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

Optimal. Leaf size=509 \[ -\frac {\log \left (a+b x+c x^2\right ) \left (e^3 \left (a^2 h-a b g+b^2 f\right )-c \left (a e \left (3 d^2 h-3 d e g+e^2 f\right )+b \left (3 d e^2 f-d^3 h\right )\right )+c^2 d^2 (3 e f-d g)\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {\log (d+e x) \left (e^3 \left (a^2 h-a b g+b^2 f\right )-c \left (a e \left (3 d^2 h-3 d e g+e^2 f\right )+b \left (3 d e^2 f-d^3 h\right )\right )+c^2 d^2 (3 e f-d g)\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c \left (2 a^2 e^2 (e g-3 d h)-3 a b e \left (d^2 (-h)-d e g+e^2 f\right )-\left (b^2 \left (d^3 h+3 d e^2 f\right )\right )\right )-b e^3 \left (a^2 h-a b g+b^2 f\right )-c^2 d \left (2 a \left (d^2 h-3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )+2 c^3 d^3 f\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {d^2 h-d e g+e^2 f}{2 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )+c d (2 e f-d g)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.25, antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1628, 634, 618, 206, 628} \[ -\frac {\log \left (a+b x+c x^2\right ) \left (e^3 \left (a^2 h-a b g+b^2 f\right )-a c e \left (3 d^2 h-3 d e g+e^2 f\right )-b c \left (3 d e^2 f-d^3 h\right )+c^2 d^2 (3 e f-d g)\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {\log (d+e x) \left (e^3 \left (a^2 h-a b g+b^2 f\right )-a c e \left (3 d^2 h-3 d e g+e^2 f\right )-b c \left (3 d e^2 f-d^3 h\right )+c^2 d^2 (3 e f-d g)\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c \left (2 a^2 e^2 (e g-3 d h)-3 a b e \left (d^2 (-h)-d e g+e^2 f\right )+b^2 \left (-\left (d^3 h+3 d e^2 f\right )\right )\right )-b e^3 \left (a^2 h-a b g+b^2 f\right )-c^2 d \left (2 a \left (d^2 h-3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )+2 c^3 d^3 f\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {d^2 h-d e g+e^2 f}{2 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )+c d (2 e f-d g)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {f+g x+h x^2}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e^2 f-d e g+d^2 h}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {e \left (c d (2 e f-d g)+a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {e \left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {c^3 d^3 f-b e^3 \left (b^2 f-a b g+a^2 h\right )+c e^2 \left (3 b^2 d f+a b (2 e f-3 d g)-a^2 (e g-3 d h)\right )-c^2 d \left (3 b d e f+a \left (3 e^2 f-3 d e g+d^2 h\right )\right )-c \left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 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\\ &=-\frac {e^2 f-d e g+d^2 h}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {c d (2 e f-d g)+a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {c^3 d^3 f-b e^3 \left (b^2 f-a b g+a^2 h\right )+c e^2 \left (3 b^2 d f+a b (2 e f-3 d g)-a^2 (e g-3 d h)\right )-c^2 d \left (3 b d e f+a \left (3 e^2 f-3 d e g+d^2 h\right )\right )-c \left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right ) x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e^2 f-d e g+d^2 h}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {c d (2 e f-d g)+a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\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 (2 c^3 d^3 f-b e^3 \left (b^2 f-a b g+a^2 h\right )-c^2 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f-3 d e g+d^2 h\right )\right )-c \left (2 a^2 e^2 (e g-3 d h)-3 a b e \left (e^2 f-d e g-d^2 h\right )-b^2 \left (3 d e^2 f+d^3 h\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e^2 f-d e g+d^2 h}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {c d (2 e f-d g)+a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 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 (2 c^3 d^3 f-b e^3 \left (b^2 f-a b g+a^2 h\right )-c^2 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f-3 