\(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)^2} \, dx\) [2198]

Optimal result

Integrand size = 20, antiderivative size = 485 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4} \]

[Out]

-1/2*e*(4*c^2*d^2+3*b^2*e^2-4*c*e*(2*a*e+b*d))/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2-e*(-b*e+2*c*d)*(c^
2*d^2+3*b^2*e^2-c*e*(11*a*e+b*d))/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)+(-b*c*d+b^2*e-2*a*c*e-c*(-b*e+2*c
*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)+(-b*e+2*c*d)*(2*c^4*d^4-3*b^4*e^4-4*c^3*d^2*e*
(-5*a*e+b*d)+4*b^2*c*e^3*(5*a*e+b*d)-2*c^2*e^2*(15*a^2*e^2+10*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)
^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^4+e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(a*e+5*b*d))*ln(e*x+d)/(a*e^2
-b*d*e+c*d^2)^4-1/2*e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(a*e+5*b*d))*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^4

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {754, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac {e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac {e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-1/2*(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) -
(e*(2*c*d - b*e)*(c^2*d^2 + 3*b^2*e^2 - c*e*(b*d + 11*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*
x)) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + b*
x + c*x^2)) + ((2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) -
2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*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)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e +
a*e^2)^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2
)^4)

Rule 212

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

Rule 632

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 642

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 648

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 754

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

Rule 814

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {2 c^2 d^2-3 b^2 e^2+c e (b d+8 a e)+3 c e (2 c d-b e) x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {e^2 \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {e^2 (2 c d-b e) \left (-c^2 d^2-3 b^2 e^2+c e (b d+11 a e)\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^4 \left (-10 c^2 d^2-3 b^2 e^2+2 c e (5 b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\int \frac {2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\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 )^4} \\ & = -\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\left (e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\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 )^4}-\frac {\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\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 )^4} \\ & = -\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}+\frac {\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \text {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 )^4} \\ & = -\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\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 )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \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] (verified)

Time = 0.62 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e^3}{2 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^2}+\frac {2 e^3 (-2 c d+b e)}{\left (c d^2+e (-b d+a e)\right )^3 (d+e x)}+\frac {-b^4 e^3+b^3 c e^2 (3 d-e x)+b^2 c e \left (4 a e^2-3 c d (d-e x)\right )+2 c^2 \left (-a^2 e^3+c^2 d^3 x+3 a c d e (d-e x)\right )+b c^2 \left (c d^2 (d-3 e x)+3 a e^2 (-3 d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right )^3 (a+x (b+c x))}+\frac {(-2 c d+b e) \left (-2 c^4 d^4+3 b^4 e^4+4 c^3 d^2 e (b d-5 a e)-4 b^2 c e^3 (b d+5 a e)+2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^4} \]

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-1/2*e^3/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^2) + (2*e^3*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^3*(d
+ e*x)) + (-(b^4*e^3) + b^3*c*e^2*(3*d - e*x) + b^2*c*e*(4*a*e^2 - 3*c*d*(d - e*x)) + 2*c^2*(-(a^2*e^3) + c^2*
d^3*x + 3*a*c*d*e*(d - e*x)) + b*c^2*(c*d^2*(d - 3*e*x) + 3*a*e^2*(-3*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*
(b*d - a*e))^3*(a + x*(b + c*x))) + ((-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^
2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*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))^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log
[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + x*(b + c
*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^4)

Maple [A] (verified)

Time = 21.84 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {\frac {\frac {c \left (3 a^{2} b c \,e^{5}-6 d \,e^{4} a^{2} c^{2}-a \,b^{3} e^{5}+6 a b \,c^{2} d^{2} e^{3}-4 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}-4 b^{3} c \,d^{2} e^{3}+6 b^{2} c^{2} d^{3} e^{2}-5 b \,c^{3} d^{4} e +2 c^{4} d^{5}\right ) x}{4 a c -b^{2}}-\frac {2 a^{3} c^{2} e^{5}-4 a^{2} b^{2} c \,e^{5}+7 a^{2} b \,c^{2} d \,e^{4}-4 a^{2} c^{3} d^{2} e^{3}+a \,b^{4} e^{5}+a \,b^{3} c d \,e^{4}-10 a \,b^{2} c^{2} d^{2} e^{3}+14 a b \,c^{3} d^{3} e^{2}-6 a \,c^{4} d^{4} e -b^{5} d \,e^{4}+4 b^{4} c \,d^{2} e^{3}-6 b^{3} c^{2} d^{3} e^{2}+4 b^{2} c^{3} d^{4} e -b \,c^{4} d^{5}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (8 a^{2} c^{3} e^{5}-14 a \,b^{2} c^{2} e^{5}+40 a b \,c^{3} d \,e^{4}-40 a \,c^{4} d^{2} e^{3}+3 b^{4} c \,e^{5}-10 b^{3} c^{2} d \,e^{4}+10 b^{2} c^{3} d^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (19 a^{2} b \,c^{2} e^{5}-30 a^{2} c^{3} d \,e^{4}-17 a \,b^{3} c \,e^{5}+50 a \,b^{2} c^{2} d \,e^{4}-50 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}-10 b^{4} c d \,e^{4}+10 b^{3} c^{2} d^{2} e^{3}-5 b \,c^{4} d^{4} e +2 c^{5} d^{5}-\frac {\left (8 a^{2} c^{3} e^{5}-14 a \,b^{2} c^{2} e^{5}+40 a b \,c^{3} d \,e^{4}-40 a \,c^{4} d^{2} e^{3}+3 b^{4} c \,e^{5}-10 b^{3} c^{2} d \,e^{4}+10 b^{2} c^{3} d^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{4}}-\frac {e^{3}}{2 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}+\frac {2 e^{3} \left (b e -2 c d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {e^{3} \left (2 a c \,e^{2}-3 b^{2} e^{2}+10 b c d e -10 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{4}}\)	\(821\)
risch	\(\text {Expression too large to display}\)	\(6775\)

