\(\int (g+h x) (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [310]

Optimal result

Integrand size = 31, antiderivative size = 466 \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {3 B^2 (b c-a d)^2 h n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^2 d^2}-\frac {3 B (b c-a d) h n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b^2 d}+\frac {3 B (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b^2 d^2}-\frac {(b g-a h)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 b^2 h}+\frac {(g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 h}-\frac {3 B^3 (b c-a d)^2 h n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {3 B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2573, 2553, 2398, 2404, 2339, 30, 2355, 2354, 2438, 2421, 6724} \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\frac {3 B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d^2}-\frac {3 B^2 h n^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d^2}+\frac {3 B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b^2 d^2}-\frac {(b g-a h)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 b^2 h}-\frac {3 B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b^2 d}+\frac {(g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 h}-\frac {3 B^3 n^3 (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 B^3 h n^3 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \]

[In]

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

[Out]

(-3*B^2*(b*c - a*d)^2*h*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^2*d^2)
- (3*B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b^2*d) + (3*B*(b*c - a*d)*(2*
b*d*g - b*c*h - a*d*h)*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b^2*d^2
) - ((b*g - a*h)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*b^2*h) + ((g + h*x)^2*(A + B*Log[(e*(a + b*x
)^n)/(c + d*x)^n])^3)/(2*h) - (3*B^3*(b*c - a*d)^2*h*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2) +
(3*B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b
*x))/(b*(c + d*x))])/(b^2*d^2) - (3*B^3*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^3*PolyLog[3, (d*(a + b*x))/(b*
(c + d*x))])/(b^2*d^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 2398

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

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m +
2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && Inte
gerQ[m] && IGtQ[p, 0]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3890\) vs. \(2(466)=932\).