d e g+d^2 h\right )\right )-c \left (2 a^2 e^2 (e g-3 d h)-3 a b e \left (e^2 f-d e g-d^2 h\right )-b^2 \left (3 d e^2 f+d^3 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 (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {e^2 f-d e g+d^2 h}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {c d (2 e f-d g)+a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {\left (2 c^3 d^3 f-b e^3 \left (b^2 f-a b g+a^2 h\right )-c^2 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f-3 d e g+d^2 h\right )\right )-c \left (2 a^2 e^2 (e g-3 d h)-3 a b e \left (e^2 f-d e g-d^2 h\right )-b^2 \left (3 d e^2 f+d^3 h\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c^2 d^2 (3 e f-d g)+e^3 \left (b^2 f-a b g+a^2 h\right )-a c e \left (e^2 f-3 d e g+3 d^2 h\right )-b c \left (3 d e^2 f-d^3 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*}

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Mathematica [A]  time = 0.77, size = 504, normalized size = 0.99 \[ -\frac {\log (d+e x) \left (-\left (e^3 \left (a^2 h-a b g+b^2 f\right )\right )+a c e \left (3 d^2 h-3 d e g+e^2 f\right )+b c \left (3 d e^2 f-d^3 h\right )+c^2 d^2 (d g-3 e f)\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac {\log (a+x (b+c x)) \left (-\left (e^3 \left (a^2 h-a b g+b^2 f\right )\right )+a c e \left (3 d^2 h-3 d e g+e^2 f\right )+b c \left (3 d e^2 f-d^3 h\right )+c^2 d^2 (d g-3 e f)\right )}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac {\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (-c \left (-2 a^2 e^2 (e g-3 d h)+3 a b e \left (d^2 (-h)-d e g+e^2 f\right )+b^2 \left (d^3 h+3 d e^2 f\right )\right )+b e^3 \left (a^2 h-a b g+b^2 f\right )+c^2 d \left (2 a \left (d^2 h-3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )-2 c^3 d^3 f\right )}{\sqrt {4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}-\frac {d^2 h-d e g+e^2 f}{2 e (d+e x)^2 \left (e (a e-b d)+c d^2\right )}+\frac {a e (2 d h-e g)+b \left (e^2 f-d^2 h\right )+c d (d g-2 e f)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(e^2*f - d*e*g + d^2*h)/(e*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (c*d*(-2*e*f + d*g) + a*e*(-(e*g) +
2*d*h) + b*(e^2*f - d^2*h))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + ((-2*c^3*d^3*f + b*e^3*(b^2*f - a*b*g +
a^2*h) + c^2*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f - 3*d*e*g + d^2*h)) - c*(-2*a^2*e^2*(e*g - 3*d*h) + 3*a*b*e*
(e^2*f - d*e*g - d^2*h) + b^2*(3*d*e^2*f + d^3*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c
]*(-(c*d^2) + e*(b*d - a*e))^3) - ((c^2*d^2*(-3*e*f + d*g) - e^3*(b^2*f - a*b*g + a^2*h) + a*c*e*(e^2*f - 3*d*
e*g + 3*d^2*h) + b*c*(3*d*e^2*f - d^3*h))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + ((c^2*d^2*(-3*e*f + d*g
) - e^3*(b^2*f - a*b*g + a^2*h) + a*c*e*(e^2*f - 3*d*e*g + 3*d^2*h) + b*c*(3*d*e^2*f - d^3*h))*Log[a + x*(b +
c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.21, size = 1002, normalized size = 1.97 \[ \frac {{\left (c^{2} d^{3} g - b c d^{3} h - 3 \, c^{2} d^{2} f e + 3 \, a c d^{2} h e + 3 \, b c d f e^{2} - 3 \, a c d g e^{2} - b^{2} f e^{3} + a c f e^{3} + a b g e^{3} - a^{2} h e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (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}\right )}} - \frac {{\left (c^{2} d^{3} g e - b c d^{3} h e - 3 \, c^{2} d^{2} f e^{2} + 3 \, a c d^{2} h e^{2} + 3 \, b c d f e^{3} - 3 \, a c d g e^{3} - b^{2} f e^{4} + a c f e^{4} + a b g e^{4} - a^{2} h e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac {{\left (2 \, c^{3} d^{3} f - b c^{2} d^{3} g + b^{2} c d^{3} h - 2 \, a c^{2} d^{3} h - 3 \, b c^{2} d^{2} f e + 6 \, a c^{2} d^{2} g e - 3 \, a b c d^{2} h e + 3 \, b^{2} c d f e^{2} - 6 \, a c^{2} d f e^{2} - 3 \, a b c d g e^{2} + 6 \, a^{2} c d h e^{2} - b^{3} f e^{3} + 3 \, a b c f e^{3} + a b^{2} g e^{3} - 2 \, a^{2} c g e^{3} - a^{2} b h