log ^3(x) \log \left (c x^n\right )+3 a b^3 m n^2 \log ^4(x) \log \left (c x^n\right )-\frac {6}{5} b^4 m n^3 \log ^5(x) \log \left (c x^n\right )-2 a b^3 m n \log ^3(x) \log ^2\left (c x^n\right )+\frac {3}{2} b^4 m n^2 \log ^4(x) \log ^2\left (c x^n\right )-\frac {2}{3} b^4 m n \log ^3(x) \log ^3\left (c x^n\right )-2 a^3 b n \log ^2(x) \log \left (1+\frac {x^{-m}}{d f}\right )+4 a^2 b^2 n^2 \log ^3(x) \log \left (1+\frac {x^{-m}}{d f}\right )-3 a b^3 n^3 \log ^4(x) \log \left (1+\frac {x^{-m}}{d f}\right )+\frac {4}{5} b^4 n^4 \log ^5(x) \log \left (1+\frac {x^{-m}}{d f}\right )-6 a^2 b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+8 a b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )-3 b^4 n^3 \log ^4(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )-6 a b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+4 b^4 n^2 \log ^3(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )-2 b^4 n \log ^2(x) \log ^3\left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+2 a^3 b n \log ^2(x) \log \left (1+d f x^m\right )-4 a^2 b^2 n^2 \log ^3(x) \log \left (1+d f x^m\right )+3 a b^3 n^3 \log ^4(x) \log \left (1+d f x^m\right )-\frac {4}{5} b^4 n^4 \log ^5(x) \log \left (1+d f x^m\right )+\frac {a^4 \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {4 a^3 b n \log (x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+\frac {6 a^2 b^2 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {4 a b^3 n^3 \log ^3(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^4 n^4 \log ^4(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+6 a^2 b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )-8 a b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )+3 b^4 n^3 \log ^4(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {4 a^3 b \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {12 a^2 b^2 n \log (x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {12 a b^3 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {4 b^4 n^3 \log ^3(x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+6 a b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )-4 b^4 n^2 \log ^3(x) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {6 a^2 b^2 \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {12 a b^3 n \log (x) \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {6 b^4 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+2 b^4 n \log ^2(x) \log ^3\left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {4 a b^3 \log \left (-d f x^m\right ) \log ^3\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {4 b^4 n \log (x) \log \left (-d f x^m\right ) \log ^3\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^4 \log \left (-d f x^m\right ) \log ^4\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b n \log (x) \left (-b^3 n^3 \log ^3(x)+4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )-6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2+4 \left (a+b \log \left (c x^n\right )\right )^3\right ) \text {Li}_2\left (-\frac {x^{-m}}{d f}\right )}{m}+\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right )^4 \text {Li}_2\left (1+d f x^m\right )}{m}+\frac {4 a^3 b n \text {Li}_3\left (-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {12 a^2 b^2 n \log \left (c x^n\right ) \text {Li}_3\left (-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {12 a b^3 n \log ^2\left (c x^n\right ) \text {Li}_3\left (-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {4 b^4 n \log ^3\left (c x^n\right ) \text {Li}_3\left (-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {12 a^2 b^2 n^2 \text {Li}_4\left (-\frac {x^{-m}}{d f}\right )}{m^3}+\frac {24 a b^3 n^2 \log \left (c x^n\right ) \text {Li}_4\left (-\frac {x^{-m}}{d f}\right )}{m^3}+\frac {12 b^4 n^2 \log ^2\left (c x^n\right ) \text {Li}_4\left (-\frac {x^{-m}}{d f}\right )}{m^3}+\frac {24 a b^3 n^3 \text {Li}_5\left (-\frac {x^{-m}}{d f}\right )}{m^4}+\frac {24 b^4 n^3 \log \left (c x^n\right ) \text {Li}_5\left (-\frac {x^{-m}}{d f}\right )}{m^4}+\frac {24 b^4 n^4 \text {Li}_6\left (-\frac {x^{-m}}{d f}\right )}{m^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^4*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

(-2*a^3*b*m*n*Log[x]^3)/3 + (3*a^2*b^2*m*n^2*Log[x]^4)/2 - (6*a*b^3*m*n^3*Log[x]^5)/5 + (b^4*m*n^4*Log[x]^6)/3
- 2*a^2*b^2*m*n*Log[x]^3*Log[c*x^n] + 3*a*b^3*m*n^2*Log[x]^4*Log[c*x^n] - (6*b^4*m*n^3*Log[x]^5*Log[c*x^n])/5
- 2*a*b^3*m*n*Log[x]^3*Log[c*x^n]^2 + (3*b^4*m*n^2*Log[x]^4*Log[c*x^n]^2)/2 - (2*b^4*m*n*Log[x]^3*Log[c*x^n]^
3)/3 - 2*a^3*b*n*Log[x]^2*Log[1 + 1/(d*f*x^m)] + 4*a^2*b^2*n^2*Log[x]^3*Log[1 + 1/(d*f*x^m)] - 3*a*b^3*n^3*Log
[x]^4*Log[1 + 1/(d*f*x^m)] + (4*b^4*n^4*Log[x]^5*Log[1 + 1/(d*f*x^m)])/5 - 6*a^2*b^2*n*Log[x]^2*Log[c*x^n]*Log
[1 + 1/(d*f*x^m)] + 8*a*b^3*n^2*Log[x]^3*Log[c*x^n]*Log[1 + 1/(d*f*x^m)] - 3*b^4*n^3*Log[x]^4*Log[c*x^n]*Log[1
+ 1/(d*f*x^m)] - 6*a*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[1 + 1/(d*f*x^m)] + 4*b^4*n^2*Log[x]^3*Log[c*x^n]^2*Log[1
+ 1/(d*f*x^m)] - 2*b^4*n*Log[x]^2*Log[c*x^n]^3*Log[1 + 1/(d*f*x^m)] + 2*a^3*b*n*Log[x]^2*Log[1 + d*f*x^m] - 4
*a^2*b^2*n^2*Log[x]^3*Log[1 + d*f*x^m] + 3*a*b^3*n^3*Log[x]^4*Log[1 + d*f*x^m] - (4*b^4*n^4*Log[x]^5*Log[1 + d
*f*x^m])/5 + (a^4*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m - (4*a^3*b*n*Log[x]*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m
+ (6*a^2*b^2*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m - (4*a*b^3*n^3*Log[x]^3*Log[-(d*f*x^m)]*Log[1 +
d*f*x^m])/m + (b^4*n^4*Log[x]^4*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + 6*a^2*b^2*n*Log[x]^2*Log[c*x^n]*Log[1 +
d*f*x^m] - 8*a*b^3*n^2*Log[x]^3*Log[c*x^n]*Log[1 + d*f*x^m] + 3*b^4*n^3*Log[x]^4*Log[c*x^n]*Log[1 + d*f*x^m] +
(4*a^3*b*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m - (12*a^2*b^2*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]*Log
[1 + d*f*x^m])/m + (12*a*b^3*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m - (4*b^4*n^3*Log[x]^3