Time = 0.94 (sec) , antiderivative size = 3890, normalized size of antiderivative = 8.35 \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-12*A*b^2*B^2*c*d*g*n^2 - 12*a*A*b*B^2*d^2*g*n^2 + 12*a*A*b*B^2*c*d*h*n^2 + 6*a*b*B^3*c*d*h*n^3 - 6*a^2*B^3*d
^2*h*n^3 + 2*A^3*b^2*d^2*g*x - 3*A^2*b^2*B*c*d*h*n*x + 3*a*A^2*b*B*d^2*h*n*x + A^3*b^2*d^2*h*x^2 + 6*a*A^2*b*B
*d^2*g*n*Log[a + b*x] - 3*a^2*A^2*B*d^2*h*n*Log[a + b*x] - 6*a*A*b*B^2*c*d*h*n^2*Log[a + b*x] + 6*a^2*A*B^2*d^
2*h*n^2*Log[a + b*x] + 12*b^2*B^3*c*d*g*n^3*Log[a + b*x] + 12*a*b*B^3*d^2*g*n^3*Log[a + b*x] - 12*a*b*B^3*c*d*
h*n^3*Log[a + b*x] - 6*a*A*b*B^2*d^2*g*n^2*Log[a + b*x]^2 + 3*a^2*A*B^2*d^2*h*n^2*Log[a + b*x]^2 + 3*a*b*B^3*c
*d*h*n^3*Log[a + b*x]^2 - 3*a^2*B^3*d^2*h*n^3*Log[a + b*x]^2 + 2*a*b*B^3*d^2*g*n^3*Log[a + b*x]^3 - a^2*B^3*d^
2*h*n^3*Log[a + b*x]^3 - 6*A^2*b^2*B*c*d*g*n*Log[c + d*x] + 3*A^2*b^2*B*c^2*h*n*Log[c + d*x] + 6*A*b^2*B^2*c^2
*h*n^2*Log[c + d*x] - 6*a*A*b*B^2*c*d*h*n^2*Log[c + d*x] - 12*b^2*B^3*c*d*g*n^3*Log[c + d*x] - 12*a*b*B^3*d^2*
g*n^3*Log[c + d*x] + 12*a*b*B^3*c*d*h*n^3*Log[c + d*x] + 12*A*b^2*B^2*c*d*g*n^2*Log[a + b*x]*Log[c + d*x] + 12
*a*A*b*B^2*d^2*g*n^2*Log[a + b*x]*Log[c + d*x] - 6*A*b^2*B^2*c^2*h*n^2*Log[a + b*x]*Log[c + d*x] - 6*a^2*A*B^2
*d^2*h*n^2*Log[a + b*x]*Log[c + d*x] - 6*b^2*B^3*c^2*h*n^3*Log[a + b*x]*Log[c + d*x] + 6*a*b*B^3*c*d*h*n^3*Log
[a + b*x]*Log[c + d*x] - 6*b^2*B^3*c*d*g*n^3*Log[a + b*x]^2*Log[c + d*x] - 12*a*b*B^3*d^2*g*n^3*Log[a + b*x]^2
*Log[c + d*x] + 3*b^2*B^3*c^2*h*n^3*Log[a + b*x]^2*Log[c + d*x] + 6*a^2*B^3*d^2*h*n^3*Log[a + b*x]^2*Log[c + d
*x] - 12*a*A*b*B^2*d^2*g*n^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 6*a^2*A*B^2*d^2*h*n^2*Log[(d*(a
+ b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 12*a*b*B^3*d^2*g*n^3*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log
[c + d*x] - 6*a^2*B^3*d^2*h*n^3*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 6*A*b^2*B^2*c*d*
g*n^2*Log[c + d*x]^2 + 3*A*b^2*B^2*c^2*h*n^2*Log[c + d*x]^2 + 3*b^2*B^3*c^2*h*n^3*Log[c + d*x]^2 - 3*a*b*B^3*c
*d*h*n^3*Log[c + d*x]^2 + 12*b^2*B^3*c*d*g*n^3*Log[a + b*x]*Log[c + d*x]^2 + 6*a*b*B^3*d^2*g*n^3*Log[a + b*x]*
Log[c + d*x]^2 - 6*b^2*B^3*c^2*h*n^3*Log[a + b*x]*Log[c + d*x]^2 - 3*a^2*B^3*d^2*h*n^3*Log[a + b*x]*Log[c + d*
x]^2 - 6*b^2*B^3*c*d*g*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - 6*a*b*B^3*d^2*g*n^3*Log[(d*(a +
b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 + 3*b^2*B^3*c^2*h*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 +
3*a^2*B^3*d^2*h*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - 2*b^2*B^3*c*d*g*n^3*Log[c + d*x]^3 + b^
2*B^3*c^2*h*n^3*Log[c + d*x]^3 - 12*A*b^2*B^2*c*d*g*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*A*b^2*
B^2*c^2*h*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*b^2*B^3*c^2*h*n^3*Log[a + b*x]*Log[(b*(c + d*x))
/(b*c - a*d)] - 12*a*b*B^3*c*d*h*n^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*a^2*B^3*d^2*h*n^3*Log[a +
b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*b^2*B^3*c*d*g*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 6*a
*b*B^3*d^2*g*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 3*b^2*B^3*c^2*h*n^3*Log[a + b*x]^2*Log[(b*(c
+ d*x))/(b*c - a*d)] - 3*a^2*B^3*d^2*h*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 12*b^2*B^3*c*d*g*n^
3*Log[a + b*x]*Log[c + d*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*b^2*B^3*c^2*h*n^3*Log[a + b*x]*Log[c + d*x]*Log
[(b*(c + d*x))/(b*c - a*d)] - 12*b^2*B^3*c*d*g*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 12*a*b*B^3*d^2*g*n^2*Log
[(e*(a + b*x)^n)/(c + d*x)^n] + 12*a*b*B^3*c*d*h*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*A^2*b^2*B*d^2*g*x*Lo
