3.2377 \(\int \frac{f+g x}{(d+e x)^2 (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=1043 \[ \frac{(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x) e^4}{\left (c d^2-b e d+a e^2\right )^4}-\frac{(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (c x^2+b x+a\right ) e^4}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac{\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e}{\left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}-\frac{\left (12 c^6 f d^6+2 c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4+10 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (-d (6 e f-5 d g) b^2-10 a e (3 e f-2 d g) b+10 a^2 e^2 g\right )-10 a b c^2 e^4 \left (-d (6 e f-5 d g) b^2-3 a e (3 e f-2 d g) b+3 a^2 e^2 g\right )-10 c^3 e^2 \left (2 b^3 g d^4-8 a b^2 e g d^3+3 a^2 b e^2 (6 e f-d g) d+6 a^3 e^3 (e f-2 d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b e d+a e^2\right )^4}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 (d+e x) \left (c x^2+b x+a\right )}-\frac{-e f b^2+c d f b+a e g b+2 a c e f-2 a c d g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2} \]

[Out]

(e*(6*c^4*d^4*f - b^3*e^3*(3*b*e*f - 2*b*d*g - a*e*g) - b*c*e^2*(7*a^2*e^2*g - a*b*e*(21*e*f - 13*d*g) - 3*b^2
*d*(e*f - d*g)) + c^3*d^2*(4*a*e*(6*e*f - d*g) - 3*b*d*(4*e*f + d*g)) - c^2*e*(2*a^2*e^2*(15*e*f - 22*d*g) + 6
*a*b*d*e*(4*e*f + d*g) - b^2*d^2*(3*e*f + 7*d*g))))/((b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (b
*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x)/(2*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^2) - (4*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e*f +
d*g)) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - b*e*(3*b*e*f - 2*b*d*g - a*e*g) + c*(2*a*e*(5*e*f - 2*d*g) -
b*d*(2*e*f + 3*d*g))) - c*(12*c^3*d^3*f + b^2*e^2*(3*b*e*f - 2*b*d*g - a*e*g) + 2*c^2*d*(2*a*e*(9*e*f - 2*d*g
) - 3*b*d*(3*e*f + d*g)) + c*e*(11*b^2*d^2*g + 16*a^2*e^2*g - 2*a*b*e*(9*e*f + 5*d*g)))*x)/(2*(b^2 - 4*a*c)^2*
(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)) - ((12*c^6*d^6*f + b^5*e^5*(3*b*e*f - 2*b*d*g - a*e*g)
+ b^3*c*e^4*(10*a^2*e^2*g - b^2*d*(6*e*f - 5*d*g) - 10*a*b*e*(3*e*f - 2*d*g)) - 10*a*b*c^2*e^4*(3*a^2*e^2*g -
b^2*d*(6*e*f - 5*d*g) - 3*a*b*e*(3*e*f - 2*d*g)) - 10*c^3*e^2*(2*b^3*d^4*g - 8*a*b^2*d^3*e*g + 6*a^3*e^3*(e*f
- 2*d*g) + 3*a^2*b*d*e^2*(6*e*f - d*g)) + 2*c^5*d^4*(2*a*e*(15*e*f - 2*d*g) - 3*b*d*(6*e*f + d*g)) + 10*c^4*d^
2*e*(2*a^2*e^2*(9*e*f - 4*d*g) - a*b*d*e*(12*e*f + d*g) + b^2*d^2*(3*e*f + 2*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^4) + (e^4*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d
