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

Optimal. Leaf size=673 \[ -\frac{c x \left (2 a^2 e^2 h-c \left (2 a \left (d^2 h-2 d e g+e^2 f\right )+b d (d g+2 e f)\right )-a b e (2 d h+e g)+b^2 \left (d^2 h+e^2 f\right )+2 c^2 d^2 f\right )+b \left (a^2 e^2 h-a c \left (d^2 (-h)-2 d e g+3 e^2 f\right )+c^2 d^2 f\right )-b^2 e (a e g+2 c d f)+2 a c (a e (e g-2 d h)+c d (2 e f-d g))+b^3 e^2 f}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-6 c^2 e \left (2 a^2 e \left (2 d^2 h-2 d e g+e^2 f\right )+4 a b d e^2 f-b^2 d^3 g\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-6 a b^2 e \left (2 d^2 h-d e g+2 e^2 f\right )+b^3 (-d) \left (-2 d^2 h-3 d e g+4 e^2 f\right )\right )-b^3 e^3 (2 a d h-a e g-b d g+2 b e f)-2 c^3 d^2 \left (b d (d g+4 e f)-2 a \left (d^2 h-2 d e g+6 e^2 f\right )\right )+4 c^4 d^4 f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}+\frac{e \log \left (a+b x+c x^2\right ) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (d^2 h-d e g+e^2 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \log (d+e x) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.55858, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1646, 1628, 634, 618, 206, 628} \[ -\frac{c x \left (2 a^2 e^2 h-c \left (2 a \left (d^2 h-2 d e g+e^2 f\right )+b d (d g+2 e f)\right )-a b e (2 d h+e g)+b^2 \left (d^2 h+e^2 f\right )+2 c^2 d^2 f\right )+b \left (a^2 e^2 h-a c \left (d^2 (-h)-2 d e g+3 e^2 f\right )+c^2 d^2 f\right )-b^2 e (a e g+2 c d f)+2 a c (a e (e g-2 d h)+c d (2 e f-d g))+b^3 e^2 f}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-6 c^2 e \left (2 a^2 e \left (2 d^2 h-2 d e g+e^2 f\right )+4 a b d e^2 f-b^2 d^3 g\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-6 a b^2 e \left (2 d^2 h-d e g+2 e^2 f\right )+b^3 (-d) \left (-2 d^2 h-3 d e g+4 e^2 f\right )\right )-b^3 e^3 (2 a d h-a e g-b d g+2 b e f)-2 c^3 d^2 \left (b d (d g+4 e f)-2 a \left (d^2 h-2 d e g+6 e^2 f\right )\right )+4 c^4 d^4 f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}+\frac{e \log \left (a+b x+c x^2\right ) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (d^2 h-d e g+e^2 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \log (d+e x) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -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{f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx &=-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{\frac{2 c^3 d^4 f-b^2 e^2 \left (a e^2 f-b d (2 e f-d g)-a d^2 h\right )-c^2 d^2 \left (b d (2 e f+d g)-2 a \left (5 e^2 f-2 d e g+d^2 h\right )\right )-c e \left (2 b^2 d^2 (e f-d g)-2 a^2 e \left (2 e^2 f-d^2 h\right )+a b d \left (8 e^2 f-3 d e g+2 d^2 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^2}+\frac{e \left (4 c^3 d^3 f-c \left (2 a b e^2 (2 e f+d g)-4 a^2 e^2 (e g-d h)-b^2 d^2 (e g+2 d h)\right )-b^2 e \left (a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )\right )-2 c^2 d \left (b d (2 e f+d g)-2 a \left (e^2 f+d e g-d^2 h\right )\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^2}+\frac{c e^2 \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x^2}{\left (c d^2-b d e+a e^2\right )^2}}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{-b^2+4 a c}\\ &=-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \left (-\frac{\left (b^2-4 a c\right ) e^2 \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\left (b^2-4 a c\right ) e^2 \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{2 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )+c^2 e \left (3 b^2 d^3 g-2 a b d \left (10 e^2 f-3 d e g+2 d^2 h\right )-6 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )+c e^2 \left (b^3 d (4 e f-3 d g)+2 a^3 e^2 h-a^2 b e (5 e g-4 d h)+a b^2 \left (10 e^2 f-5 d e g+6 d^2 h\right )\right )-c \left (b^2-4 a c\right ) e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=-\frac{e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{\int \frac{2 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )+c^2 e \left (3 b^2 d^3 g-2 a b d \left (10 e^2 f-3 d e g+2 d^2 h\right )-6 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )+c e^2 \left (b^3 d (4 e f-3 d g)+2 a^3 e^2 h-a^2 b e (5 e g-4 d h)+a b^2 \left (10 e^2 f-5 d e g+6 d^2 h\right )\right )-c \left (b^2-4 a c\right ) e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right )\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 )^3}-\frac{\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\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 ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 2.47566, size = 650, normalized size = 0.97 \[ \frac{b \left (-a^2 e^2 h+a c \left (d^2 (-h)-2 d e (g-h x)+e^2 (3 f+g x)\right )+c^2 d (-d f+d g x+2 e f x)\right )+2 c \left (a^2 (-e) (e (g+h x)-2 d h)+a c \left (d^2 (g+h x)-2 d e (f+g x)+e^2 f x\right )-c^2 d^2 f x\right )+b^2 \left (a e^2 g-c \left (d^2 h x-2 d e f+e^2 f x\right )\right )+b^3 \left (-e^2\right ) f}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}-\frac{\tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-6 c^2 e \left (2 a^2 e \left (2 d^2 h-2 d e g+e^2 f\right )+4 a b d e^2 f-b^2 d^3 g\right )+c e \left (-6 a^2 b e^3 g+4 a^3 e^3 h+6 a b^2 e \left (2 d^2 h-d e g+2 e^2 f\right )+b^3 d \left (-2 d^2 h-3 d e g+4 e^2 f\right )\right )+b^3 e^3 (-2 a d h+a e g+b d g-2 b e f)-2 c^3 d^2 \left (b d (d g+4 e f)-2 a \left (d^2 h-2 d e g+6 e^2 f\right )\right )+4 c^4 d^4 f\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (b d-a e)-c d^2\right )^3}-\frac{e \left (d^2 h-d e g+e^2 f\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{\log (d+e x) \left (e^3 (-2 a d h+a e g+b d g-2 b e f)+c d e \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{\left (e (a e-b d)+c d^2\right )^3}-\frac{\log (a+x (b+c x)) \left (e^3 (-2 a d h+a e g+b d g-2 b e f)+c d e \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{2 \left (e (a e-b d)+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*(e^2*f - d*e*g + d^2*h))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))) + (-(b^3*e^2*f) + b^2*(a*e^2*g - c*(-2
*d*e*f + e^2*f*x + d^2*h*x)) + b*(-(a^2*e^2*h) + c^2*d*(-(d*f) + 2*e*f*x + d*g*x) + a*c*(-(d^2*h) + e^2*(3*f +
g*x) - 2*d*e*(g - h*x))) + 2*c*(-(c^2*d^2*f*x) + a*c*(e^2*f*x - 2*d*e*(f + g*x) + d^2*(g + h*x)) - a^2*e*(-2*
d*h + e*(g + h*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - ((4*c^4*d^4*f + b^3*e^3*
(-2*b*e*f + b*d*g + a*e*g - 2*a*d*h) - 2*c^3*d^2*(b*d*(4*e*f + d*g) - 2*a*(6*e^2*f - 2*d*e*g + d^2*h)) - 6*c^2
*e*(4*a*b*d*e^2*f - b^2*d^3*g + 2*a^2*e*(e^2*f - 2*d*e*g + 2*d^2*h)) + c*e*(-6*a^2*b*e^3*g + 4*a^3*e^3*h + b^3
*d*(4*e^2*f - 3*d*e*g - 2*d^2*h) + 6*a*b^2*e*(2*e^2*f - d*e*g + 2*d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*
c]])/((-b^2 + 4*a*c)^(3/2)*(-(c*d^2) + e*(b*d - a*e))^3) + ((e^3*(-2*b*e*f + b*d*g + a*e*g - 2*a*d*h) + c*d*e*
(4*e^2*f - 3*d*e*g + 2*d^2*h))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - ((e^3*(-2*b*e*f + b*d*g + a*e*g -
2*a*d*h) + c*d*e*(4*e^2*f - 3*d*e*g + 2*d^2*h))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)

