3.91 \(\int \frac{\log (d (a+b x+c x^2)^n)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=519 \[ \frac{n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 e \left (a e^2-b d e+c d^2\right )^4}-\frac{n \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )}{4 e \left (a e^2-b d e+c d^2\right )^4}+\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{4 e (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{8 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 \left (a e^2-b d e+c d^2\right )^4}+\frac{n (2 c d-b e)}{12 e (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4} \]

[Out]

((2*c*d - b*e)*n)/(12*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n)/(
8*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n)/(4*e*(c
*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)
)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*(c*d^2 - b*d*e + a*e^2)^4) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^
3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[d + e*x])/(4*e*
(c*d^2 - b*d*e + a*e^2)^4) + ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c
^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[a + b*x + c*x^2])/(8*e*(c*d^2 - b*d*e + a*e^2)^4) - Log[d*(a +
b*x + c*x^2)^n]/(4*e*(d + e*x)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.00557, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac{n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 e \left (a e^2-b d e+c d^2\right )^4}-\frac{n \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )}{4 e \left (a e^2-b d e+c d^2\right )^4}+\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{4 e (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{8 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 \left (a e^2-b d e+c d^2\right )^4}+\frac{n (2 c d-b e)}{12 e (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^5,x]

[Out]

((2*c*d - b*e)*n)/(12*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n)/(
8*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n)/(4*e*(c
*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)
)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*(c*d^2 - b*d*e + a*e^2)^4) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^
3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[d + e*x])/(4*e*
(c*d^2 - b*d*e + a*e^2)^4) + ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c
^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[a + b*x + c*x^2])/(8*e*(c*d^2 - b*d*e + a*e^2)^4) - Log[d*(a +
b*x + c*x^2)^n]/(4*e*(d + e*x)^4)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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{\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac{n \int \frac{b+2 c x}{(d+e x)^4 \left (a+b x+c x^2\right )} \, dx}{4 e}\\ &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac{n \int \left (\frac{e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{e \left (-2 c^2 d^2-b^2 e^2+2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac{e (2 c d-b e) \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{e \left (-2 c^4 d^4-b^4 e^4+4 b^2 c e^3 (b d+a e)+4 c^3 d^2 e (b d+3 a e)-2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right )}{\left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac{-4 b^4 c d e^3+b^5 e^4+b^3 c e^2 \left (6 c d^2-5 a e^2\right )-4 b^2 c^2 d e \left (c d^2-4 a e^2\right )+8 a c^3 d e \left (c d^2-a e^2\right )+b c^2 \left (c^2 d^4-18 a c d^2 e^2+5 a^2 e^4\right )+c \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}\right ) \, dx}{4 e}\\ &=\frac{(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac{n \int \frac{-4 b^4 c d e^3+b^5 e^4+b^3 c e^2 \left (6 c d^2-5 a e^2\right )-4 b^2 c^2 d e \left (c d^2-4 a e^2\right )+8 a c^3 d e \left (c d^2-a e^2\right )+b c^2 \left (c^2 d^4-18 a c d^2 e^2+5 a^2 e^4\right )+c \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{4 e \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{8 \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{8 e \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 e \left (c d^2-b d e+a e^2\right )^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\sqrt{b^2-4 a c} (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 \left (c d^2-b d e+a e^2\right )^4}-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 e \left (c d^2-b d e+a e^2\right )^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 2.06492, size = 469, normalized size = 0.9 \[ \frac{\frac{n (d+e x) \left (-6 (d+e x)^3 \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )+3 (d+e x)^3 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log (a+x (b+c x))+3 (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+6 (d+e x)^2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )+6 e \sqrt{b^2-4 a c} (d+e x)^3 (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3\right )}{\left (e (a e-b d)+c d^2\right )^4}-6 \log \left (d (a+x (b+c x))^n\right )}{24 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^5,x]

[Out]

((n*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3 + 3*(c*d^2 + e*(-(b*d) + a*e))^2*(2*c^2*d^2 + b^2*
e^2 - 2*c*e*(b*d + a*e))*(d + e*x) + 6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*d
+ 3*a*e))*(d + e*x)^2 + 6*Sqrt[b^2 - 4*a*c]*e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x
)^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 6*(2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b
*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^3*Log[d + e*x] + 3*(2*c^4*d^4 + b^4*e^4 -
4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^
3*Log[a + x*(b + c*x)]))/(c*d^2 + e*(-(b*d) + a*e))^4 - 6*Log[d*(a + x*(b + c*x))^n])/(24*e*(d + e*x)^4)

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Maple [B]  time = 0.443, size = 1137077, normalized size = 2190.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x)