g[(e*(a + b*x)^n)/(c + d*x)^n] - 6*A*b^2*B^2*c*d*h*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a*A*b*B^2*d^2*h*n*
x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*A^2*b^2*B*d^2*h*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 12*a*A*b*B^2*d^2
*g*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a^2*A*B^2*d^2*h*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c +
d*x)^n] - 6*a*b*B^3*c*d*h*n^2*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a^2*B^3*d^2*h*n^2*Log[a + b*x
]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a*b*B^3*d^2*g*n^2*Log[a + b*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*a
^2*B^3*d^2*h*n^2*Log[a + b*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 12*A*b^2*B^2*c*d*g*n*Log[c + d*x]*Log[(e*(a
+ b*x)^n)/(c + d*x)^n] + 6*A*b^2*B^2*c^2*h*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*b^2*B^3*c^2*h*
n^2*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a*b*B^3*c*d*h*n^2*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c +
d*x)^n] + 12*b^2*B^3*c*d*g*n^2*Log[a + b*x]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 12*a*b*B^3*d^2*g*n
^2*Log[a + b*x]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*b^2*B^3*c^2*h*n^2*Log[a + b*x]*Log[c + d*x]*
Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a^2*B^3*d^2*h*n^2*Log[a + b*x]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)
^n] - 12*a*b*B^3*d^2*g*n^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6
*a^2*B^3*d^2*h*n^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*b^2*B^3
*c*d*g*n^2*Log[c + d*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*b^2*B^3*c^2*h*n^2*Log[c + d*x]^2*Log[(e*(a + b*
x)^n)/(c + d*x)^n] - 12*b^2*B^3*c*d*g*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^n)/(c +
d*x)^n] + 6*b^2*B^3*c^2*h*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^n)/(c + d*x)^n] +
6*A*b^2*B^2*d^2*g*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 3*b^2*B^3*c*d*h*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^
2 + 3*a*b*B^3*d^2*h*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*A*b^2*B^2*d^2*h*x^2*Log[(e*(a + b*x)^n)/(c + d*
x)^n]^2 + 6*a*b*B^3*d^2*g*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 3*a^2*B^3*d^2*h*n*Log[a + b*x]*L
og[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 6*b^2*B^3*c*d*g*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*b^2*
B^3*c^2*h*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*b^2*B^3*d^2*g*x*Log[(e*(a + b*x)^n)/(c + d*x)^
n]^3 + b^2*B^3*d^2*h*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3 + 6*B^2*n^2*(-2*A*b^2*c*d*g + A*b^2*c^2*h + b^2*B*
c^2*h*n - 2*a*b*B*c*d*h*n + a^2*B*d^2*h*n + a*B*d^2*(2*b*g - a*h)*n*Log[a + b*x] + b^2*B*c*(-2*d*g + c*h)*n*Lo
g[c + d*x] - 2*b^2*B*c*d*g*Log[(e*(a + b*x)^n)/(c + d*x)^n] + b^2*B*c^2*h*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Po
lyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 6*B^2*n^2*(a*B*d^2*(2*b*g - a*h)*n*Log[a + b*x] + b^2*B*c*(-2*d*g + c
*h)*n*Log[c + d*x] + a*d^2*(-2*b*g + a*h)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))*PolyLog[2, (b*(c + d*x))/(
b*c - a*d)] + 12*b^2*B^3*c*d*g*n^3*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 12*a*b*B^3*d^2*g*n^3*PolyLog[3,
(d*(a + b*x))/(-(b*c) + a*d)] - 6*b^2*B^3*c^2*h*n^3*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] + 6*a^2*B^3*d^2*h
*n^3*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] + 12*b^2*B^3*c*d*g*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 1
2*a*b*B^3*d^2*g*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 6*b^2*B^3*c^2*h*n^3*PolyLog[3, (b*(c + d*x))/(b*c
- a*d)] + 6*a^2*B^3*d^2*h*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(2*b^2*d^2)