*g - a*e*g))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^4*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*
g))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

________________________________________________________________________________________

Rubi [A]  time = 6.29436, antiderivative size = 1043, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {822, 800, 634, 618, 206, 628} \[ \frac{(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x) e^4}{\left (c d^2-b e d+a e^2\right )^4}-\frac{(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (c x^2+b x+a\right ) e^4}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac{\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e}{\left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}-\frac{\left (12 c^6 f d^6+2 c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4+10 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (-d (6 e f-5 d g) b^2-10 a e (3 e f-2 d g) b+10 a^2 e^2 g\right )-10 a b c^2 e^4 \left (-d (6 e f-5 d g) b^2-3 a e (3 e f-2 d g) b+3 a^2 e^2 g\right )-10 c^3 e^2 \left (2 b^3 g d^4-8 a b^2 e g d^3+3 a^2 b e^2 (6 e f-d g) d+6 a^3 e^3 (e f-2 d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b e d+a e^2\right )^4}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 (d+e x) \left (c x^2+b x+a\right )}-\frac{-e f b^2+c d f b+a e g b+2 a c e f-2 a c d g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(6*c^4*d^4*f - b^3*e^3*(3*b*e*f - 2*b*d*g - a*e*g) - b*c*e^2*(7*a^2*e^2*g - a*b*e*(21*e*f - 13*d*g) - 3*b^2
*d*(e*f - d*g)) + c^3*d^2*(4*a*e*(6*e*f - d*g) - 3*b*d*(4*e*f + d*g)) - c^2*e*(2*a^2*e^2*(15*e*f - 22*d*g) + 6
*a*b*d*e*(4*e*f + d*g) - b^2*d^2*(3*e*f + 7*d*g))))/((b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (b
*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x)/(2*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^2) - (4*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e*f +
d*g)) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - b*e*(3*b*e*f - 2*b*d*g - a*e*g) + c*(2*a*e*(5*e*f - 2*d*g) -
b*d*(2*e*f + 3*d*g))) - c*(12*c^3*d^3*f + b^2*e^2*(3*b*e*f - 2*b*d*g - a*e*g) + 2*c^2*d*(2*a*e*(9*e*f - 2*d*g
) - 3*b*d*(3*e*f + d*g)) + c*e*(11*b^2*d^2*g + 16*a^2*e^2*g - 2*a*b*e*(9*e*f + 5*d*g)))*x)/(2*(b^2 - 4*a*c)^2*
(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)) - ((12*c^6*d^6*f + b^5*e^5*(3*b*e*f - 2*b*d*g - a*e*g)
+ b^3*c*e^4*(10*a^2*e^2*g - b^2*d*(6*e*f - 5*d*g) - 10*a*b*e*(3*e*f - 2*d*g)) - 10*a*b*c^2*e^4*(3*a^2*e^2*g -
b^2*d*(6*e*f - 5*d*g) - 3*a*b*e*(3*e*f - 2*d*g)) - 10*c^3*e^2*(2*b^3*d^4*g - 8*a*b^2*d^3*e*g + 6*a^3*e^3*(e*f