[Out]

result too large to display

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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(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="fricas")

[Out]

Timed out

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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(ln(d*(c*x**2+b*x+a)**n)/(e*x+d)**5,x)

[Out]

Timed out

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Giac [B]  time = 2.58396, size = 5075, normalized size = 9.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="giac")

[Out]

1/8*(2*c^4*d^4*n - 4*b*c^3*d^3*n*e + 6*b^2*c^2*d^2*n*e^2 - 12*a*c^3*d^2*n*e^2 - 4*b^3*c*d*n*e^3 + 12*a*b*c^2*d
*n*e^3 + b^4*n*e^4 - 4*a*b^2*c*n*e^4 + 2*a^2*c^2*n*e^4)*log(c*x^2 + b*x + a)/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*
b^2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*c*d^4*e^5 +
6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8
+ a^4*e^9) - 1/4*(4*b^2*c^3*d^3*n - 16*a*c^4*d^3*n - 6*b^3*c^2*d^2*n*e + 24*a*b*c^3*d^2*n*e + 4*b^4*c*d*n*e^2
- 20*a*b^2*c^2*d*n*e^2 + 16*a^2*c^3*d*n*e^2 - b^5*n*e^3 + 6*a*b^3*c*n*e^3 - 8*a^2*b*c^2*n*e^3)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 1
2*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^
5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-b^2 + 4*a*c)) - 1/24*(12*c^4*d^4*n*x^
4*e^4*log(x*e + d) + 48*c^4*d^5*n*x^3*e^3*log(x*e + d) + 72*c^4*d^6*n*x^2*e^2*log(x*e + d) + 48*c^4*d^7*n*x*e*
log(x*e + d) - 12*c^4*d^5*n*x^3*e^3 - 42*c^4*d^6*n*x^2*e^2 - 52*c^4*d^7*n*x*e + 6*c^4*d^8*n*log(c*x^2 + b*x +
a) - 24*b*c^3*d^7*n*e*log(c*x^2 + b*x + a) + 12*c^4*d^8*n*log(x*e + d) - 24*b*c^3*d^3*n*x^4*e^5*log(x*e + d) -
96*b*c^3*d^4*n*x^3*e^4*log(x*e + d) - 144*b*c^3*d^5*n*x^2*e^3*log(x*e + d) - 96*b*c^3*d^6*n*x*e^2*log(x*e + d
) - 24*b*c^3*d^7*n*e*log(x*e + d) - 22*c^4*d^8*n + 30*b*c^3*d^4*n*x^3*e^4 + 108*b*c^3*d^5*n*x^2*e^3 + 140*b*c^
3*d^6*n*x*e^2 + 62*b*c^3*d^7*n*e + 36*b^2*c^2*d^6*n*e^2*log(c*x^2 + b*x + a) + 24*a*c^3*d^6*n*e^2*log(c*x^2 +
b*x + a) + 36*b^2*c^2*d^2*n*x^4*e^6*log(x*e + d) - 72*a*c^3*d^2*n*x^4*e^6*log(x*e + d) + 144*b^2*c^2*d^3*n*x^3
*e^5*log(x*e + d) - 288*a*c^3*d^3*n*x^3*e^5*log(x*e + d) + 216*b^2*c^2*d^4*n*x^2*e^4*log(x*e + d) - 432*a*c^3*
