\(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x)^4} \, dx\) [59]

Optimal result

Integrand size = 24, antiderivative size = 564 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\frac {b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^3 (f+g x)}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac {b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac {b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac {b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}-\frac {b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}-\frac {b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac {b^2 e^3 n^2 \log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac {b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (e f-d g)^3}-\frac {b e^3 n \log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g (e f-d g)^3}-\frac {b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (e f-d g)^3}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2 (e f-d g)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac {b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}-\frac {b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3} \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^4,x]

[Out]

(b^2*e^2*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)^3*(f + g*x)) + (b*e*n*(a + b*Log[c*(d + e*x)^n
])^2)/(2*g*(e*f - d*g)*(f + g*x)^2) - (b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/((e*f - d*g)^3*(f + g*x
)) - (a + b*Log[c*(d + e*x)^n])^3/(3*g*(f + g*x)^3) - (b^3*e^3*n^3*Log[f + g*x])/(g*(e*f - d*g)^3) + (2*b^2*e^
3*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)^3) + (b^2*e^3*n^2*(a + b*Log[c
*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b*e^3*n*(a + b*Log[c*(d + e*x)^n])^2*L
og[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b^3*e^3*n^3*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/
(g*(e*f - d*g)^3) + (2*b^2*e^3*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*
f - d*g)^3) + (2*b^3*e^3*n^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)^3) + (2*b^3*e^3*n^3*Poly
Log[3, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^3)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

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 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\frac {3 b e (e f-d g)^2 n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 b e^2 (e f-d g) n (f+g x)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-6 b (e f-d g)^3 n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 b e^3 n (f+g x)^3 \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 (e f-d g)^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3-6 b e^3 n (f+g x)^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e^2 g (d+e x) (f+g x)^2+g \left (3 d e^2 f^2-3 d^2 e f g+d^3 g^2+e^3 x \left (3 f^2+3 f g x+g^2 x^2\right )\right ) \log ^2(d+e x)+3 e^3 (f+g x)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+e (f+g x) \log (d+e x) \left (g^2 (d+e x)^2-4 e g (d+e x) (f+g x)-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e^3 (f+g x)^3 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (2 g \left (3 d e^2 f^2-3 d^2 e f g+d^3 g^2+e^3 x \left (3 f^2+3 f g x+g^2 x^2\right )\right ) \log ^3(d+e x)-6 e^3 (f+g x)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 e (f+g x) \log ^2(d+e x) \left (g^2 (d+e x)^2-4 e g (d+e x) (f+g x)-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+18 e^3 (f+g x)^3 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e^2 (f+g x)^2 \log (d+e x) \left (g (d+e x)+3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+12 e^3 (f+g x)^3 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{6 g (e f-d g)^3 (f+g x)^3} \]

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^4,x]

[Out]

(3*b*e*(e*f - d*g)^2*n*(f + g*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 6*b*e^2*(e*f - d*g)*n*(f +
g*x)^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 6*b*(e*f - d*g)^3*n*Log[d + e*x]*(a - b*n*Log[d + e*x
] + b*Log[c*(d + e*x)^n])^2 + 6*b*e^3*n*(f + g*x)^3*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])
^2 - 2*(e*f - d*g)^3*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 - 6*b*e^3*n*(f + g*x)^3*(a - b*n*Log[d +
e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] + 6*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e^2*g*(
d + e*x)*(f + g*x)^2 + g*(3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2 + e^3*x*(3*f^2 + 3*f*g*x + g^2*x^2))*Log[d + e*x
]^2 + 3*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + e*(f + g*x)*Log[d + e*x]*(g^2*(d + e*x)^2 - 4*e*g*(d
+ e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e*(f + g*x))/(e*f - d*g)]) - 2*e^3*(f + g*x)^3*PolyLog[2, (g*(d + e*
x))/(-(e*f) + d*g)]) + b^3*n^3*(2*g*(3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2 + e^3*x*(3*f^2 + 3*f*g*x + g^2*x^2))*
Log[d + e*x]^3 - 6*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*e*(f + g*x)*Log[d + e*x]^2*(g^2*(d + e*x
)^2 - 4*e*g*(d + e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e*(f + g*x))/(e*f - d*g)]) + 18*e^3*(f + g*x)^3*PolyL
og[2, (g*(d + e*x))/(-(e*f) + d*g)] + 6*e^2*(f + g*x)^2*Log[d + e*x]*(g*(d + e*x) + 3*e*(f + g*x)*Log[(e*(f +
g*x))/(e*f - d*g)] - 2*e*(f + g*x)*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 12*e^3*(f + g*x)^3*PolyLog[3, (
g*(d + e*x))/(-(e*f) + d*g)]))/(6*g*(e*f - d*g)^3*(f + g*x)^3)

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}{\left (g x +f \right )^{4}}d x\]

[In]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)^4,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)^4,x)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^4,x, algorithm="fricas")

[Out]

integral((b^3*log((e*x + d)^n*c)^3 + 3*a*b^2*log((e*x + d)^n*c)^2 + 3*a^2*b*log((e*x + d)^n*c) + a^3)/(g^4*x^4
+ 4*f*g^3*x^3 + 6*f^2*g^2*x^2 + 4*f^3*g*x + f^4), x)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right )^{4}}\, dx \]

[In]

integrate((a+b*ln(c*(e*x+d)**n))**3/(g*x+f)**4,x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**3/(f + g*x)**4, x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^4,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^4,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3/(g*x + f)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{{\left (f+g\,x\right )}^4} \,d x \]

[In]

int((a + b*log(c*(d + e*x)^n))^3/(f + g*x)^4,x)

[Out]

int((a + b*log(c*(d + e*x)^n))^3/(f + g*x)^4, x)