- 2*d*g) + 3*a^2*b*d*e^2*(6*e*f - d*g)) + 2*c^5*d^4*(2*a*e*(15*e*f - 2*d*g) - 3*b*d*(6*e*f + d*g)) + 10*c^4*d^
2*e*(2*a^2*e^2*(9*e*f - 4*d*g) - a*b*d*e*(12*e*f + d*g) + b^2*d^2*(3*e*f + 2*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^4) + (e^4*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d
*g - a*e*g))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^4*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*
g))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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{f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))+4 c e (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}+\frac{\int \frac{-2 \left (2 c e (b d (c d-b e)+a e (4 c d-b e)) (2 c d f+2 a e g-b (e f+d g))-\frac{1}{2} \left (2 c^2 d^2-2 b^2 e^2+6 a c e^2\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )\right )+2 c e \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{2 e^2 \left (-6 c^4 d^4 f+b^3 e^3 (3 b e f-2 b d g-a e g)+b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )-c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))+c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{2 \left (b^2-4 a c\right )^2 e^5 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g))}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{2 \left (6 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)-c^3 e^2 \left (10 b^3 d^4 g-40 a b^2 d^3 e g+a^2 b d e^2 (138 e f-55 d g)+30 a^3 e^3 (e f-2 d g)\right )-a b c^2 e^4 \left (23 a^2 e^2 g-9 b^2 d (6 e f-5 d g)-23 a b e (3 e f-2 d g)\right )+b^3 c e^4 \left (9 a^2 e^2 g-b^2 d (6 e f-5 d g)-9 a b e (3 e f-2 d g)\right )+c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+5 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}+\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}+\frac{\int \frac{6 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)-c^3 e^2 \left (10 b^3 d^4 g-40 a b^2 d^3 e g+a^2 b d e^2 (138 e f-55 d g)+30 a^3 e^3 (e f-2 d g)\right )-a b c^2 e^4 \left (23 a^2 e^2 g-9 b^2 d (6 e f-5 d g)-23 a b e (3 e f-2 d g)\right )+b^3 c e^4 \left (9 a^2 e^2 g-b^2 d (6 e f-5 d g)-9 a b e (3 e f-2 d g)\right )+c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+5 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}+\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{\left (e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g))\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}+\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}-\frac{\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac{4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \left (a+b x+c x^2\right )}-\frac{\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^4}+\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}\\ \end{align*}