e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (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}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (c^{2} d^{6} h - 3 \, c^{2} d^{5} g e + 5 \, c^{2} d^{4} f e^{2} + 4 \, b c d^{4} g e^{2} - b^{2} d^{4} h e^{2} - 2 \, a c d^{4} h e^{2} - 8 \, b c d^{3} f e^{3} - b^{2} d^{3} g e^{3} - 2 \, a c d^{3} g e^{3} + 4 \, a b d^{3} h e^{3} + 3 \, b^{2} d^{2} f e^{4} + 6 \, a c d^{2} f e^{4} - 3 \, a^{2} d^{2} h e^{4} - 4 \, a b d f e^{5} + a^{2} d g e^{5} + a^{2} f e^{6} - 2 \, {\left (c^{2} d^{4} g e^{2} - b c d^{4} h e^{2} - 2 \, c^{2} d^{3} f e^{3} - b c d^{3} g e^{3} + b^{2} d^{3} h e^{3} + 2 \, a c d^{3} h e^{3} + 3 \, b c d^{2} f e^{4} - 3 \, a b d^{2} h e^{4} - b^{2} d f e^{5} - 2 \, a c d f e^{5} + a b d g e^{5} + 2 \, a^{2} d h e^{5} + a b f e^{6} - a^{2} g e^{6}\right )} x\right )} e^{\left (-1\right )}}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 1945, normalized size = 3.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^2*e*f+6/(a*e^2-b*d*e+c*
d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c*d*e^2*h+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c*e^3*f+6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x
+b)/(4*a*c-b^2)^(1/2))*a*c^2*d^2*e*g-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*a*c^2*d*e^2*f+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^2*f
-1/2/(a*e^2-b*d*e+c*d^2)*e/(e*x+d)^2*f+1/2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2*d*g+1/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^
2+b*x+a)*g*e^3*b*a+1/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*f*e^3*a-1/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*
x+a)*b*d^3*h-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c*d*e^2*g-3/(a*
e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c*d^2*e*h+1/(a*e^2-b*d*e+c*d^2)^3
*ln(e*x+d)*a^2*e^3*h+1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b^2*e^3*f-1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^2*d^3*g-1
/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*a*e^2*g-1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*b*d^2*h+1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)
*b*e^2*f+1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*c*g*d^2-3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*d*g*e^2*a+3/2/(a*
e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*d*f*e^2*b-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*a^2*b*e^3*h-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c
*e^3*g+1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*e^3*g-2/(a*e^2-b*d*
e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^2*d^3*h+1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d^3*h-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^3*g+3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*a*d^2*e*h-3/(a*e^2-b*d*e+c*d^
2)^3*ln(e*x+d)*a*c*d^2*e*h+3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*c*d*e^2*g-3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*c
*d*e^2*f-3/2/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*d^2*f*e-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3*f+2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2
)^(1/2))*c^3*d^3*f-1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*b*e^3*g-1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*c*e^3*f+1/(
a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*c*d^3*h+3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^2*d^2*f*e+2/(a*e^2-b*d*e+c*d^2)^2
/(e*x+d)*a*d*e*h-2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*c*d*e*f+1/2/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*g*d^3-1
/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*a^2*e^3*h-1/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*f*e^3*b^2-1/2/(a*
e^2-b*d*e+c*d^2)/e/(e*x+d)^2*d^2*h