Maple [F]

\[\int \left (h x +g \right ) {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]

[In]

int((h*x+g)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

[Out]

int((h*x+g)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

Fricas [F]

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

[In]

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

[Out]

integral(A^3*h*x + A^3*g + (B^3*h*x + B^3*g)*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*(A*B^2*h*x + A*B^2*g)*log((b
*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B*h*x + A^2*B*g)*log((b*x + a)^n*e/(d*x + c)^n), x)

Sympy [F(-2)]

Exception generated. \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((h*x+g)*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

3/2*A^2*B*h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 1/2*A^3*h*x^2 + 3*A^2*B*g*x*log((b*x + a)^n*e/(d*x + c)^n) +
A^3*g*x + 3*(a*e*n*log(b*x + a)/b - c*e*n*log(d*x + c)/d)*A^2*B*g/e - 3/2*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*
log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A^2*B*h/e - 1/2*((B^3*b^2*d^2*h*x^2 + 2*B^3*b^2*d^2*g*x)*log((
d*x + c)^n)^3 + 3*((2*c*d*g*n - c^2*h*n)*B^3*b^2*log(d*x + c) - (2*a*b*d^2*g*n - a^2*d^2*h*n)*B^3*log(b*x + a)
- (B^3*b^2*d^2*h*log(e) + A*B^2*b^2*d^2*h)*x^2 - (2*A*B^2*b^2*d^2*g + (a*b*d^2*h*n - (c*d*h*n - 2*d^2*g*log(e
))*b^2)*B^3)*x - (B^3*b^2*d^2*h*x^2 + 2*B^3*b^2*d^2*g*x)*log((b*x + a)^n))*log((d*x + c)^n)^2)/(b^2*d^2) - int
egrate(-(B^3*b^2*c*d*g*log(e)^3 + 3*A*B^2*b^2*c*d*g*log(e)^2 + (B^3*b^2*d^2*h*x^2 + B^3*b^2*c*d*g + (d^2*g + c
*d*h)*B^3*b^2*x)*log((b*x + a)^n)^3 + (B^3*b^2*d^2*h*log(e)^3 + 3*A*B^2*b^2*d^2*h*log(e)^2)*x^2 + 3*(B^3*b^2*c
*d*g*log(e) + A*B^2*b^2*c*d*g + (B^3*b^2*d^2*h*log(e) + A*B^2*b^2*d^2*h)*x^2 + ((d^2*g + c*d*h)*A*B^2*b^2 + (d
^2*g*log(e) + c*d*h*log(e))*B^3*b^2)*x)*log((b*x + a)^n)^2 + (3*(d^2*g*log(e)^2 + c*d*h*log(e)^2)*A*B^2*b^2 +
(d^2*g*log(e)^3 + c*d*h*log(e)^3)*B^3*b^2)*x + 3*(B^3*b^2*c*d*g*log(e)^2 + 2*A*B^2*b^2*c*d*g*log(e) + (B^3*b^2
*d^2*h*log(e)^2 + 2*A*B^2*b^2*d^2*h*log(e))*x^2 + (2*(d^2*g*log(e) + c*d*h*log(e))*A*B^2*b^2 + (d^2*g*log(e)^2
+ c*d*h*log(e)^2)*B^3*b^2)*x)*log((b*x + a)^n) - 3*(B^3*b^2*c*d*g*log(e)^2 + 2*A*B^2*b^2*c*d*g*log(e) - (2*c*
d*g*n^2 - c^2*h*n^2)*B^3*b^2*log(d*x + c) + (2*a*b*d^2*g*n^2 - a^2*d^2*h*n^2)*B^3*log(b*x + a) + ((h*n + 2*h*l
og(e))*A*B^2*b^2*d^2 + (h*n*log(e) + h*log(e)^2)*B^3*b^2*d^2)*x^2 + (B^3*b^2*d^2*h*x^2 + B^3*b^2*c*d*g + (d^2*
g + c*d*h)*B^3*b^2*x)*log((b*x + a)^n)^2 + (2*(c*d*h*log(e) + (g*n + g*log(e))*d^2)*A*B^2*b^2 + (a*b*d^2*h*n^2
- ((h*n^2 - h*log(e)^2)*c*d - (2*g*n*log(e) + g*log(e)^2)*d^2)*b^2)*B^3)*x + (2*B^3*b^2*c*d*g*log(e) + 2*A*B^
2*b^2*c*d*g + ((h*n + 2*h*log(e))*B^3*b^2*d^2 + 2*A*B^2*b^2*d^2*h)*x^2 + 2*((d^2*g + c*d*h)*A*B^2*b^2 + (c*d*h
*log(e) + (g*n + g*log(e))*d^2)*B^3*b^2)*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^2*d^2*x + b^2*c*d), x)

Giac [F]

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

[In]

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

[Out]

integrate((h*x + g)*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3, x)

Mupad [F(-1)]

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

[In]

int((g + h*x)*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3,x)

[Out]

int((g + h*x)*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3, x)