Mathematica [A]  time = 6.93197, size = 1409, normalized size = 1.35 \[ -\frac{(e f-d g) e^4}{\left (c d^2-b e d+a e^2\right )^3 (d+e x)}+\frac{\left (3 e^6 f b^6-2 d e^5 g b^6-6 c d e^5 f b^5-a e^6 g b^5+5 c d^2 e^4 g b^5-30 a c e^6 f b^4+20 a c d e^5 g b^4+60 a c^2 d e^5 f b^3+10 a^2 c e^6 g b^3-50 a c^2 d^2 e^4 g b^3-20 c^3 d^4 e^2 g b^3+90 a^2 c^2 e^6 f b^2+30 c^4 d^4 e^2 f b^2-60 a^2 c^2 d e^5 g b^2+80 a c^3 d^3 e^3 g b^2+20 c^4 d^5 e g b^2-180 a^2 c^3 d e^5 f b-120 a c^4 d^3 e^3 f b-36 c^5 d^5 e f b-6 c^5 d^6 g b-30 a^3 c^2 e^6 g b+30 a^2 c^3 d^2 e^4 g b-10 a c^4 d^4 e^2 g b+12 c^6 d^6 f-60 a^3 c^3 e^6 f+180 a^2 c^4 d^2 e^4 f+60 a c^5 d^4 e^2 f+120 a^3 c^3 d e^5 g-80 a^2 c^4 d^3 e^3 g-8 a c^5 d^5 e g\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (b^2-4 a c\right )^2 \sqrt{4 a c-b^2} \left (-c d^2+b e d-a e^2\right )^4}+\frac{\left (-3 b f e^6+a g e^6+6 c d f e^5+2 b d g e^5-5 c d^2 g e^4\right ) \log (d+e x)}{\left (c d^2-b e d+a e^2\right )^4}+\frac{\left (3 b f e^6-a g e^6-6 c d f e^5-2 b d g e^5+5 c d^2 g e^4\right ) \log \left (c x^2+b x+a\right )}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac{-4 e^4 f b^5+2 d e^3 g b^5+7 c d e^3 f b^4+2 a e^4 g b^4-6 c d^2 e^2 g b^4-4 c e^4 f x b^4+2 c d e^3 g x b^4+29 a c e^4 f b^3+3 c^2 d^2 e^2 f b^3-13 a c d e^3 g b^3+7 c^2 d^3 e g b^3+6 c^2 d e^3 f x b^3+2 a c e^4 g x b^3-6 c^2 d^2 e^2 g x b^3-56 a c^2 d e^3 f b^2-12 c^3 d^3 e f b^2-3 c^3 d^4 g b^2-15 a^2 c e^4 g b^2+18 a c^2 d^2 e^2 g b^2+26 a c^2 e^4 f x b^2+6 c^3 d^2 e^2 f x b^2-10 a c^2 d e^3 g x b^2+14 c^3 d^3 e g x b^2+6 c^4 d^4 f b-46 a^2 c^2 e^4 f b+24 a c^3 d^2 e^2 f b+44 a^2 c^2 d e^3 g b-4 a c^3 d^3 e g b-48 a c^3 d e^3 f x b-24 c^4 d^3 e f x b-6 c^4 d^4 g x b-14 a^2 c^2 e^4 g x b-12 a c^3 d^2 e^2 g x b+64 a^2 c^3 d e^3 f+16 a^3 c^2 e^4 g-48 a^2 c^3 d^2 e^2 g+12 c^5 d^4 f x-28 a^2 c^3 e^4 f x+48 a c^4 d^2 e^2 f x+56 a^2 c^3 d e^3 g x-8 a c^4 d^3 e g x}{2 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )}+\frac{e^2 f b^3-2 c d e f b^2-a e^2 g b^2+c e^2 f x b^2+c^2 d^2 f b-3 a c e^2 f b+2 a c d e g b-2 c^2 d e f x b-c^2 d^2 g x b-a c e^2 g x b+4 a c^2 d e f-2 a c^2 d^2 g+2 a^2 c e^2 g+2 c^3 d^2 f x-2 a c^2 e^2 f x+4 a c^2 d e g x}{2 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^4*(e*f - d*g))/((c*d^2 - b*d*e + a*e^2)^3*(d + e*x))) + (b*c^2*d^2*f - 2*b^2*c*d*e*f + 4*a*c^2*d*e*f + b^
3*e^2*f - 3*a*b*c*e^2*f - 2*a*c^2*d^2*g + 2*a*b*c*d*e*g - a*b^2*e^2*g + 2*a^2*c*e^2*g + 2*c^3*d^2*f*x - 2*b*c^
2*d*e*f*x + b^2*c*e^2*f*x - 2*a*c^2*e^2*f*x - b*c^2*d^2*g*x + 4*a*c^2*d*e*g*x - a*b*c*e^2*g*x)/(2*(-b^2 + 4*a*