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.82, size = 12784, normalized size = 25.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(root(24*a^6*b*c*d*e^11*z^3 + 24*a*b*c^6*d^11*e*z^3 + 240*a^4*b*c^3*d^5*e^7*z^3 + 240*a^3*b*c^4*d^7*
e^5*z^3 + 120*a^5*b*c^2*d^3*e^9*z^3 + 120*a^2*b*c^5*d^9*e^3*z^3 - 54*a^5*b^2*c*d^2*e^10*z^3 - 54*a*b^2*c^5*d^1
0*e^2*z^3 + 50*a^4*b^3*c*d^3*e^9*z^3 + 50*a*b^3*c^4*d^9*e^3*z^3 - 36*a^2*b^5*c*d^5*e^7*z^3 - 36*a*b^5*c^2*d^7*
e^5*z^3 + 26*a*b^6*c*d^6*e^6*z^3 - 340*a^3*b^2*c^3*d^6*e^6*z^3 - 225*a^4*b^2*c^2*d^4*e^8*z^3 - 225*a^2*b^2*c^4
*d^8*e^4*z^3 + 180*a^3*b^3*c^2*d^5*e^7*z^3 + 180*a^2*b^3*c^3*d^7*e^5*z^3 - 30*a^2*b^4*c^2*d^6*e^6*z^3 - 6*b^7*
c*d^7*e^5*z^3 - 6*b^3*c^5*d^11*e*z^3 - 6*a^5*b^3*d*e^11*z^3 - 6*a*b^7*d^5*e^7*z^3 - 20*b^5*c^3*d^9*e^3*z^3 + 1
5*b^6*c^2*d^8*e^4*z^3 + 15*b^4*c^4*d^10*e^2*z^3 - 80*a^4*c^4*d^6*e^6*z^3 - 60*a^5*c^3*d^4*e^8*z^3 - 60*a^3*c^5
*d^8*e^4*z^3 - 24*a^6*c^2*d^2*e^10*z^3 - 24*a^2*c^6*d^10*e^2*z^3 - 20*a^3*b^5*d^3*e^9*z^3 + 15*a^4*b^4*d^2*e^1
0*z^3 + 15*a^2*b^6*d^4*e^8*z^3 - 4*a^7*c*e^12*z^3 - 4*a*c^7*d^12*z^3 + b^8*d^6*e^6*z^3 + b^2*c^6*d^12*z^3 + a^
6*b^2*e^12*z^3 - 9*a^3*b^2*c*d*e^5*g*h*z - 9*a*b^2*c^3*d^5*e*g*h*z - 30*a^3*b*c^2*d*e^5*f*h*z + 9*a^2*b^3*c*d*
e^5*f*h*z + 3*a*b^4*c*d^2*e^4*f*h*z + 27*a*b*c^4*d^4*e^2*f*g*z + 6*a^2*b^2*c^2*d^3*e^3*g*h*z - 33*a^2*b^2*c^2*
d^2*e^4*f*h*z + 18*a*b*c^4*d^5*e*f*h*z - 12*a*b^4*c*d*e^5*f*g*z + 27*a^3*b*c^2*d^2*e^4*g*h*z + 27*a^2*b*c^3*d^
4*e^2*g*h*z - 3*a^2*b^3*c*d^2*e^4*g*h*z - 3*a*b^3*c^2*d^4*e^2*g*h*z + 52*a^2*b*c^3*d^3*e^3*f*h*z - 4*a*b^3*c^2
*d^3*e^3*f*h*z - 3*a*b^2*c^3*d^4*e^2*f*h*z - 93*a^2*b*c^3*d^2*e^4*f*g*z + 51*a^2*b^2*c^2*d*e^5*f*g*z - 34*a*b^
2*c^3*d^3*e^3*f*g*z + 27*a*b^3*c^2*d^2*e^4*f*g*z - 24*a*c^5*d^5*e*f*g*z - 7*a^4*b*c*e^6*g*h*z - 7*a*b*c^4*d^6*
g*h*z + a*b^4*c*d^3*e^3*g*h*z - 80*a^3*c^3*d^3*e^3*g*h*z + 3*b^4*c^2*d^4*e^2*f*h*z - 66*a^2*c^4*d^4*e^2*f*h*z
+ 54*a^3*c^3*d^2*e^4*f*h*z - 3*b^3*c^3*d^4*e^2*f*g*z + 80*a^2*c^4*d^3*e^3*f*g*z - 21*a^2*b*c^3*d^5*e*h^2*z + 6
*a*b^3*c^2*d^5*e*h^2*z - 21*a^3*b*c^2*d*e^5*g^2*z + 6*a^2*b^3*c*d*e^5*g^2*z - 66*a*b*c^4*d^3*e^3*f^2*z - 30*a*
b^3*c^2*d*e^5*f^2*z + 27*a^2*b*c^3*d*e^5*f^2*z - 12*a^2*b^2*c^2*d^4*e^2*h^2*z - 12*a^2*b^2*c^2*d^2*e^4*g^2*z +
24*a^4*c^2*d*e^5*g*h*z + 24*a^2*c^4*d^5*e*g*h*z - 3*b^3*c^3*d^5*e*f*h*z - b^5*c*d^3*e^3*f*h*z + 3*b^2*c^4*d^5
*e*f*g*z - 24*a^3*c^3*d*e^5*f*g*z + 9*a^3*b^2*c*e^6*f*h*z - 10*a^2*b^3*c*e^6*f*g*z + 9*a^3*b*c^2*e^6*f*g*z + 3
*a^4*b*c*d*e^5*h^2*z + 3*a*b*c^4*d^5*e*g^2*z + 14*a^3*b*c^2*d^3*e^3*h^2*z + 3*a^3*b^2*c*d^2*e^4*h^2*z - a^2*b^
3*c*d^3*e^3*h^2*z + 14*a^2*b*c^3*d^3*e^3*g^2*z + 3*a*b^2*c^3*d^4*e^2*g^2*z - a*b^3*c^2*d^3*e^3*g^2*z + 63*a*b^
2*c^3*d^2*e^4*f^2*z + 2*b^3*c^3*d^6*g*h*z - 6*a^4*c^2*e^6*f*h*z + 2*a^3*b^3*e^6*g*h*z - b^2*c^4*d^6*f*h*z - 2*
a^2*b^4*e^6*f*h*z + 6*b^5*c*d*e^5*f^2*z + 3*b*c^5*d^5*e*f^2*z + 6*a*b^4*c*e^6*f^2*z + b^4*c^2*d^3*e^3*f*g*z +
33*a^3*c^3*d^4*e^2*h^2*z - 27*a^4*c^2*d^2*e^4*h^2*z + 33*a^3*c^3*d^2*e^4*g^2*z - 27*a^2*c^4*d^4*e^2*g^2*z + 19
*b^3*c^3*d^3*e^3*f^2*z - 15*b^4*c^2*d^2*e^4*f^2*z - 12*b^2*c^4*d^4*e^2*f^2*z - 27*a^2*c^4*d^2*e^4*f^2*z - 9*a^
2*b^2*c^2*e^6*f^2*z + 2*a*c^5*d^6*f*h*z + 2*a*b^5*e^6*f*g*z + 33*a*c^5*d^4*e^2*f^2*z + 4*a^3*b^2*c*e^6*g^2*z +
4*a*b^2*c^3*d^6*h^2*z - b^4*c^2*d^6*h^2*z - b^2*c^4*d^6*g^2*z - a^4*c^2*e^6*g^2*z - a^4*b^2*e^6*h^2*z - a^2*c
^4*d^6*h^2*z + 3*a^3*c^3*e^6*f^2*z - a^2*b^4*e^6*g^2*z + b*c^5*d^6*f*g*z + 3*a^5*c*e^6*h^2*z + 3*a*c^5*d^6*g^2
*z - c^6*d^6*f^2*z - b^6*e^6*f^2*z + 6*a*b^2*c^2*d*e^2*f*g*h - 2*a*b^3*c*e^3*f*g*h + 3*a^2*b*c^2*d^2*e*g*h^2 -
3*a^2*b*c^2*d*e^2*g^2*h - 3*a^2*b*c^2*d*e^2*f*h^2 - 3*a*b^2*c^2*d^2*e*f*h^2 - 6*a^2*c^3*d*e^2*f*g*h + 2*a^2*b
*c^2*e^3*f*g*h + 6*a*b*c^3*d*e^2*f^2*h - 6*a*b*c^3*d*e^2*f*g^2 - 2*b^2*c^3*d^3*f*g*h - 9*a*c^4*d^2*e*f^2*h - 3
*b*c^4*d^2*e*f^2*g + 3*a*c^4*d^2*e*f*g^2 + 3*a*c^4*d*e^2*f^2*g - 2*a^3*b*c*e^3*g*h^2 + 2*a*b*c^3*d^3*g^2*h - 2