*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m + 6*a*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[1 + d*f*x^m] - 4*b^4*n^2
*Log[x]^3*Log[c*x^n]^2*Log[1 + d*f*x^m] + (6*a^2*b^2*Log[-(d*f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m])/m - (12*a*
b^3*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m])/m + (6*b^4*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[c*x^n]
^2*Log[1 + d*f*x^m])/m + 2*b^4*n*Log[x]^2*Log[c*x^n]^3*Log[1 + d*f*x^m] + (4*a*b^3*Log[-(d*f*x^m)]*Log[c*x^n]^
3*Log[1 + d*f*x^m])/m - (4*b^4*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]^3*Log[1 + d*f*x^m])/m + (b^4*Log[-(d*f*x^m)
]*Log[c*x^n]^4*Log[1 + d*f*x^m])/m + (b*n*Log[x]*(-(b^3*n^3*Log[x]^3) + 4*b^2*n^2*Log[x]^2*(a + b*Log[c*x^n])
- 6*b*n*Log[x]*(a + b*Log[c*x^n])^2 + 4*(a + b*Log[c*x^n])^3)*PolyLog[2, -(1/(d*f*x^m))])/m + ((a - b*n*Log[x]
+ b*Log[c*x^n])^4*PolyLog[2, 1 + d*f*x^m])/m + (4*a^3*b*n*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (12*a^2*b^2*n*Log
[c*x^n]*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (12*a*b^3*n*Log[c*x^n]^2*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (4*b^4*n*
Log[c*x^n]^3*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (12*a^2*b^2*n^2*PolyLog[4, -(1/(d*f*x^m))])/m^3 + (24*a*b^3*n^2
*Log[c*x^n]*PolyLog[4, -(1/(d*f*x^m))])/m^3 + (12*b^4*n^2*Log[c*x^n]^2*PolyLog[4, -(1/(d*f*x^m))])/m^3 + (24*a
*b^3*n^3*PolyLog[5, -(1/(d*f*x^m))])/m^4 + (24*b^4*n^3*Log[c*x^n]*PolyLog[5, -(1/(d*f*x^m))])/m^4 + (24*b^4*n^
4*PolyLog[6, -(1/(d*f*x^m))])/m^5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.59, size = 38574, normalized size = 281.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(38574\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^4*ln(d*(1/d+f*x^m))/x,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="maxima")

[Out]

1/5*(b^4*n^4*log(x)^5 + 5*b^4*log(x)*log(x^n)^4 - 5*(b^4*n^3*log(c) + a*b^3*n^3)*log(x)^4 + 10*(b^4*n^2*log(c)
^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2)*log(x)^3 - 10*(b^4*n*log(x)^2 - 2*(b^4*log(c) + a*b^3)*log(x))*log(x^n)
^3 + 10*(b^4*n^2*log(x)^3 - 3*(b^4*n*log(c) + a*b^3*n)*log(x)^2 + 3*(b^4*log(c)^2 + 2*a*b^3*log(c) + a^2*b^2)*
log(x))*log(x^n)^2 - 10*(b^4*n*log(c)^3 + 3*a*b^3*n*log(c)^2 + 3*a^2*b^2*n*log(c) + a^3*b*n)*log(x)^2 - 5*(b^4
*n^3*log(x)^4 - 4*(b^4*n^2*log(c) + a*b^3*n^2)*log(x)^3 + 6*(b^4*n*log(c)^2 + 2*a*b^3*n*log(c) + a^2*b^2*n)*lo
g(x)^2 - 4*(b^4*log(c)^3 + 3*a*b^3*log(c)^2 + 3*a^2*b^2*log(c) + a^3*b)*log(x))*log(x^n) + 5*(b^4*log(c)^4 + 4
*a*b^3*log(c)^3 + 6*a^2*b^2*log(c)^2 + 4*a^3*b*log(c) + a^4)*log(x))*log(d*f*x^m + 1) - integrate(1/5*(5*b^4*d
*f*m*x^m*log(x)*log(x^n)^4 - 10*(b^4*d*f*m*n*log(x)^2 - 2*(b^4*d*f*m*log(c) + a*b^3*d*f*m)*log(x))*x^m*log(x^n
)^3 + 10*(b^4*d*f*m*n^2*log(x)^3 - 3*(b^4*d*f*m*n*log(c) + a*b^3*d*f*m*n)*log(x)^2 + 3*(b^4*d*f*m*log(c)^2 + 2
*a*b^3*d*f*m*log(c) + a^2*b^2*d*f*m)*log(x))*x^m*log(x^n)^2 - 5*(b^4*d*f*m*n^3*log(x)^4 - 4*(b^4*d*f*m*n^2*log
(c) + a*b^3*d*f*m*n^2)*log(x)^3 + 6*(b^4*d*f*m*n*log(c)^2 + 2*a*b^3*d*f*m*n*log(c) + a^2*b^2*d*f*m*n)*log(x)^2