c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^2) + (6*b*c^4*d^4*f - 12*b^2*c^3*d^3*e*f + 3*b^3*c^2*d^2*e^2*f
+ 24*a*b*c^3*d^2*e^2*f + 7*b^4*c*d*e^3*f - 56*a*b^2*c^2*d*e^3*f + 64*a^2*c^3*d*e^3*f - 4*b^5*e^4*f + 29*a*b^3*
c*e^4*f - 46*a^2*b*c^2*e^4*f - 3*b^2*c^3*d^4*g + 7*b^3*c^2*d^3*e*g - 4*a*b*c^3*d^3*e*g - 6*b^4*c*d^2*e^2*g + 1
8*a*b^2*c^2*d^2*e^2*g - 48*a^2*c^3*d^2*e^2*g + 2*b^5*d*e^3*g - 13*a*b^3*c*d*e^3*g + 44*a^2*b*c^2*d*e^3*g + 2*a
*b^4*e^4*g - 15*a^2*b^2*c*e^4*g + 16*a^3*c^2*e^4*g + 12*c^5*d^4*f*x - 24*b*c^4*d^3*e*f*x + 6*b^2*c^3*d^2*e^2*f
*x + 48*a*c^4*d^2*e^2*f*x + 6*b^3*c^2*d*e^3*f*x - 48*a*b*c^3*d*e^3*f*x - 4*b^4*c*e^4*f*x + 26*a*b^2*c^2*e^4*f*
x - 28*a^2*c^3*e^4*f*x - 6*b*c^4*d^4*g*x + 14*b^2*c^3*d^3*e*g*x - 8*a*c^4*d^3*e*g*x - 6*b^3*c^2*d^2*e^2*g*x -
12*a*b*c^3*d^2*e^2*g*x + 2*b^4*c*d*e^3*g*x - 10*a*b^2*c^2*d*e^3*g*x + 56*a^2*c^3*d*e^3*g*x + 2*a*b^3*c*e^4*g*x
- 14*a^2*b*c^2*e^4*g*x)/(2*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)) + ((12*c^6*d^6*f - 3
6*b*c^5*d^5*e*f + 30*b^2*c^4*d^4*e^2*f + 60*a*c^5*d^4*e^2*f - 120*a*b*c^4*d^3*e^3*f + 180*a^2*c^4*d^2*e^4*f -
6*b^5*c*d*e^5*f + 60*a*b^3*c^2*d*e^5*f - 180*a^2*b*c^3*d*e^5*f + 3*b^6*e^6*f - 30*a*b^4*c*e^6*f + 90*a^2*b^2*c
^2*e^6*f - 60*a^3*c^3*e^6*f - 6*b*c^5*d^6*g + 20*b^2*c^4*d^5*e*g - 8*a*c^5*d^5*e*g - 20*b^3*c^3*d^4*e^2*g - 10
*a*b*c^4*d^4*e^2*g + 80*a*b^2*c^3*d^3*e^3*g - 80*a^2*c^4*d^3*e^3*g + 5*b^5*c*d^2*e^4*g - 50*a*b^3*c^2*d^2*e^4*
g + 30*a^2*b*c^3*d^2*e^4*g - 2*b^6*d*e^5*g + 20*a*b^4*c*d*e^5*g - 60*a^2*b^2*c^2*d*e^5*g + 120*a^3*c^3*d*e^5*g
- a*b^5*e^6*g + 10*a^2*b^3*c*e^6*g - 30*a^3*b*c^2*e^6*g)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*
c)^2*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^4) + ((6*c*d*e^5*f - 3*b*e^6*f - 5*c*d^2*e^4*g + 2*b*d*e^5*
g + a*e^6*g)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 + ((-6*c*d*e^5*f + 3*b*e^6*f + 5*c*d^2*e^4*g - 2*b*d*e^5*
g - a*e^6*g)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 13679, normalized size = 13.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

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Giac [B]  time = 3.65167, size = 4613, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*c^6*d^6*f*e^2 - 6*b*c^5*d^6*g*e^2 - 36*b*c^5*d^5*f*e^3 + 20*b^2*c^4*d^5*g*e^3 - 8*a*c^5*d^5*g*e^3 + 30*b^2
*c^4*d^4*f*e^4 + 60*a*c^5*d^4*f*e^4 - 20*b^3*c^3*d^4*g*e^4 - 10*a*b*c^4*d^4*g*e^4 - 120*a*b*c^4*d^3*f*e^5 + 80
*a*b^2*c^3*d^3*g*e^5 - 80*a^2*c^4*d^3*g*e^5 + 180*a^2*c^4*d^2*f*e^6 + 5*b^5*c*d^2*g*e^6 - 50*a*b^3*c^2*d^2*g*e