*a*b*c^3*d^3*f*h^2 + 2*a*c^4*d^3*f*g*h - 3*b^3*c^2*d*e^2*f^2*h + 3*b^2*c^3*d^2*e*f^2*h + 3*a^3*c^2*d*e^2*g*h^2
- 3*a^2*c^3*d^2*e*g^2*h + 9*a^2*c^3*d^2*e*f*h^2 + 3*b^2*c^3*d*e^2*f^2*g - 3*a*b^2*c^2*e^3*f^2*h + 2*a^2*b^2*c
*e^3*f*h^2 - a*b^2*c^2*d^3*g*h^2 + 2*a*b^2*c^2*e^3*f*g^2 - 3*a^3*c^2*e^3*f*h^2 + 3*a^2*c^3*e^3*f^2*h - b^3*c^2
*e^3*f^2*g - a^2*c^3*d^3*g*h^2 - a^2*c^3*e^3*f*g^2 - 3*a^3*c^2*d^2*e*h^3 + 3*a^2*c^3*d*e^2*g^3 - a^2*b*c^2*e^3
*g^3 - 3*b*c^4*d*e^2*f^3 + a^2*b^2*c*e^3*g^2*h + a^3*c^2*e^3*g^2*h + b^3*c^2*d^3*f*h^2 + a^2*b*c^2*d^3*h^3 + b
^4*c*e^3*f^2*h + b*c^4*d^3*f^2*h + b*c^4*d^3*f*g^2 - c^5*d^3*f^2*g + 3*c^5*d^2*e*f^3 - a*c^4*e^3*f^3 - a*c^4*d
^3*g^3 + b^2*c^3*e^3*f^3 + a^4*c*e^3*h^3, z, k)*(root(24*a^6*b*c*d*e^11*z^3 + 24*a*b*c^6*d^11*e*z^3 + 240*a^4*
b*c^3*d^5*e^7*z^3 + 240*a^3*b*c^4*d^7*e^5*z^3 + 120*a^5*b*c^2*d^3*e^9*z^3 + 120*a^2*b*c^5*d^9*e^3*z^3 - 54*a^5
*b^2*c*d^2*e^10*z^3 - 54*a*b^2*c^5*d^10*e^2*z^3 + 50*a^4*b^3*c*d^3*e^9*z^3 + 50*a*b^3*c^4*d^9*e^3*z^3 - 36*a^2
*b^5*c*d^5*e^7*z^3 - 36*a*b^5*c^2*d^7*e^5*z^3 + 26*a*b^6*c*d^6*e^6*z^3 - 340*a^3*b^2*c^3*d^6*e^6*z^3 - 225*a^4
*b^2*c^2*d^4*e^8*z^3 - 225*a^2*b^2*c^4*d^8*e^4*z^3 + 180*a^3*b^3*c^2*d^5*e^7*z^3 + 180*a^2*b^3*c^3*d^7*e^5*z^3
- 30*a^2*b^4*c^2*d^6*e^6*z^3 - 6*b^7*c*d^7*e^5*z^3 - 6*b^3*c^5*d^11*e*z^3 - 6*a^5*b^3*d*e^11*z^3 - 6*a*b^7*d^
5*e^7*z^3 - 20*b^5*c^3*d^9*e^3*z^3 + 15*b^6*c^2*d^8*e^4*z^3 + 15*b^4*c^4*d^10*e^2*z^3 - 80*a^4*c^4*d^6*e^6*z^3
- 60*a^5*c^3*d^4*e^8*z^3 - 60*a^3*c^5*d^8*e^4*z^3 - 24*a^6*c^2*d^2*e^10*z^3 - 24*a^2*c^6*d^10*e^2*z^3 - 20*a^
3*b^5*d^3*e^9*z^3 + 15*a^4*b^4*d^2*e^10*z^3 + 15*a^2*b^6*d^4*e^8*z^3 - 4*a^7*c*e^12*z^3 - 4*a*c^7*d^12*z^3 + b
^8*d^6*e^6*z^3 + b^2*c^6*d^12*z^3 + a^6*b^2*e^12*z^3 - 9*a^3*b^2*c*d*e^5*g*h*z - 9*a*b^2*c^3*d^5*e*g*h*z - 30*
a^3*b*c^2*d*e^5*f*h*z + 9*a^2*b^3*c*d*e^5*f*h*z + 3*a*b^4*c*d^2*e^4*f*h*z + 27*a*b*c^4*d^4*e^2*f*g*z + 6*a^2*b
^2*c^2*d^3*e^3*g*h*z - 33*a^2*b^2*c^2*d^2*e^4*f*h*z + 18*a*b*c^4*d^5*e*f*h*z - 12*a*b^4*c*d*e^5*f*g*z + 27*a^3
*b*c^2*d^2*e^4*g*h*z + 27*a^2*b*c^3*d^4*e^2*g*h*z - 3*a^2*b^3*c*d^2*e^4*g*h*z - 3*a*b^3*c^2*d^4*e^2*g*h*z + 52
*a^2*b*c^3*d^3*e^3*f*h*z - 4*a*b^3*c^2*d^3*e^3*f*h*z - 3*a*b^2*c^3*d^4*e^2*f*h*z - 93*a^2*b*c^3*d^2*e^4*f*g*z
+ 51*a^2*b^2*c^2*d*e^5*f*g*z - 34*a*b^2*c^3*d^3*e^3*f*g*z + 27*a*b^3*c^2*d^2*e^4*f*g*z - 24*a*c^5*d^5*e*f*g*z
- 7*a^4*b*c*e^6*g*h*z - 7*a*b*c^4*d^6*g*h*z + a*b^4*c*d^3*e^3*g*h*z - 80*a^3*c^3*d^3*e^3*g*h*z + 3*b^4*c^2*d^4
*e^2*f*h*z - 66*a^2*c^4*d^4*e^2*f*h*z + 54*a^3*c^3*d^2*e^4*f*h*z - 3*b^3*c^3*d^4*e^2*f*g*z + 80*a^2*c^4*d^3*e^
3*f*g*z - 21*a^2*b*c^3*d^5*e*h^2*z + 6*a*b^3*c^2*d^5*e*h^2*z - 21*a^3*b*c^2*d*e^5*g^2*z + 6*a^2*b^3*c*d*e^5*g^
2*z - 66*a*b*c^4*d^3*e^3*f^2*z - 30*a*b^3*c^2*d*e^5*f^2*z + 27*a^2*b*c^3*d*e^5*f^2*z - 12*a^2*b^2*c^2*d^4*e^2*
h^2*z - 12*a^2*b^2*c^2*d^2*e^4*g^2*z + 24*a^4*c^2*d*e^5*g*h*z + 24*a^2*c^4*d^5*e*g*h*z - 3*b^3*c^3*d^5*e*f*h*z
- b^5*c*d^3*e^3*f*h*z + 3*b^2*c^4*d^5*e*f*g*z - 24*a^3*c^3*d*e^5*f*g*z + 9*a^3*b^2*c*e^6*f*h*z - 10*a^2*b^3*c
*e^6*f*g*z + 9*a^3*b*c^2*e^6*f*g*z + 3*a^4*b*c*d*e^5*h^2*z + 3*a*b*c^4*d^5*e*g^2*z + 14*a^3*b*c^2*d^3*e^3*h^2*
z + 3*a^3*b^2*c*d^2*e^4*h^2*z - a^2*b^3*c*d^3*e^3*h^2*z + 14*a^2*b*c^3*d^3*e^3*g^2*z + 3*a*b^2*c^3*d^4*e^2*g^2
*z - a*b^3*c^2*d^3*e^3*g^2*z + 63*a*b^2*c^3*d^2*e^4*f^2*z + 2*b^3*c^3*d^6*g*h*z - 6*a^4*c^2*e^6*f*h*z + 2*a^3*
b^3*e^6*g*h*z - b^2*c^4*d^6*f*h*z - 2*a^2*b^4*e^6*f*h*z + 6*b^5*c*d*e^5*f^2*z + 3*b*c^5*d^5*e*f^2*z + 6*a*b^4*
c*e^6*f^2*z + b^4*c^2*d^3*e^3*f*g*z + 33*a^3*c^3*d^4*e^2*h^2*z - 27*a^4*c^2*d^2*e^4*h^2*z + 33*a^3*c^3*d^2*e^4
*g^2*z - 27*a^2*c^4*d^4*e^2*g^2*z + 19*b^3*c^3*d^3*e^3*f^2*z - 15*b^4*c^2*d^2*e^4*f^2*z - 12*b^2*c^4*d^4*e^2*f
^2*z - 27*a^2*c^4*d^2*e^4*f^2*z - 9*a^2*b^2*c^2*e^6*f^2*z + 2*a*c^5*d^6*f*h*z + 2*a*b^5*e^6*f*g*z + 33*a*c^5*d
^4*e^2*f^2*z + 4*a^3*b^2*c*e^6*g^2*z + 4*a*b^2*c^3*d^6*h^2*z - b^4*c^2*d^6*h^2*z - b^2*c^4*d^6*g^2*z - a^4*c^2
