\(\int \frac {x^3 \log (e (\frac {a+b x}{c+d x})^n)}{f+g x+h x^2} \, dx\) [82]

Optimal result

Integrand size = 34, antiderivative size = 1046 \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3} \]

[Out]

1/2*a*n*x/b/h-1/2*c*n*x/d/h-1/2*a^2*n*ln(b*x+a)/b^2/h+1/2*n*x^2*ln(b*x+a)/h-g*n*(b*x+a)*ln(b*x+a)/b/h^2+1/2*c^
2*n*ln(d*x+c)/d^2/h-1/2*n*x^2*ln(d*x+c)/h+g*n*(d*x+c)*ln(d*x+c)/d/h^2+g*x*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^
n)-n*ln(d*x+c))/h^2-1/2*x^2*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))/h-1/2*(-f*h+g^2)*(n*ln(b*x+a)-
ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))*ln(h*x^2+g*x+f)/h^3+1/2*n*ln(b*x+a)*ln(-b*(g+2*h*x-(-4*f*h+g^2)^(1/2))/
(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-1/2*n*ln(d*x+c)*ln(-d*(g+2*h
*x-(-4*f*h+g^2)^(1/2))/(2*c*h-d*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n
*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-1/2*n
*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n
*ln(b*x+a)*ln(-b*(g+2*h*x+(-4*f*h+g^2)^(1/2))/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*h+g*(-3*f*h+g^2)/(-4*f*
h+g^2)^(1/2))/h^3-1/2*n*ln(d*x+c)*ln(-d*(g+2*h*x+(-4*f*h+g^2)^(1/2))/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*
h+g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*
h+g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-1/2*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*
h+g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-g*(-3*f*h+g^2)*arctanh((2*h*x+g)/(-4*f*h+g^2)^(1/2))*(n*ln(b*x+a)-ln(
e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))/h^3/(-4*f*h+g^2)^(1/2)

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 1046, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2593, 2465, 2436, 2332, 2442, 45, 2441, 2440, 2438, 715, 648, 632, 212, 642} \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=-\frac {n \log (a+b x) a^2}{2 b^2 h}+\frac {n x a}{2 b h}-\frac {c n x}{2 d h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {c^2 n \log (c+d x)}{2 d^2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {g \left (g^2-3 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 h^3}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3} \]

[In]

Int[(x^3*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x + h*x^2),x]

[Out]

(a*n*x)/(2*b*h) - (c*n*x)/(2*d*h) - (a^2*n*Log[a + b*x])/(2*b^2*h) + (n*x^2*Log[a + b*x])/(2*h) - (g*n*(a + b*
x)*Log[a + b*x])/(b*h^2) + (c^2*n*Log[c + d*x])/(2*d^2*h) - (n*x^2*Log[c + d*x])/(2*h) + (g*n*(c + d*x)*Log[c
+ d*x])/(d*h^2) + (g*x*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/h^2 - (x^2*(n*Log[a
+ b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(2*h) - (g*(g^2 - 3*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[
g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(h^3*Sqrt[g^2 - 4*f*h]) + ((
g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a
*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]
*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) + ((g^2 - f*h + (g*(
g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt
[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + S
qrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h)*(n*Log[a + b*x] - Log
[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*h^3) + ((g^2 - f*h - (g*(g^2 - 3*f*h))/
Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^3) + ((g^2 - f*h +
(g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^3
) - ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 -
4*f*h]))])/(2*h^3) - ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h -
d*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^3)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 715

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2,
0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