^6 + 30*a^2*b*c^3*d^2*g*e^6 - 6*b^5*c*d*f*e^7 + 60*a*b^3*c^2*d*f*e^7 - 180*a^2*b*c^3*d*f*e^7 - 2*b^6*d*g*e^7 +
20*a*b^4*c*d*g*e^7 - 60*a^2*b^2*c^2*d*g*e^7 + 120*a^3*c^3*d*g*e^7 + 3*b^6*f*e^8 - 30*a*b^4*c*f*e^8 + 90*a^2*b
^2*c^2*f*e^8 - 60*a^3*c^3*f*e^8 - a*b^5*g*e^8 + 10*a^2*b^3*c*g*e^8 - 30*a^3*b*c^2*g*e^8)*arctan((2*c*d - 2*c*d
^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*a*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/((b^4*c^4*d^8 -
8*a*b^2*c^5*d^8 + 16*a^2*c^6*d^8 - 4*b^5*c^3*d^7*e + 32*a*b^3*c^4*d^7*e - 64*a^2*b*c^5*d^7*e + 6*b^6*c^2*d^6*e
^2 - 44*a*b^4*c^3*d^6*e^2 + 64*a^2*b^2*c^4*d^6*e^2 + 64*a^3*c^5*d^6*e^2 - 4*b^7*c*d^5*e^3 + 20*a*b^5*c^2*d^5*e
^3 + 32*a^2*b^3*c^3*d^5*e^3 - 192*a^3*b*c^4*d^5*e^3 + b^8*d^4*e^4 + 4*a*b^6*c*d^4*e^4 - 74*a^2*b^4*c^2*d^4*e^4
+ 144*a^3*b^2*c^3*d^4*e^4 + 96*a^4*c^4*d^4*e^4 - 4*a*b^7*d^3*e^5 + 20*a^2*b^5*c*d^3*e^5 + 32*a^3*b^3*c^2*d^3*
e^5 - 192*a^4*b*c^3*d^3*e^5 + 6*a^2*b^6*d^2*e^6 - 44*a^3*b^4*c*d^2*e^6 + 64*a^4*b^2*c^2*d^2*e^6 + 64*a^5*c^3*d
^2*e^6 - 4*a^3*b^5*d*e^7 + 32*a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 + a^4*b^4*e^8 - 8*a^5*b^2*c*e^8 + 16*a^6*c^
2*e^8)*sqrt(-b^2 + 4*a*c)) + 1/2*(5*c*d^2*g*e^4 - 6*c*d*f*e^5 - 2*b*d*g*e^5 + 3*b*f*e^6 - a*g*e^6)*log(c - 2*c
*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(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 - 12*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) + (d*g*e^10/(x*e + d) - f*e^11/(x*e + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*c*d^4*e^
8 + 3*a*c^2*d^4*e^8 - b^3*d^3*e^9 - 6*a*b*c*d^3*e^9 + 3*a*b^2*d^2*e^10 + 3*a^2*c*d^2*e^10 - 3*a^2*b*d*e^11 + a
^3*e^12) + 1/2*(12*c^7*d^5*f*e - 6*b*c^6*d^5*g*e - 30*b*c^6*d^4*f*e^2 + 17*b^2*c^5*d^4*g*e^2 - 8*a*c^6*d^4*g*e
^2 + 16*b^2*c^5*d^3*f*e^3 + 56*a*c^6*d^3*f*e^3 - 12*b^3*c^4*d^3*g*e^3 - 12*a*b*c^5*d^3*g*e^3 + 6*b^3*c^4*d^2*f
*e^4 - 84*a*b*c^5*d^2*f*e^4 + 8*b^4*c^3*d^2*g*e^4 - 34*a*b^2*c^4*d^2*g*e^4 + 128*a^2*c^5*d^2*g*e^4 - 14*b^4*c^
3*d*f*e^5 + 100*a*b^2*c^4*d*f*e^5 - 116*a^2*c^5*d*f*e^5 - 2*b^5*c^2*d*g*e^5 + 18*a*b^3*c^3*d*g*e^5 - 70*a^2*b*
c^4*d*g*e^5 + 5*b^5*c^2*f*e^6 - 36*a*b^3*c^3*f*e^6 + 58*a^2*b*c^4*f*e^6 - 3*a*b^4*c^2*g*e^6 + 21*a^2*b^2*c^3*g
*e^6 - 24*a^3*c^4*g*e^6 - 2*(18*c^7*d^6*f*e^2 - 9*b*c^6*d^6*g*e^2 - 54*b*c^6*d^5*f*e^3 + 30*b^2*c^5*d^5*g*e^3
- 12*a*c^6*d^5*g*e^3 + 47*b^2*c^5*d^4*f*e^4 + 82*a*c^6*d^4*f*e^4 - 31*b^3*c^4*d^4*g*e^4 - 11*a*b*c^5*d^4*g*e^4
- 4*b^3*c^4*d^3*f*e^5 - 164*a*b*c^5*d^3*f*e^5 + 24*b^4*c^3*d^3*g*e^5 - 64*a*b^2*c^4*d^3*g*e^5 + 232*a^2*c^5*d
^3*g*e^5 - 29*b^4*c^3*d^2*f*e^6 + 244*a*b^2*c^4*d^2*f*e^6 - 242*a^2*c^5*d^2*f*e^6 - 11*b^5*c^2*d^2*g*e^6 + 67*
a*b^3*c^3*d^2*g*e^6 - 227*a^2*b*c^4*d^2*g*e^6 + 22*b^5*c^2*d*f*e^7 - 162*a*b^3*c^3*d*f*e^7 + 242*a^2*b*c^4*d*f
