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

Optimal. Leaf size=1046 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.71701, antiderivative size = 1046, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 14, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {2513, 2418, 2389, 2295, 2395, 43, 2394, 2393, 2391, 701, 634, 618, 206, 628} \[ -\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 ) \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-\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 \text{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 \text{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 \text{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 \text{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} \]

Antiderivative was successfully verified.

[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 2513

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]]

Rule 2418

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 2389

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 2295

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

Rule 2395

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 43

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 2394

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 2393

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 2391

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 701

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 634

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 618

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 206

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

Rule 628

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]

Rubi steps

\begin{align*} \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 &=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) \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{b h^2}+\frac{(g n) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 1.37733, size = 1240, normalized size = 1.19 \[ \frac{x^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) h^2+\frac{n \left (b \left (b \log (c+d x) c^2+d (a d-b c) x\right )-a^2 d^2 \log (a+b x)\right ) h^2}{b^2 d^2}-\frac{2 g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) h}{b}+\frac{2 (b c-a d) g n \log (c+d x) h}{b d}+\frac{2 f g \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right ) h}{\sqrt{g^2-4 f h}}-\frac{2 f g \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x+\sqrt{g^2-4 f h}\right ) h}{\sqrt{g^2-4 f h}}-\frac{2 f g n \left (\left (\log \left (\frac{2 h (a+b x)}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\log \left (\frac{2 h (c+d x)}{-g d+\sqrt{g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{2 c h+d \left (\sqrt{g^2-4 f h}-g\right )}\right )\right ) h}{\sqrt{g^2-4 f h}}+\frac{2 f g 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+2 h x+\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{b \left (g+\sqrt{g^2-4 f h}\right )-2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{d \left (g+\sqrt{g^2-4 f h}\right )-2 c h}\right )\right ) h}{\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+2 h x-\sqrt{g^2-4 f h}\right )+\left (g^2-f h\right ) \left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x+\sqrt{g^2-4 f h}\right )-\frac{\left (g^2-f h\right ) \left (\sqrt{g^2-4 f h}-g\right ) n \left (\left (\log \left (\frac{2 h (a+b x)}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\log \left (\frac{2 h (c+d x)}{-g d+\sqrt{g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{2 c h+d \left (\sqrt{g^2-4 f h}-g\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+2 h x+\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{b \left (g+\sqrt{g^2-4 f h}\right )-2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{d \left (g+\sqrt{g^2-4 f h}\right )-2 c h}\right )\right )}{\sqrt{g^2-4 f h}}}{2 h^3} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 1.511, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{h{x}^{2}+gx+f}\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)