3.4.12 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{g+h x} \, dx\) [312]

Optimal. Leaf size=425 \[ -\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{h}+\frac {3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \text {Li}_2\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {6 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{h}-\frac {6 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_3\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 B^3 n^3 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right )}{h}+\frac {6 B^3 n^3 \text {Li}_4\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h} \]

[Out]

-ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/h+(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3*ln(1-(-c*h
+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h-3*B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*polylog(2,d*(b*x+a)/b/(d*x+c))/h
+3*B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*polylog(2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h+6*B^2*n^2*(A+B*l
n(e*(b*x+a)^n/((d*x+c)^n)))*polylog(3,d*(b*x+a)/b/(d*x+c))/h-6*B^2*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*polyl
og(3,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h-6*B^3*n^3*polylog(4,d*(b*x+a)/b/(d*x+c))/h+6*B^3*n^3*polylog(4,(
-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h

________________________________________________________________________________________

Rubi [A]
time = 0.46, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2573, 2553, 2404, 2354, 2421, 2430, 6724} \begin {gather*} -\frac {6 B^2 n^2 \text {PolyLog}\left (3,\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}+\frac {6 B^2 n^2 \text {PolyLog}\left (3,\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 )}{h}+\frac {3 B n \text {PolyLog}\left (2,\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h}-\frac {3 B n \text {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 )^2}{h}+\frac {6 B^3 n^3 \text {PolyLog}\left (4,\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{h}-\frac {6 B^3 n^3 \text {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{h}+\frac {\log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h}-\frac {\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 )^3}{h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/h) + ((A + B*Log[(e*(a + b*x)^n)
/(c + d*x)^n])^3*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h - (3*B*n*(A + B*Log[(e*(a + b*x)^
n)/(c + d*x)^n])^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/h + (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]
)^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c
+ d*x)^n])*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/h - (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Po
lyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h - (6*B^3*n^3*PolyLog[4, (d*(a + b*x))/(b*(c + d*x
))])/h + (6*B^3*n^3*PolyLog[4, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h

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 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 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 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*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx &=\int \left (\frac {A^3}{g+h x}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}\right ) \, dx\\ &=\frac {A^3 \log (g+h x)}{h}+\left (3 A^2 B\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx+\left (3 A B^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx+B^3 \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx\\ &=\frac {A^3 \log (g+h x)}{h}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {\left (3 A B^2 d\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{h}+\frac {\left (B^3 d\right ) \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{h}-\frac {\left (3 A B^2 (d g-c h)\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{h}-\frac {\left (B^3 (d g-c h)\right ) \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{h}-\frac {\left (3 A^2 b B n\right ) \int \frac {\log (g+h x)}{a+b x} \, dx}{h}+\frac {\left (3 A^2 B d n\right ) \int \frac {\log (g+h x)}{c+d x} \, dx}{h}\\ &=-\frac {3 A B^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac {B^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A^3 \log (g+h x)}{h}-\frac {3 A^2 B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\left (3 A^2 B n\right ) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx-\left (3 A^2 B n\right ) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx+\frac {\left (6 A B^2 (b c-a d) n\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (6 A B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\frac {\left (3 B^3 (b c-a d) n\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (3 B^3 (b c-a d) n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac {3 A B^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac {B^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A^3 \log (g+h x)}{h}-\frac {3 A^2 B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}-\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {\left (3 A^2 B n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}-\frac {\left (3 A^2 B n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}+\frac {\left (6 A B^2 (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (6 A B^2 (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\frac {\left (6 B^3 (b c-a d) n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (6 B^3 (b c-a d) n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac {3 A B^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac {B^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A^3 \log (g+h x)}{h}-\frac {3 A^2 B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {3 A^2 B n \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {3 A^2 B n \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}-\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {6 A B^2 n^2 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}-\frac {6 A B^2 n^2 \text {Li}_3\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {\left (6 B^3 (b c-a d) n^3\right ) \int \frac {\text {Li}_3\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\frac {\left (6 B^3 (b c-a d) n^3\right ) \int \frac {\text {Li}_3\left (1+\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac {3 A B^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac {B^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A^3 \log (g+h x)}{h}-\frac {3 A^2 B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {3 A^2 B n \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {3 A^2 B n \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}-\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {6 A B^2 n^2 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}-\frac {6 A B^2 n^2 \text {Li}_3\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 B^3 n^3 \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{h}+\frac {6 B^3 n^3 \text {Li}_4\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}\\ \end {align*}

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Mathematica [F]
time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{3}}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [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(e*(b*x+a)^n/((d*x+c)^n)))^3/(h*x+g),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3}{g+h\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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