3.65 \(\int \frac{(a+b \log (c x^n))^3 \log (d (\frac{1}{d}+f x^m))}{x} \, dx\)

Optimal. Leaf size=105 \[ -\frac{6 b^2 n^2 \text{PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m^3}+\frac{3 b n \text{PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}+\frac{6 b^3 n^3 \text{PolyLog}\left (5,-d f x^m\right )}{m^4} \]

[Out]

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

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Rubi [A]  time = 0.113773, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2374, 2383, 6589} \[ -\frac{6 b^2 n^2 \text{PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m^3}+\frac{3 b n \text{PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}+\frac{6 b^3 n^3 \text{PolyLog}\left (5,-d f x^m\right )}{m^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

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

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

Mathematica [B]  time = 0.39454, size = 1035, normalized size = 9.86 \[ -\frac{3}{10} b^3 m n^3 \log ^5(x)+\frac{3}{4} a b^2 m n^2 \log ^4(x)+\frac{3}{4} b^3 m n^2 \log \left (c x^n\right ) \log ^4(x)-\frac{3}{4} b^3 n^3 \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^4(x)+\frac{3}{4} b^3 n^3 \log \left (d f x^m+1\right ) \log ^4(x)-\frac{1}{2} b^3 m n \log ^2\left (c x^n\right ) \log ^3(x)-\frac{1}{2} a^2 b m n \log ^3(x)-a b^2 m n \log \left (c x^n\right ) \log ^3(x)+2 a b^2 n^2 \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^3(x)+2 b^3 n^2 \log \left (c x^n\right ) \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^3(x)-2 a b^2 n^2 \log \left (d f x^m+1\right ) \log ^3(x)-\frac{b^3 n^3 \log \left (-d f x^m\right ) \log \left (d f x^m+1\right ) \log ^3(x)}{m}-2 b^3 n^2 \log \left (c x^n\right ) \log \left (d f x^m+1\right ) \log ^3(x)-\frac{3}{2} b^3 n \log ^2\left (c x^n\right ) \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^2(x)-\frac{3}{2} a^2 b n \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^2(x)-3 a b^2 n \log \left (c x^n\right ) \log \left (\frac{x^{-m}}{d f}+1\right ) \log ^2(x)+\frac{3}{2} b^3 n \log ^2\left (c x^n\right ) \log \left (d f x^m+1\right ) \log ^2(x)+\frac{3}{2} a^2 b n \log \left (d f x^m+1\right ) \log ^2(x)+\frac{3 a b^2 n^2 \log \left (-d f x^m\right ) \log \left (d f x^m+1\right ) \log ^2(x)}{m}+3 a b^2 n \log \left (c x^n\right ) \log \left (d f x^m+1\right ) \log ^2(x)+\frac{3 b^3 n^2 \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (d f x^m+1\right ) \log ^2(x)}{m}-\frac{3 b^3 n \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (d f x^m+1\right ) \log (x)}{m}-\frac{3 a^2 b n \log \left (-d f x^m\right ) \log \left (d f x^m+1\right ) \log (x)}{m}-\frac{6 a b^2 n \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (d f x^m+1\right ) \log (x)}{m}+\frac{b n \left (b^2 n^2 \log ^2(x)-3 b n \left (a+b \log \left (c x^n\right )\right ) \log (x)+3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \text{PolyLog}\left (2,-\frac{x^{-m}}{d f}\right ) \log (x)}{m}+\frac{b^3 \log \left (-d f x^m\right ) \log ^3\left (c x^n\right ) \log \left (d f x^m+1\right )}{m}+\frac{3 a b^2 \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (d f x^m+1\right )}{m}+\frac{a^3 \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}+\frac{3 a^2 b \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (d f x^m+1\right )}{m}+\frac{\left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \text{PolyLog}\left (2,d f x^m+1\right )}{m}+\frac{3 b^3 n \log ^2\left (c x^n\right ) \text{PolyLog}\left (3,-\frac{x^{-m}}{d f}\right )}{m^2}+\frac{3 a^2 b n \text{PolyLog}\left (3,-\frac{x^{-m}}{d f}\right )}{m^2}+\frac{6 a b^2 n \log \left (c x^n\right ) \text{PolyLog}\left (3,-\frac{x^{-m}}{d f}\right )}{m^2}+\frac{6 a b^2 n^2 \text{PolyLog}\left (4,-\frac{x^{-m}}{d f}\right )}{m^3}+\frac{6 b^3 n^2 \log \left (c x^n\right ) \text{PolyLog}\left (4,-\frac{x^{-m}}{d f}\right )}{m^3}+\frac{6 b^3 n^3 \text{PolyLog}\left (5,-\frac{x^{-m}}{d f}\right )}{m^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

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

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Maple [C]  time = 0.25, size = 11734, normalized size = 111.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^m))/x,x)

[Out]

result too large to display

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

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Fricas [C]  time = 1.42618, size = 678, normalized size = 6.46 \begin{align*} \frac{6 \, b^{3} n^{3}{\rm polylog}\left (5, -d f x^{m}\right ) -{\left (b^{3} m^{3} n^{3} \log \left (x\right )^{3} + b^{3} m^{3} \log \left (c\right )^{3} + 3 \, a b^{2} m^{3} \log \left (c\right )^{2} + 3 \, a^{2} b m^{3} \log \left (c\right ) + a^{3} m^{3} + 3 \,{\left (b^{3} m^{3} n^{2} \log \left (c\right ) + a b^{2} m^{3} n^{2}\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{3} m^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} m^{3} n \log \left (c\right ) + a^{2} b m^{3} n\right )} \log \left (x\right )\right )}{\rm Li}_2\left (-d f x^{m}\right ) - 6 \,{\left (b^{3} m n^{3} \log \left (x\right ) + b^{3} m n^{2} \log \left (c\right ) + a b^{2} m n^{2}\right )}{\rm polylog}\left (4, -d f x^{m}\right ) + 3 \,{\left (b^{3} m^{2} n^{3} \log \left (x\right )^{2} + b^{3} m^{2} n \log \left (c\right )^{2} + 2 \, a b^{2} m^{2} n \log \left (c\right ) + a^{2} b m^{2} n + 2 \,{\left (b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} \log \left (x\right )\right )}{\rm polylog}\left (3, -d f x^{m}\right )}{m^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")

[Out]

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

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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*(1/d+f*x**m))/x,x)

[Out]

Timed out

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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} + \frac{1}{d}\right )} 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*(1/d+f*x^m))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*log((f*x^m + 1/d)*d)/x, x)