d^4*n*x^2*e^4*log(x*e + d) + 144*b^2*c^2*d^5*n*x*e^3*log(x*e + d) - 288*a*c^3*d^5*n*x*e^3*log(x*e + d) + 36*b^
2*c^2*d^6*n*e^2*log(x*e + d) - 72*a*c^3*d^6*n*e^2*log(x*e + d) + 6*c^4*d^8*log(d) - 24*b*c^3*d^7*e*log(d) - 36
*b^2*c^2*d^3*n*x^3*e^5 + 24*a*c^3*d^3*n*x^3*e^5 - 129*b^2*c^2*d^4*n*x^2*e^4 + 66*a*c^3*d^4*n*x^2*e^4 - 168*b^2
*c^2*d^5*n*x*e^3 + 48*a*c^3*d^5*n*x*e^3 - 75*b^2*c^2*d^6*n*e^2 + 6*a*c^3*d^6*n*e^2 - 24*b^3*c*d^5*n*e^3*log(c*
x^2 + b*x + a) - 72*a*b*c^2*d^5*n*e^3*log(c*x^2 + b*x + a) - 24*b^3*c*d*n*x^4*e^7*log(x*e + d) + 72*a*b*c^2*d*
n*x^4*e^7*log(x*e + d) - 96*b^3*c*d^2*n*x^3*e^6*log(x*e + d) + 288*a*b*c^2*d^2*n*x^3*e^6*log(x*e + d) - 144*b^
3*c*d^3*n*x^2*e^5*log(x*e + d) + 432*a*b*c^2*d^3*n*x^2*e^5*log(x*e + d) - 96*b^3*c*d^4*n*x*e^4*log(x*e + d) +
288*a*b*c^2*d^4*n*x*e^4*log(x*e + d) - 24*b^3*c*d^5*n*e^3*log(x*e + d) + 72*a*b*c^2*d^5*n*e^3*log(x*e + d) + 3
6*b^2*c^2*d^6*e^2*log(d) + 24*a*c^3*d^6*e^2*log(d) + 24*b^3*c*d^2*n*x^3*e^6 - 36*a*b*c^2*d^2*n*x^3*e^6 + 84*b^
3*c*d^3*n*x^2*e^5 - 96*a*b*c^2*d^3*n*x^2*e^5 + 106*b^3*c*d^4*n*x*e^4 - 54*a*b*c^2*d^4*n*x*e^4 + 46*b^3*c*d^5*n
*e^3 + 6*a*b*c^2*d^5*n*e^3 + 6*b^4*d^4*n*e^4*log(c*x^2 + b*x + a) + 72*a*b^2*c*d^4*n*e^4*log(c*x^2 + b*x + a)
+ 36*a^2*c^2*d^4*n*e^4*log(c*x^2 + b*x + a) + 6*b^4*n*x^4*e^8*log(x*e + d) - 24*a*b^2*c*n*x^4*e^8*log(x*e + d)
+ 12*a^2*c^2*n*x^4*e^8*log(x*e + d) + 24*b^4*d*n*x^3*e^7*log(x*e + d) - 96*a*b^2*c*d*n*x^3*e^7*log(x*e + d) +
48*a^2*c^2*d*n*x^3*e^7*log(x*e + d) + 36*b^4*d^2*n*x^2*e^6*log(x*e + d) - 144*a*b^2*c*d^2*n*x^2*e^6*log(x*e +
d) + 72*a^2*c^2*d^2*n*x^2*e^6*log(x*e + d) + 24*b^4*d^3*n*x*e^5*log(x*e + d) - 96*a*b^2*c*d^3*n*x*e^5*log(x*e
+ d) + 48*a^2*c^2*d^3*n*x*e^5*log(x*e + d) + 6*b^4*d^4*n*e^4*log(x*e + d) - 24*a*b^2*c*d^4*n*e^4*log(x*e + d)
+ 12*a^2*c^2*d^4*n*e^4*log(x*e + d) - 24*b^3*c*d^5*e^3*log(d) - 72*a*b*c^2*d^5*e^3*log(d) - 6*b^4*d*n*x^3*e^7
+ 36*a^2*c^2*d*n*x^3*e^7 - 21*b^4*d^2*n*x^2*e^6 - 12*a*b^2*c*d^2*n*x^2*e^6 + 114*a^2*c^2*d^2*n*x^2*e^6 - 26*b
^4*d^3*n*x*e^5 - 48*a*b^2*c*d^3*n*x*e^5 + 108*a^2*c^2*d^3*n*x*e^5 - 11*b^4*d^4*n*e^4 - 36*a*b^2*c*d^4*n*e^4 +
30*a^2*c^2*d^4*n*e^4 - 24*a*b^3*d^3*n*e^5*log(c*x^2 + b*x + a) - 72*a^2*b*c*d^3*n*e^5*log(c*x^2 + b*x + a) + 6
*b^4*d^4*e^4*log(d) + 72*a*b^2*c*d^4*e^4*log(d) + 36*a^2*c^2*d^4*e^4*log(d) + 6*a*b^3*n*x^3*e^8 - 18*a^2*b*c*n