*e^6*g^2*z - a^4*b^2*e^6*h^2*z - a^2*c^4*d^6*h^2*z + 3*a^3*c^3*e^6*f^2*z - a^2*b^4*e^6*g^2*z + b*c^5*d^6*f*g*z
+ 3*a^5*c*e^6*h^2*z + 3*a*c^5*d^6*g^2*z - c^6*d^6*f^2*z - b^6*e^6*f^2*z + 6*a*b^2*c^2*d*e^2*f*g*h - 2*a*b^3*c
*e^3*f*g*h + 3*a^2*b*c^2*d^2*e*g*h^2 - 3*a^2*b*c^2*d*e^2*g^2*h - 3*a^2*b*c^2*d*e^2*f*h^2 - 3*a*b^2*c^2*d^2*e*f
*h^2 - 6*a^2*c^3*d*e^2*f*g*h + 2*a^2*b*c^2*e^3*f*g*h + 6*a*b*c^3*d*e^2*f^2*h - 6*a*b*c^3*d*e^2*f*g^2 - 2*b^2*c
^3*d^3*f*g*h - 9*a*c^4*d^2*e*f^2*h - 3*b*c^4*d^2*e*f^2*g + 3*a*c^4*d^2*e*f*g^2 + 3*a*c^4*d*e^2*f^2*g - 2*a^3*b
*c*e^3*g*h^2 + 2*a*b*c^3*d^3*g^2*h - 2*a*b*c^3*d^3*f*h^2 + 2*a*c^4*d^3*f*g*h - 3*b^3*c^2*d*e^2*f^2*h + 3*b^2*c
^3*d^2*e*f^2*h + 3*a^3*c^2*d*e^2*g*h^2 - 3*a^2*c^3*d^2*e*g^2*h + 9*a^2*c^3*d^2*e*f*h^2 + 3*b^2*c^3*d*e^2*f^2*g
- 3*a*b^2*c^2*e^3*f^2*h + 2*a^2*b^2*c*e^3*f*h^2 - a*b^2*c^2*d^3*g*h^2 + 2*a*b^2*c^2*e^3*f*g^2 - 3*a^3*c^2*e^3
*f*h^2 + 3*a^2*c^3*e^3*f^2*h - b^3*c^2*e^3*f^2*g - a^2*c^3*d^3*g*h^2 - a^2*c^3*e^3*f*g^2 - 3*a^3*c^2*d^2*e*h^3
+ 3*a^2*c^3*d*e^2*g^3 - a^2*b*c^2*e^3*g^3 - 3*b*c^4*d*e^2*f^3 + a^2*b^2*c*e^3*g^2*h + a^3*c^2*e^3*g^2*h + b^3
*c^2*d^3*f*h^2 + a^2*b*c^2*d^3*h^3 + b^4*c*e^3*f^2*h + b*c^4*d^3*f^2*h + b*c^4*d^3*f*g^2 - c^5*d^3*f^2*g + 3*c
^5*d^2*e*f^3 - a*c^4*e^3*f^3 - a*c^4*d^3*g^3 + b^2*c^3*e^3*f^3 + a^4*c*e^3*h^3, z, k)*((8*a*c^6*d^9*e^2 + 8*a^
5*c^2*d*e^10 - b^6*c*d^5*e^6 + 32*a^2*c^5*d^7*e^4 + 48*a^3*c^4*d^5*e^6 + 32*a^4*c^3*d^3*e^8 + 3*b^2*c^5*d^9*e^
2 - 2*b^3*c^4*d^8*e^3 - 2*b^4*c^3*d^7*e^4 + 3*b^5*c^2*d^6*e^5 - a^5*b*c*e^11 - b*c^6*d^10*e + 114*a^2*b^2*c^3*
d^5*e^6 - 38*a^2*b^3*c^2*d^4*e^7 + 60*a^3*b^2*c^2*d^3*e^8 - 37*a*b*c^5*d^8*e^3 + 3*a*b^5*c*d^4*e^7 + 3*a^4*b^2
*c*d*e^10 + 60*a*b^2*c^4*d^7*e^4 - 38*a*b^3*c^3*d^6*e^5 + 4*a*b^4*c^2*d^5*e^6 - 106*a^2*b*c^4*d^6*e^5 - 2*a^2*
b^4*c*d^3*e^8 - 106*a^3*b*c^3*d^4*e^7 - 2*a^3*b^3*c*d^2*e^9 - 37*a^4*b*c^2*d^2*e^9)/(a^4*e^8 + c^4*d^8 + b^4*d
^4*e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2
*d^4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^
2*b*c*d^3*e^5) + (x*(6*a^5*c^2*e^11 - 2*c^7*d^10*e - 2*a^4*b^2*c*e^11 - 2*a*c^6*d^8*e^3 + 10*b*c^6*d^9*e^2 - 2
*b^6*c*d^4*e^7 + 12*a^2*c^5*d^6*e^5 + 28*a^3*c^4*d^4*e^7 + 22*a^4*c^3*d^2*e^9 - 22*b^2*c^5*d^8*e^3 + 28*b^3*c^
4*d^7*e^4 - 22*b^4*c^3*d^6*e^5 + 10*b^5*c^2*d^5*e^6 + 24*a^2*b^2*c^3*d^4*e^7 + 12*a^2*b^3*c^2*d^3*e^8 + 20*a^3
*b^2*c^2*d^2*e^9 + 8*a*b*c^5*d^7*e^4 + 8*a*b^5*c*d^3*e^8 + 8*a^3*b^3*c*d*e^10 - 22*a^4*b*c^2*d*e^10 - 20*a*b^2
*c^4*d^6*e^5 + 32*a*b^3*c^3*d^5*e^6 - 26*a*b^4*c^2*d^4*e^7 - 36*a^2*b*c^4*d^5*e^6 - 12*a^2*b^4*c*d^2*e^9 - 56*
a^3*b*c^3*d^3*e^8))/(a^4*e^8 + c^4*d^8 + b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4
*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 1
2*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b*c*d^3*e^5)) + (a^4*c^2*e^8*g + c^6*d^7*e*f + a^4*b*c*e^8*h -
a*c^5*d^7*e*h - b*c^5*d^7*e*g + a^2*b^3*c*e^8*f - 2*a^3*b*c^2*e^8*f - a^3*b^2*c*e^8*g + 3*a*c^5*d^5*e^3*f + a
^3*c^3*d*e^7*f + a*c^5*d^6*e^2*g - b*c^5*d^6*e^2*f + b^5*c*d^2*e^6*f - a^4*c^2*d*e^7*h + b^2*c^4*d^7*e*h + 3*a
^2*c^4*d^3*e^5*f + 3*a^2*c^4*d^4*e^4*g + 3*a^3*c^3*d^2*e^6*g - 3*b^2*c^4*d^5*e^3*f + 6*b^3*c^3*d^4*e^4*f - 4*b
^4*c^2*d^3*e^5*f - 3*a^2*c^4*d^5*e^3*h - 3*a^3*c^3*d^3*e^5*h + 2*b^2*c^4*d^6*e^2*g - b^3*c^3*d^5*e^3*g - 2*b^3
*c^3*d^6*e^2*h + b^4*c^2*d^5*e^3*h - a*b^2*c^3*d^3*e^5*f + 4*a*b^3*c^2*d^2*e^6*f - 5*a^2*b*c^3*d^2*e^6*f + 2*a
^2*b^2*c^2*d*e^7*f - 2*a*b^2*c^3*d^4*e^4*g + 4*a*b^3*c^2*d^3*e^5*g - a^2*b*c^3*d^3*e^5*g + 5*a*b^2*c^3*d^5*e^3
*h - 4*a*b^3*c^2*d^4*e^4*h + a^2*b*c^3*d^4*e^4*h + a^2*b^3*c*d^2*e^6*h + 2*a^3*b*c^2*d^2*e^6*h - 2*a*b^4*c*d*e
^7*f - 5*a^2*b^2*c^2*d^2*e^6*g + 2*a^2*b^2*c^2*d^3*e^5*h - 4*a*b*c^4*d^4*e^4*f - 2*a*b*c^4*d^5*e^3*g - a*b^4*c
*d^2*e^6*g + 2*a^2*b^3*c*d*e^7*g - 2*a^3*b^2*c*d*e^7*h)/(a^4*e^8 + c^4*d^8 + b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 4