________________________________________________________________________________________

Maple [B]  time = 0.215, size = 4716, normalized size = 7. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*f+4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*a*b*e^4*f+1/(a*e^2-
b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b^2*d^4*h-8/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+
a)*a*d*e^3*f-24/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^2*d*e^3*f-2/
(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*a*b*d*e^3*g+12/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*c*d^2*e^2*h-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*a*b^2*c*d*e^3*g+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*c^2*d^3*e*g-6/(a*e^2-
b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*c^2*d^2*e^2*f+3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^
2)*x*b^2*d^2*e^2*f+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b^2*d^3*e*g-3/2/(a*e^2-b*d*e+c*d^2)
^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d^2*e^2*g+1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d^3*e
*h-4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*a*d^3*e*h+6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c^2*l
n(c*x^2+b*x+a)*a*d^2*e^2*g-8/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^3
*d^3*e*g+4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*a^2*d*e^3*h+6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)
^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^2*d^3*e*g-24/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((
2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^2*d^2*e^2*h+24/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*a*c^3*d^2*e^2*f-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b
^3*c*d^3*e*h-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d^2*e^2*g+4/(
a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d*e^3*f+2/(a*e^2-b*d*e+c*d^2)
^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d*e^3*f-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b^2*d*e^3*h
+24/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^2*d*e^3*g-2/(a*e^2-b*d*e
+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*d*e^3*h+12/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-
b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*c*e^4*f-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*c*e^4*g-8/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b
^2)^(1/2))*b*c^3*d^3*e*f-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*c^3*d^4*g+1/(a*e^2-b*d*e+c*d^2)^3
/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a*b^2*e^4*f-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a^2*b*e^4*g+4
/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a^2*d*e^3*g-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*
a*c-b^2)*x*b^3*d*e^3*f-6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a*b*d^2*e^2*g-4/(a*e^2-b*d*e+c*
d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a^2*b*d*e^3*h+3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a*b^2
*d^2*e^2*h+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a*b^2*d*e^3*g-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-
b^2)*c*ln(c*x^2+b*x+a)*a^2*e^4*g+1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
*a*b^3*e^4*g+4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^3*d^4*h-1/(a*e^
2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b^2*e^4*g+1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*b^4*d*e^3*g+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*e^4*f-4/(a*e^2-b*
d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*b*d^3*e*f+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b
*c*d^2*e^2*h-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*c*d^3*e*h-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+
b*x+a)/(4*a*c-b^2)*a*b^2*c*d^2*e^2*g-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*b*c^3*d^4*g+1/2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b^2*e^4*g+4/(a*e^2-b*d*e+c*d^2)^3/(4*
a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*c*e^4*h-12/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^2*e^4*f-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*d*e^3*f+1/
(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b*c^3*d^4*f+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*
a^3*c*e^4*g+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^4/(4*a*c-b^2)*x*d^4*f+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)/(4*a*c-b^2)*a^3*b*e^4*h-e/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*d^2*h+e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*d*g+e^4/(a*e
^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*g-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*f+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)/(4*a*c-b^2)*a*b*c^2*d^4*h+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*c^3*d^3*e*f+3/(a*e^2-b*d*e+c*d
^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^3*c*d^2*e^2*f-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c^2*d^3*
e*f+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a^3*e^4*h-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/
(4*a*c-b^2)*x*a^2*e^4*f-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*b*d^4*g-4/(a*e^2-b*d*e+c*d^2)^
3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^3*c*d*e^3*h-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b^2*d*e^3*h-3/
(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*c*e^4*f-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2
)*a^2*c^2*d^3*e*h+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*c^2*d*e^3*f+1/(a*e^2-b*d*e+c*d^2)^3/(c
*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*d*e^3*g-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*a*d^4*h+1/(a*e^2
-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*c*d*e^3*f-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*
x*b^3*d^3*e*h+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*a*d^3*e*g+1/2/(a*e^2-b*d*e+c*d^2)^3/(4*a
*c-b^2)*ln(c*x^2+b*x+a)*b^3*d*e^3*g-3*e^2/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c*d^2*g+4*e^3/(a*e^2-b*d*e+c*d^2)^3*
ln(e*x+d)*c*d*f-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3*e^4*f+4/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b
^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^4*f-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(3/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*b^4*e^4*f-2*e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*d*h+e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x
+d)*b*d*g+2*e/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c*d^3*h