- 4*(b^4*d*f*m*log(c)^3 + 3*a*b^3*d*f*m*log(c)^2 + 3*a^2*b^2*d*f*m*log(c) + a^3*b*d*f*m)*log(x))*x^m*log(x^n)
+ (b^4*d*f*m*n^4*log(x)^5 - 5*(b^4*d*f*m*n^3*log(c) + a*b^3*d*f*m*n^3)*log(x)^4 + 10*(b^4*d*f*m*n^2*log(c)^2
+ 2*a*b^3*d*f*m*n^2*log(c) + a^2*b^2*d*f*m*n^2)*log(x)^3 - 10*(b^4*d*f*m*n*log(c)^3 + 3*a*b^3*d*f*m*n*log(c)^2
+ 3*a^2*b^2*d*f*m*n*log(c) + a^3*b*d*f*m*n)*log(x)^2 + 5*(b^4*d*f*m*log(c)^4 + 4*a*b^3*d*f*m*log(c)^3 + 6*a^2
*b^2*d*f*m*log(c)^2 + 4*a^3*b*d*f*m*log(c) + a^4*d*f*m)*log(x))*x^m)/(d*f*x*x^m + x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (136) = 272\).
time = 0.36, size = 523, normalized size = 3.82 \begin {gather*} -\frac {24 \, b^{4} n^{4} {\rm polylog}\left (6, -d f x^{m}\right ) + {\left (b^{4} m^{4} n^{4} \log \left (x\right )^{4} + b^{4} m^{4} \log \left (c\right )^{4} + 4 \, a b^{3} m^{4} \log \left (c\right )^{3} + 6 \, a^{2} b^{2} m^{4} \log \left (c\right )^{2} + 4 \, a^{3} b m^{4} \log \left (c\right ) + a^{4} m^{4} + 4 \, {\left (b^{4} m^{4} n^{3} \log \left (c\right ) + a b^{3} m^{4} n^{3}\right )} \log \left (x\right )^{3} + 6 \, {\left (b^{4} m^{4} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{4} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{4} n^{2}\right )} \log \left (x\right )^{2} + 4 \, {\left (b^{4} m^{4} n \log \left (c\right )^{3} + 3 \, a b^{3} m^{4} n \log \left (c\right )^{2} + 3 \, a^{2} b^{2} m^{4} n \log \left (c\right ) + a^{3} b m^{4} n\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-d f x^{m}\right ) - 24 \, {\left (b^{4} m n^{4} \log \left (x\right ) + b^{4} m n^{3} \log \left (c\right ) + a b^{3} m n^{3}\right )} {\rm polylog}\left (5, -d f x^{m}\right ) + 12 \, {\left (b^{4} m^{2} n^{4} \log \left (x\right )^{2} + b^{4} m^{2} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{2} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{2} n^{2} + 2 \, {\left (b^{4} m^{2} n^{3} \log \left (c\right ) + a b^{3} m^{2} n^{3}\right )} \log \left (x\right )\right )} {\rm polylog}\left (4, -d f x^{m}\right ) - 4 \, {\left (b^{4} m^{3} n^{4} \log \left (x\right )^{3} + b^{4} m^{3} n \log \left (c\right )^{3} + 3 \, a b^{3} m^{3} n \log \left (c\right )^{2} + 3 \, a^{2} b^{2} m^{3} n \log \left (c\right ) + a^{3} b m^{3} n + 3 \, {\left (b^{4} m^{3} n^{3} \log \left (c\right ) + a b^{3} m^{3} n^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{4} m^{3} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{3} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{3} n^{2}\right )} \log \left (x\right )\right )} {\rm polylog}\left (3, -d f x^{m}\right )}{m^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**4*ln(d*(1/d+f*x**m))/x,x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^4*log((f*x^m + 1/d)*d)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^4}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n))^4)/x,x)

[Out]

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n))^4)/x, x)

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