*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 6*b^2*c^2*d^6*e^2
- 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b*c*d^3*e^5) + (x*(3*a^4*c
^2*e^8*h - 3*a^3*c^3*e^8*f + 5*c^6*d^6*e^2*f - c^6*d^7*e*g + b*c^5*d^7*e*h - 2*a^3*b*c^2*e^8*g + 7*a*c^5*d^4*e
^4*f + 5*a*c^5*d^5*e^3*g - 15*b*c^5*d^5*e^3*f + 7*a^3*c^3*d*e^7*g - 5*a*c^5*d^6*e^2*h + b*c^5*d^6*e^2*g + 2*a^
2*b^2*c^2*e^8*f - a^2*c^4*d^2*e^6*f + 13*a^2*c^4*d^3*e^5*g + 17*b^2*c^4*d^4*e^4*f - 9*b^3*c^3*d^3*e^5*f + 2*b^
4*c^2*d^2*e^6*f - 7*a^2*c^4*d^4*e^4*h + a^3*c^3*d^2*e^6*h + b^2*c^4*d^5*e^3*g - b^3*c^3*d^4*e^4*g - b^2*c^4*d^
6*e^2*h - b^3*c^3*d^5*e^3*h + b^4*c^2*d^4*e^4*h + 11*a*b^2*c^3*d^2*e^6*f + 13*a*b^2*c^3*d^3*e^5*g - 2*a*b^3*c^
2*d^2*e^6*g - 19*a^2*b*c^3*d^2*e^6*g + 4*a^2*b^2*c^2*d*e^7*g - a*b^2*c^3*d^4*e^4*h - 4*a*b^3*c^2*d^3*e^5*h + a
^2*b*c^3*d^3*e^5*h + 8*a^2*b^2*c^2*d^2*e^6*h - 14*a*b*c^4*d^3*e^5*f - 4*a*b^3*c^2*d*e^7*f + a^2*b*c^3*d*e^7*f
- 16*a*b*c^4*d^4*e^4*g + 10*a*b*c^4*d^5*e^3*h - 8*a^3*b*c^2*d*e^7*h))/(a^4*e^8 + c^4*d^8 + b^4*d^4*e^4 - 4*a*b
^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 6*b
^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b*c*d^3*e^5)
) - (2*c^5*d^3*e^2*f^2 - b^3*c^2*e^5*f^2 - c^5*d^4*e*f*g + 2*a^2*c^3*d^3*e^2*h^2 + a*b*c^3*e^5*f^2 - 2*a*c^4*d
*e^4*f^2 - a^2*c^3*e^5*f*g + a^3*c^2*e^5*g*h - a^2*b*c^2*e^5*g^2 - 2*a*c^4*d^3*e^2*g^2 - 5*b*c^4*d^2*e^3*f^2 +
2*a^2*c^3*d*e^4*g^2 + 4*b^2*c^3*d*e^4*f^2 - 2*a^3*c^2*d*e^4*h^2 + a*b*c^3*d^2*e^3*g^2 - b^2*c^3*d^2*e^3*f*g -
6*a^2*c^3*d^2*e^3*g*h - 2*b^2*c^3*d^3*e^2*f*h + b^3*c^2*d^2*e^3*f*h + a*c^4*d^4*e*g*h + b*c^4*d^4*e*f*h + a^2
*b*c^2*d^2*e^3*h^2 - a*b*c^3*d^4*e*h^2 + 2*a*b^2*c^2*e^5*f*g - a^2*b*c^2*e^5*f*h + 6*a*c^4*d^2*e^3*f*g - 4*a*c
^4*d^3*e^2*f*h + 2*b*c^4*d^3*e^2*f*g + 4*a^2*c^3*d*e^4*f*h + 4*a*b*c^3*d^2*e^3*f*h - 2*a*b^2*c^2*d*e^4*f*h + 2
*a*b*c^3*d^3*e^2*g*h + 2*a^2*b*c^2*d*e^4*g*h - a*b^2*c^2*d^2*e^3*g*h - 6*a*b*c^3*d*e^4*f*g)/(a^4*e^8 + c^4*d^8
+ b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6
*a^2*c^2*d^4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4
- 12*a^2*b*c*d^3*e^5) + (x*(c^5*d^4*e*g^2 + a^2*c^3*e^5*g^2 + b^2*c^3*e^5*f^2 + 4*c^5*d^2*e^3*f^2 + 4*a^2*c^3
*d^2*e^3*h^2 - 4*b*c^4*d*e^4*f^2 - 4*c^5*d^3*e^2*f*g - 2*a*c^4*d^2*e^3*g^2 + b^2*c^3*d^4*e*h^2 - 4*a*b*c^3*d^3
*e^2*h^2 - 2*b^2*c^3*d^2*e^3*f*h - 2*a*b*c^3*e^5*f*g + 4*a*c^4*d*e^4*f*g - 2*b*c^4*d^4*e*g*h - 8*a*c^4*d^2*e^3
*f*h + 2*b*c^4*d^2*e^3*f*g + 4*a*c^4*d^3*e^2*g*h + 4*b*c^4*d^3*e^2*f*h - 4*a^2*c^3*d*e^4*g*h + 2*a*b*c^3*d^2*e
^3*g*h + 4*a*b*c^3*d*e^4*f*h))/(a^4*e^8 + c^4*d^8 + b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*
d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^
3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b*c*d^3*e^5))*root(24*a^6*b*c*d*e^11*z^3 + 24*a*b*c
^6*d^11*e*z^3 + 240*a^4*b*c^3*d^5*e^7*z^3 + 240*a^3*b*c^4*d^7*e^5*z^3 + 120*a^5*b*c^2*d^3*e^9*z^3 + 120*a^2*b*
c^5*d^9*e^3*z^3 - 54*a^5*b^2*c*d^2*e^10*z^3 - 54*a*b^2*c^5*d^10*e^2*z^3 + 50*a^4*b^3*c*d^3*e^9*z^3 + 50*a*b^3*
c^4*d^9*e^3*z^3 - 36*a^2*b^5*c*d^5*e^7*z^3 - 36*a*b^5*c^2*d^7*e^5*z^3 + 26*a*b^6*c*d^6*e^6*z^3 - 340*a^3*b^2*c
^3*d^6*e^6*z^3 - 225*a^4*b^2*c^2*d^4*e^8*z^3 - 225*a^2*b^2*c^4*d^8*e^4*z^3 + 180*a^3*b^3*c^2*d^5*e^7*z^3 + 180
*a^2*b^3*c^3*d^7*e^5*z^3 - 30*a^2*b^4*c^2*d^6*e^6*z^3 - 6*b^7*c*d^7*e^5*z^3 - 6*b^3*c^5*d^11*e*z^3 - 6*a^5*b^3
*d*e^11*z^3 - 6*a*b^7*d^5*e^7*z^3 - 20*b^5*c^3*d^9*e^3*z^3 + 15*b^6*c^2*d^8*e^4*z^3 + 15*b^4*c^4*d^10*e^2*z^3
- 80*a^4*c^4*d^6*e^6*z^3 - 60*a^5*c^3*d^4*e^8*z^3 - 60*a^3*c^5*d^8*e^4*z^3 - 24*a^6*c^2*d^2*e^10*z^3 - 24*a^2*
c^6*d^10*e^2*z^3 - 20*a^3*b^5*d^3*e^9*z^3 + 15*a^4*b^4*d^2*e^10*z^3 + 15*a^2*b^6*d^4*e^8*z^3 - 4*a^7*c*e^12*z^
3 - 4*a*c^7*d^12*z^3 + b^8*d^6*e^6*z^3 + b^2*c^6*d^12*z^3 + a^6*b^2*e^12*z^3 - 9*a^3*b^2*c*d*e^5*g*h*z - 9*a*b
^2*c^3*d^5*e*g*h*z - 30*a^3*b*c^2*d*e^5*f*h*z + 9*a^2*b^3*c*d*e^5*f*h*z + 3*a*b^4*c*d^2*e^4*f*h*z + 27*a*b*c^4