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

[Out]

Timed out

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Giac [B]  time = 1.42511, size = 1940, normalized size = 2.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*c^4*d^4*f*e^2 - 2*b*c^3*d^4*g*e^2 + 4*a*c^3*d^4*h*e^2 - 8*b*c^3*d^3*f*e^3 + 6*b^2*c^2*d^3*g*e^3 - 8*a*c^3*
d^3*g*e^3 - 2*b^3*c*d^3*h*e^3 + 24*a*c^3*d^2*f*e^4 - 3*b^3*c*d^2*g*e^4 + 12*a*b^2*c*d^2*h*e^4 - 24*a^2*c^2*d^2
*h*e^4 + 4*b^3*c*d*f*e^5 - 24*a*b*c^2*d*f*e^5 + b^4*d*g*e^5 - 6*a*b^2*c*d*g*e^5 + 24*a^2*c^2*d*g*e^5 - 2*a*b^3
*d*h*e^5 - 2*b^4*f*e^6 + 12*a*b^2*c*f*e^6 - 12*a^2*c^2*f*e^6 + a*b^3*g*e^6 - 6*a^2*b*c*g*e^6 + 4*a^3*c*h*e^6)*
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^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2*d^4*e^2
- 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^2*e^4 - 9*a^2*b^2*c*
d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^2 + 4*a
*c)) - 1/2*(2*c*d^3*h*e - 3*c*d^2*g*e^2 + 4*c*d*f*e^3 + b*d*g*e^3 - 2*a*d*h*e^3 - 2*b*f*e^4 + a*g*e^4)*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^3*d^6 - 3*b*c
^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e
^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (d^2*h*e^5/(x*e + d) - d*g*e^6/(x*e + d) + f*e^7/(x*e + d))/(c^2*d^4*e^4 - 2*b
*c*d^3*e^5 + b^2*d^2*e^6 + 2*a*c*d^2*e^6 - 2*a*b*d*e^7 + a^2*e^8) - ((2*c^4*d^3*f*e - b*c^3*d^3*g*e + b^2*c^2*
d^3*h*e - 2*a*c^3*d^3*h*e - 3*b*c^3*d^2*f*e^2 + 6*a*c^3*d^2*g*e^2 - 3*a*b*c^2*d^2*h*e^2 + 3*b^2*c^2*d*f*e^3 -
6*a*c^3*d*f*e^3 - 3*a*b*c^2*d*g*e^3 + 6*a^2*c^2*d*h*e^3 - b^3*c*f*e^4 + 3*a*b*c^2*f*e^4 + a*b^2*c*g*e^4 - 2*a^
2*c^2*g*e^4 - a^2*b*c*h*e^4)/(c*d^2 - b*d*e + a*e^2) - (2*c^4*d^4*f*e^2 - b*c^3*d^4*g*e^2 + b^2*c^2*d^4*h*e^2
- 2*a*c^3*d^4*h*e^2 - 4*b*c^3*d^3*f*e^3 + 8*a*c^3*d^3*g*e^3 - 4*a*b*c^2*d^3*h*e^3 + 6*b^2*c^2*d^2*f*e^4 - 12*a
*c^3*d^2*f*e^4 - 6*a*b*c^2*d^2*g*e^4 + 12*a^2*c^2*d^2*h*e^4 - 4*b^3*c*d*f*e^5 + 12*a*b*c^2*d*f*e^5 + 4*a*b^2*c
*d*g*e^5 - 8*a^2*c^2*d*g*e^5 - 4*a^2*b*c*d*h*e^5 + b^4*f*e^6 - 4*a*b^2*c*f*e^6 + 2*a^2*c^2*f*e^6 - a*b^3*g*e^6
+ 3*a^2*b*c*g*e^6 + a^2*b^2*h*e^6 - 2*a^3*c*h*e^6)*e^(-1)/((c*d^2 - b*d*e + a*e^2)*(x*e + d)))/((c*d^2 - b*d*
e + a*e^2)^2*(b^2 - 4*a*c)*(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))