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

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2442

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

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2593

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rubi steps \begin{align*} \text {integral}& = n \int \frac {x^3 \log (a+b x)}{f+g x+h x^2} \, dx-n \int \frac {x^3 \log (c+d x)}{f+g x+h x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {x^3}{f+g x+h x^2} \, dx \\ & = n \int \left (-\frac {g \log (a+b x)}{h^2}+\frac {x \log (a+b x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (a+b x)}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx-n \int \left (-\frac {g \log (c+d x)}{h^2}+\frac {x \log (c+d x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (c+d x)}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \left (-\frac {g}{h^2}+\frac {x}{h}+\frac {f g+\left (g^2-f h\right ) x}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx \\ & = \frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {n \int \frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (a+b x)}{f+g x+h x^2} \, dx}{h^2}-\frac {n \int \frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (c+d x)}{f+g x+h x^2} \, dx}{h^2}-\frac {(g n) \int \log (a+b x) \, dx}{h^2}+\frac {(g n) \int \log (c+d x) \, dx}{h^2}+\frac {n \int x \log (a+b x) \, dx}{h}-\frac {n \int x \log (c+d x) \, dx}{h}-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {f g+\left (g^2-f h\right ) x}{f+g x+h x^2} \, dx}{h^2} \\ & = \frac {n x^2 \log (a+b x)}{2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {n \int \left (\frac {\left (g^2-f h+\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g^2-f h-\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h^2}-\frac {n \int \left (\frac {\left (g^2-f h+\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g^2-f h-\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h^2}-\frac {(g n) \text {Subst}(\int \log (x) \, dx,x,a+b x)}{b h^2}+\frac {(g n) \text {Subst}(\int \log (x) \, dx,x,c+d x)}{d h^2}-\frac {(b n) \int \frac {x^2}{a+b x} \, dx}{2 h}+\frac {(d n) \int \frac {x^2}{c+d x} \, dx}{2 h}+\frac {\left (g \left (g^2-3 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {1}{f+g x+h x^2} \, dx}{2 h^3}-\frac {\left (\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {g+2 h x}{f+g x+h x^2} \, dx}{2 h^3} \\ & = \frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {(b n) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 h}+\frac {(d n) \int \left (-\frac {c}{d^2}+\frac {x}{d}+\frac {c^2}{d^2 (c+d x)}\right ) \, dx}{2 h}+\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}+\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (g \left (g^2-3 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \text {Subst}\left (\int \frac {1}{g^2-4 f h-x^2} \, dx,x,g+2 h x\right )}{h^3} \\ & = \frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {\left (b \left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^3}+\frac {\left (d \left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^3}-\frac {\left (b \left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^3}+\frac {\left (d \left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^3} \\ & = \frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^3}+\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^3}-\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^3}+\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^3} \\ & = \frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 1240, normalized size of antiderivative = 1.19 \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\frac {h^2 x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\frac {2 g h (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+\frac {2 (b c-a d) g h n \log (c+d x)}{b d}+\frac {h^2 n \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )}{b^2 d^2}+\frac {2 f g h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{\sqrt {g^2-4 f h}}+\left (g^2-f h\right ) \left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )-\frac {2 f g h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{\sqrt {g^2-4 f h}}+\left (g^2-f h\right ) \left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )-\frac {2 f g h n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{2 c h+d \left (-g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}-\frac {\left (g^2-f h\right ) \left (-g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{2 c h+d \left (-g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}+\frac {2 f g h n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}-\frac {\left (g^2-f h\right ) \left (g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}}{2 h^3} \]

[In]

Integrate[(x^3*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x + h*x^2),x]

[Out]

(h^2*x^2*Log[e*((a + b*x)/(c + d*x))^n] - (2*g*h*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + (2*(b*c - a*d)*
g*h*n*Log[c + d*x])/(b*d) + (h^2*n*(-(a^2*d^2*Log[a + b*x]) + b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c + d*x])))/(b
^2*d^2) + (2*f*g*h*Log[e*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x])/Sqrt[g^2 - 4*f*h] + (g^2
- f*h)*(1 - g/Sqrt[g^2 - 4*f*h])*Log[e*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] - (2*f*g*h
*Log[e*((a + b*x)/(c + d*x))^n]*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x])/Sqrt[g^2 - 4*f*h] + (g^2 - f*h)*(1 + g/Sqr
t[g^2 - 4*f*h])*Log[e*((a + b*x)/(c + d*x))^n]*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x] - (2*f*g*h*n*((Log[(2*h*(a +
b*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - Log[(2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sqrt[g^2 - 4*f*h])])*
Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*Sqrt[
g^2 - 4*f*h])] - PolyLog[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[
g^2 - 4*f*h] - ((g^2 - f*h)*(-g + Sqrt[g^2 - 4*f*h])*n*((Log[(2*h*(a + b*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*
f*h])] - Log[(2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sqrt[g^2 - 4*f*h])])*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + Pol
yLog[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - PolyLog[2, (d*(-g + Sqr
t[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h] + (2*f*g*h*n*((Log[(2*h*(a
+ b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - Log[(2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])*Log[g
+ Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(-2*a*h + b*(g + Sqrt[g^2 - 4*f
*h]))] - PolyLog[2, (d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(-2*c*h + d*(g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f
*h] - ((g^2 - f*h)*(g + Sqrt[g^2 - 4*f*h])*n*((Log[(2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - Log[
(2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(g +
Sqrt[g^2 - 4*f*h] + 2*h*x))/(-2*a*h + b*(g + Sqrt[g^2 - 4*f*h]))] - PolyLog[2, (d*(g + Sqrt[g^2 - 4*f*h] + 2*h
*x))/(-2*c*h + d*(g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h])/(2*h^3)

Maple [F]

\[\int \frac {x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}d x\]

[In]

int(x^3*ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)

[Out]

int(x^3*ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)

Fricas [F]

\[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]

[In]

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Timed out} \]

[In]

integrate(x**3*ln(e*((b*x+a)/(d*x+c))**n)/(h*x**2+g*x+f),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*f*h-g^2>0)', see `assume?` f
or more deta

Giac [F]

\[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]

[In]

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int \frac {x^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]

[In]

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f + g*x + h*x^2),x)

[Out]

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f + g*x + h*x^2), x)