*x^3*e^8 + 24*a*b^3*d*n*x^2*e^7 - 60*a^2*b*c*d*n*x^2*e^7 + 36*a*b^3*d^2*n*x*e^6 - 48*a^2*b*c*d^2*n*x*e^6 + 18*
a*b^3*d^3*n*e^5 - 6*a^2*b*c*d^3*n*e^5 + 36*a^2*b^2*d^2*n*e^6*log(c*x^2 + b*x + a) + 24*a^3*c*d^2*n*e^6*log(c*x
^2 + b*x + a) - 24*a*b^3*d^3*e^5*log(d) - 72*a^2*b*c*d^3*e^5*log(d) - 3*a^2*b^2*n*x^2*e^8 + 6*a^3*c*n*x^2*e^8
- 12*a^2*b^2*d*n*x*e^7 + 8*a^3*c*d*n*x*e^7 - 9*a^2*b^2*d^2*n*e^6 + 2*a^3*c*d^2*n*e^6 - 24*a^3*b*d*n*e^7*log(c*
x^2 + b*x + a) + 36*a^2*b^2*d^2*e^6*log(d) + 24*a^3*c*d^2*e^6*log(d) + 2*a^3*b*n*x*e^8 + 2*a^3*b*d*n*e^7 + 6*a
^4*n*e^8*log(c*x^2 + b*x + a) - 24*a^3*b*d*e^7*log(d) + 6*a^4*e^8*log(d))/(c^4*d^8*x^4*e^5 + 4*c^4*d^9*x^3*e^4
+ 6*c^4*d^10*x^2*e^3 + 4*c^4*d^11*x*e^2 + c^4*d^12*e - 4*b*c^3*d^7*x^4*e^6 - 16*b*c^3*d^8*x^3*e^5 - 24*b*c^3*
d^9*x^2*e^4 - 16*b*c^3*d^10*x*e^3 - 4*b*c^3*d^11*e^2 + 6*b^2*c^2*d^6*x^4*e^7 + 4*a*c^3*d^6*x^4*e^7 + 24*b^2*c^
2*d^7*x^3*e^6 + 16*a*c^3*d^7*x^3*e^6 + 36*b^2*c^2*d^8*x^2*e^5 + 24*a*c^3*d^8*x^2*e^5 + 24*b^2*c^2*d^9*x*e^4 +
16*a*c^3*d^9*x*e^4 + 6*b^2*c^2*d^10*e^3 + 4*a*c^3*d^10*e^3 - 4*b^3*c*d^5*x^4*e^8 - 12*a*b*c^2*d^5*x^4*e^8 - 16
*b^3*c*d^6*x^3*e^7 - 48*a*b*c^2*d^6*x^3*e^7 - 24*b^3*c*d^7*x^2*e^6 - 72*a*b*c^2*d^7*x^2*e^6 - 16*b^3*c*d^8*x*e
^5 - 48*a*b*c^2*d^8*x*e^5 - 4*b^3*c*d^9*e^4 - 12*a*b*c^2*d^9*e^4 + b^4*d^4*x^4*e^9 + 12*a*b^2*c*d^4*x^4*e^9 +
6*a^2*c^2*d^4*x^4*e^9 + 4*b^4*d^5*x^3*e^8 + 48*a*b^2*c*d^5*x^3*e^8 + 24*a^2*c^2*d^5*x^3*e^8 + 6*b^4*d^6*x^2*e^
7 + 72*a*b^2*c*d^6*x^2*e^7 + 36*a^2*c^2*d^6*x^2*e^7 + 4*b^4*d^7*x*e^6 + 48*a*b^2*c*d^7*x*e^6 + 24*a^2*c^2*d^7*
x*e^6 + b^4*d^8*e^5 + 12*a*b^2*c*d^8*e^5 + 6*a^2*c^2*d^8*e^5 - 4*a*b^3*d^3*x^4*e^10 - 12*a^2*b*c*d^3*x^4*e^10
- 16*a*b^3*d^4*x^3*e^9 - 48*a^2*b*c*d^4*x^3*e^9 - 24*a*b^3*d^5*x^2*e^8 - 72*a^2*b*c*d^5*x^2*e^8 - 16*a*b^3*d^6
*x*e^7 - 48*a^2*b*c*d^6*x*e^7 - 4*a*b^3*d^7*e^6 - 12*a^2*b*c*d^7*e^6 + 6*a^2*b^2*d^2*x^4*e^11 + 4*a^3*c*d^2*x^
4*e^11 + 24*a^2*b^2*d^3*x^3*e^10 + 16*a^3*c*d^3*x^3*e^10 + 36*a^2*b^2*d^4*x^2*e^9 + 24*a^3*c*d^4*x^2*e^9 + 24*
a^2*b^2*d^5*x*e^8 + 16*a^3*c*d^5*x*e^8 + 6*a^2*b^2*d^6*e^7 + 4*a^3*c*d^6*e^7 - 4*a^3*b*d*x^4*e^12 - 16*a^3*b*d
^2*x^3*e^11 - 24*a^3*b*d^3*x^2*e^10 - 16*a^3*b*d^4*x*e^9 - 4*a^3*b*d^5*e^8 + a^4*x^4*e^13 + 4*a^4*d*x^3*e^12 +
6*a^4*d^2*x^2*e^11 + 4*a^4*d^3*x*e^10 + a^4*d^4*e^9)