*e^7 + 2*b^6*c*d*g*e^7 - 24*a*b^4*c^2*d*g*e^7 + 110*a^2*b^2*c^3*d*g*e^7 - 76*a^3*c^4*d*g*e^7 - 5*b^6*c*f*e^8 +
38*a*b^4*c^2*f*e^8 - 71*a^2*b^2*c^3*f*e^8 + 14*a^3*c^4*f*e^8 + 3*a*b^5*c*g*e^8 - 22*a^2*b^3*c^2*g*e^8 + 31*a^
3*b*c^3*g*e^8)*e^(-1)/(x*e + d) + (36*c^7*d^7*f*e^3 - 18*b*c^6*d^7*g*e^3 - 126*b*c^6*d^6*f*e^4 + 69*b^2*c^5*d^
6*g*e^4 - 24*a*c^6*d^6*g*e^4 + 144*b^2*c^5*d^5*f*e^5 + 180*a*c^6*d^5*f*e^5 - 90*b^3*c^4*d^5*g*e^5 - 18*a*b*c^5
*d^5*g*e^5 - 45*b^3*c^4*d^4*f*e^6 - 450*a*b*c^5*d^4*f*e^6 + 80*b^4*c^3*d^4*g*e^6 - 145*a*b^2*c^4*d^4*g*e^6 + 5
60*a^2*c^5*d^4*g*e^6 - 70*b^4*c^3*d^3*f*e^7 + 740*a*b^2*c^4*d^3*f*e^7 - 580*a^2*c^5*d^3*f*e^7 - 50*b^5*c^2*d^3
*g*e^7 + 250*a*b^3*c^3*d^3*g*e^7 - 830*a^2*b*c^4*d^3*g*e^7 + 87*b^5*c^2*d^2*f*e^8 - 660*a*b^3*c^3*d^2*f*e^8 +
870*a^2*b*c^4*d^2*f*e^8 + 16*b^6*c*d^2*g*e^8 - 129*a*b^4*c^2*d^2*g*e^8 + 471*a^2*b^2*c^3*d^2*g*e^8 - 88*a^3*c^
4*d^2*g*e^8 - 36*b^6*c*d*f*e^9 + 258*a*b^4*c^2*d*f*e^9 - 372*a^2*b^2*c^3*d*f*e^9 - 84*a^3*c^4*d*f*e^9 - 2*b^7*
d*g*e^9 + 32*a*b^5*c*d*g*e^9 - 160*a^2*b^3*c^2*d*g*e^9 + 130*a^3*b*c^3*d*g*e^9 + 5*b^7*f*e^10 - 34*a*b^5*c*f*e
^10 + 41*a^2*b^3*c^2*f*e^10 + 42*a^3*b*c^3*f*e^10 - 3*a*b^6*g*e^10 + 19*a^2*b^4*c*g*e^10 - 11*a^3*b^2*c^2*g*e^
10 - 32*a^4*c^3*g*e^10)*e^(-2)/(x*e + d)^2 - 2*(6*c^7*d^8*f*e^4 - 3*b*c^6*d^8*g*e^4 - 24*b*c^6*d^7*f*e^5 + 13*
b^2*c^5*d^7*g*e^5 - 4*a*c^6*d^7*g*e^5 + 33*b^2*c^5*d^6*f*e^6 + 36*a*c^6*d^6*f*e^6 - 20*b^3*c^4*d^6*g*e^6 - 4*a
*b*c^5*d^6*g*e^6 - 15*b^3*c^4*d^5*f*e^7 - 108*a*b*c^5*d^5*f*e^7 + 20*b^4*c^3*d^5*g*e^7 - 25*a*b^2*c^4*d^5*g*e^
7 + 116*a^2*c^5*d^5*g*e^7 - 15*b^4*c^3*d^4*f*e^8 + 195*a*b^2*c^4*d^4*f*e^8 - 120*a^2*c^5*d^4*f*e^8 - 15*b^5*c^
2*d^4*g*e^8 + 65*a*b^3*c^3*d^4*g*e^8 - 230*a^2*b*c^4*d^4*g*e^8 + 27*b^5*c^2*d^3*f*e^9 - 210*a*b^3*c^3*d^3*f*e^
9 + 240*a^2*b*c^4*d^3*f*e^9 + 6*b^6*c*d^3*g*e^9 - 39*a*b^4*c^2*d^3*g*e^9 + 131*a^2*b^2*c^3*d^3*g*e^9 + 52*a^3*
c^4*d^3*g*e^9 - 15*b^6*c*d^2*f*e^10 + 99*a*b^4*c^2*d^2*f*e^10 - 81*a^2*b^2*c^3*d^2*f*e^10 - 132*a^3*c^4*d^2*f*
e^10 - b^7*d^2*g*e^10 + 11*a*b^5*c*d^2*g*e^10 - 46*a^2*b^3*c^2*d^2*g*e^10 - 12*a^3*b*c^3*d^2*g*e^10 + 3*b^7*d*
f*e^11 - 12*a*b^5*c*d*f*e^11 - 39*a^2*b^3*c^2*d*f*e^11 + 132*a^3*b*c^3*d*f*e^11 - a*b^6*d*g*e^11 + a^2*b^4*c*d
*g*e^11 + 41*a^3*b^2*c^2*d*g*e^11 - 68*a^4*c^3*d*g*e^11 - 3*a*b^6*f*e^12 + 24*a^2*b^4*c*f*e^12 - 51*a^3*b^2*c^
2*f*e^12 + 18*a^4*c^3*f*e^12 + 2*a^2*b^5*g*e^12 - 15*a^3*b^3*c*g*e^12 + 25*a^4*b*c^2*g*e^12)*e^(-3)/(x*e + d)^
3)/((c*d^2 - b*d*e + a*e^2)^4*(b^2 - 4*a*c)^2*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e
/(x*e + d)^2 + a*e^2/(x*e + d)^2)^2)