*d^4*e^2*f*g*z + 6*a^2*b^2*c^2*d^3*e^3*g*h*z - 33*a^2*b^2*c^2*d^2*e^4*f*h*z + 18*a*b*c^4*d^5*e*f*h*z - 12*a*b^
4*c*d*e^5*f*g*z + 27*a^3*b*c^2*d^2*e^4*g*h*z + 27*a^2*b*c^3*d^4*e^2*g*h*z - 3*a^2*b^3*c*d^2*e^4*g*h*z - 3*a*b^
3*c^2*d^4*e^2*g*h*z + 52*a^2*b*c^3*d^3*e^3*f*h*z - 4*a*b^3*c^2*d^3*e^3*f*h*z - 3*a*b^2*c^3*d^4*e^2*f*h*z - 93*
a^2*b*c^3*d^2*e^4*f*g*z + 51*a^2*b^2*c^2*d*e^5*f*g*z - 34*a*b^2*c^3*d^3*e^3*f*g*z + 27*a*b^3*c^2*d^2*e^4*f*g*z
- 24*a*c^5*d^5*e*f*g*z - 7*a^4*b*c*e^6*g*h*z - 7*a*b*c^4*d^6*g*h*z + a*b^4*c*d^3*e^3*g*h*z - 80*a^3*c^3*d^3*e
^3*g*h*z + 3*b^4*c^2*d^4*e^2*f*h*z - 66*a^2*c^4*d^4*e^2*f*h*z + 54*a^3*c^3*d^2*e^4*f*h*z - 3*b^3*c^3*d^4*e^2*f
*g*z + 80*a^2*c^4*d^3*e^3*f*g*z - 21*a^2*b*c^3*d^5*e*h^2*z + 6*a*b^3*c^2*d^5*e*h^2*z - 21*a^3*b*c^2*d*e^5*g^2*
z + 6*a^2*b^3*c*d*e^5*g^2*z - 66*a*b*c^4*d^3*e^3*f^2*z - 30*a*b^3*c^2*d*e^5*f^2*z + 27*a^2*b*c^3*d*e^5*f^2*z -
12*a^2*b^2*c^2*d^4*e^2*h^2*z - 12*a^2*b^2*c^2*d^2*e^4*g^2*z + 24*a^4*c^2*d*e^5*g*h*z + 24*a^2*c^4*d^5*e*g*h*z
- 3*b^3*c^3*d^5*e*f*h*z - b^5*c*d^3*e^3*f*h*z + 3*b^2*c^4*d^5*e*f*g*z - 24*a^3*c^3*d*e^5*f*g*z + 9*a^3*b^2*c*
e^6*f*h*z - 10*a^2*b^3*c*e^6*f*g*z + 9*a^3*b*c^2*e^6*f*g*z + 3*a^4*b*c*d*e^5*h^2*z + 3*a*b*c^4*d^5*e*g^2*z + 1
4*a^3*b*c^2*d^3*e^3*h^2*z + 3*a^3*b^2*c*d^2*e^4*h^2*z - a^2*b^3*c*d^3*e^3*h^2*z + 14*a^2*b*c^3*d^3*e^3*g^2*z +
3*a*b^2*c^3*d^4*e^2*g^2*z - a*b^3*c^2*d^3*e^3*g^2*z + 63*a*b^2*c^3*d^2*e^4*f^2*z + 2*b^3*c^3*d^6*g*h*z - 6*a^
4*c^2*e^6*f*h*z + 2*a^3*b^3*e^6*g*h*z - b^2*c^4*d^6*f*h*z - 2*a^2*b^4*e^6*f*h*z + 6*b^5*c*d*e^5*f^2*z + 3*b*c^
5*d^5*e*f^2*z + 6*a*b^4*c*e^6*f^2*z + b^4*c^2*d^3*e^3*f*g*z + 33*a^3*c^3*d^4*e^2*h^2*z - 27*a^4*c^2*d^2*e^4*h^
2*z + 33*a^3*c^3*d^2*e^4*g^2*z - 27*a^2*c^4*d^4*e^2*g^2*z + 19*b^3*c^3*d^3*e^3*f^2*z - 15*b^4*c^2*d^2*e^4*f^2*
z - 12*b^2*c^4*d^4*e^2*f^2*z - 27*a^2*c^4*d^2*e^4*f^2*z - 9*a^2*b^2*c^2*e^6*f^2*z + 2*a*c^5*d^6*f*h*z + 2*a*b^
5*e^6*f*g*z + 33*a*c^5*d^4*e^2*f^2*z + 4*a^3*b^2*c*e^6*g^2*z + 4*a*b^2*c^3*d^6*h^2*z - b^4*c^2*d^6*h^2*z - b^2
*c^4*d^6*g^2*z - a^4*c^2*e^6*g^2*z - a^4*b^2*e^6*h^2*z - a^2*c^4*d^6*h^2*z + 3*a^3*c^3*e^6*f^2*z - a^2*b^4*e^6
*g^2*z + b*c^5*d^6*f*g*z + 3*a^5*c*e^6*h^2*z + 3*a*c^5*d^6*g^2*z - c^6*d^6*f^2*z - b^6*e^6*f^2*z + 6*a*b^2*c^2
*d*e^2*f*g*h - 2*a*b^3*c*e^3*f*g*h + 3*a^2*b*c^2*d^2*e*g*h^2 - 3*a^2*b*c^2*d*e^2*g^2*h - 3*a^2*b*c^2*d*e^2*f*h
^2 - 3*a*b^2*c^2*d^2*e*f*h^2 - 6*a^2*c^3*d*e^2*f*g*h + 2*a^2*b*c^2*e^3*f*g*h + 6*a*b*c^3*d*e^2*f^2*h - 6*a*b*c
^3*d*e^2*f*g^2 - 2*b^2*c^3*d^3*f*g*h - 9*a*c^4*d^2*e*f^2*h - 3*b*c^4*d^2*e*f^2*g + 3*a*c^4*d^2*e*f*g^2 + 3*a*c
^4*d*e^2*f^2*g - 2*a^3*b*c*e^3*g*h^2 + 2*a*b*c^3*d^3*g^2*h - 2*a*b*c^3*d^3*f*h^2 + 2*a*c^4*d^3*f*g*h - 3*b^3*c
^2*d*e^2*f^2*h + 3*b^2*c^3*d^2*e*f^2*h + 3*a^3*c^2*d*e^2*g*h^2 - 3*a^2*c^3*d^2*e*g^2*h + 9*a^2*c^3*d^2*e*f*h^2
+ 3*b^2*c^3*d*e^2*f^2*g - 3*a*b^2*c^2*e^3*f^2*h + 2*a^2*b^2*c*e^3*f*h^2 - a*b^2*c^2*d^3*g*h^2 + 2*a*b^2*c^2*e
^3*f*g^2 - 3*a^3*c^2*e^3*f*h^2 + 3*a^2*c^3*e^3*f^2*h - b^3*c^2*e^3*f^2*g - a^2*c^3*d^3*g*h^2 - a^2*c^3*e^3*f*g
^2 - 3*a^3*c^2*d^2*e*h^3 + 3*a^2*c^3*d*e^2*g^3 - a^2*b*c^2*e^3*g^3 - 3*b*c^4*d*e^2*f^3 + a^2*b^2*c*e^3*g^2*h +
a^3*c^2*e^3*g^2*h + b^3*c^2*d^3*f*h^2 + a^2*b*c^2*d^3*h^3 + b^4*c*e^3*f^2*h + b*c^4*d^3*f^2*h + b*c^4*d^3*f*g
^2 - c^5*d^3*f^2*g + 3*c^5*d^2*e*f^3 - a*c^4*e^3*f^3 - a*c^4*d^3*g^3 + b^2*c^3*e^3*f^3 + a^4*c*e^3*h^3, z, k),
k, 1, 3) - ((a*e^4*f + c*d^4*h + a*d*e^3*g - 3*b*d*e^3*f + b*d^3*e*h - 3*c*d^3*e*g - 3*a*d^2*e^2*h + b*d^2*e^
2*g + 5*c*d^2*e^2*f)/(2*e*(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)) + (x*
(a*e^3*g - b*e^3*f - 2*a*d*e^2*h + 2*c*d*e^2*f + b*d^2*e*h - c*d^2*e*g))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*
a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2))/(d^2 + e^2*x^2 + 2*d*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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