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/(a*e^2-b*d*e+c*d^2)^4*((c*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4-a*b^3*e^5+6*a*b*c^2*d^2*e^3-4*a*c^3*d^3*e^2+b^4*d*e
^4-4*b^3*c*d^2*e^3+6*b^2*c^2*d^3*e^2-5*b*c^3*d^4*e+2*c^4*d^5)/(4*a*c-b^2)*x-(2*a^3*c^2*e^5-4*a^2*b^2*c*e^5+7*a
^2*b*c^2*d*e^4-4*a^2*c^3*d^2*e^3+a*b^4*e^5+a*b^3*c*d*e^4-10*a*b^2*c^2*d^2*e^3+14*a*b*c^3*d^3*e^2-6*a*c^4*d^4*e
-b^5*d*e^4+4*b^4*c*d^2*e^3-6*b^3*c^2*d^3*e^2+4*b^2*c^3*d^4*e-b*c^4*d^5)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^
2)*(1/2*(8*a^2*c^3*e^5-14*a*b^2*c^2*e^5+40*a*b*c^3*d*e^4-40*a*c^4*d^2*e^3+3*b^4*c*e^5-10*b^3*c^2*d*e^4+10*b^2*
c^3*d^2*e^3)/c*ln(c*x^2+b*x+a)+2*(19*a^2*b*c^2*e^5-30*a^2*c^3*d*e^4-17*a*b^3*c*e^5+50*a*b^2*c^2*d*e^4-50*a*b*c
^3*d^2*e^3+20*a*c^4*d^3*e^2+3*b^5*e^5-10*b^4*c*d*e^4+10*b^3*c^2*d^2*e^3-5*b*c^4*d^4*e+2*c^5*d^5-1/2*(8*a^2*c^3
*e^5-14*a*b^2*c^2*e^5+40*a*b*c^3*d*e^4-40*a*c^4*d^2*e^3+3*b^4*c*e^5-10*b^3*c^2*d*e^4+10*b^2*c^3*d^2*e^3)*b/c)/
(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))-1/2*e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2+2*e^3/(a*e^2-b
*d*e+c*d^2)^3*(b*e-2*c*d)/(e*x+d)-e^3*(2*a*c*e^2-3*b^2*e^2+10*b*c*d*e-10*c^2*d^2)/(a*e^2-b*d*e+c*d^2)^4*ln(e*x
+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5338 vs. \(2 (477) = 954\).

Time = 253.27 (sec) , antiderivative size = 10696, normalized size of antiderivative = 22.05 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^2,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 deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1720 vs. \(2 (477) = 954\).

Time = 0.30 (sec) , antiderivative size = 1720, normalized size of antiderivative = 3.55 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-1/2*(10*c^2*d^2*e^3 - 10*b*c*d*e^4 + 3*b^2*e^5 - 2*a*c*e^5)*log(c*x^2 + b*x + a)/(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) + (10*c^2*d^2*e^4 - 10*b*c*d*e^5 + 3*b^2*e^6 - 2*a*c*e^6)*log(abs(e*x + d))/(c^4*d^8*e - 4*b*c^3*
d^7*e^2 + 6*b^2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*
c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4
*a^3*b*d*e^8 + a^4*e^9) - (4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60*a*b*c^3*d^2
*e^3 - 10*b^4*c*d*e^4 + 60*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 3*b^5*e^5 - 20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)
*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4*d^8 - 4*a*c^5*d^8 - 4*b^3*c^3*d^7*e + 16*a*b*c^4*d^7*e + 6*b
^4*c^2*d^6*e^2 - 20*a*b^2*c^3*d^6*e^2 - 16*a^2*c^4*d^6*e^2 - 4*b^5*c*d^5*e^3 + 4*a*b^3*c^2*d^5*e^3 + 48*a^2*b*
c^3*d^5*e^3 + b^6*d^4*e^4 + 8*a*b^4*c*d^4*e^4 - 42*a^2*b^2*c^2*d^4*e^4 - 24*a^3*c^3*d^4*e^4 - 4*a*b^5*d^3*e^5
+ 4*a^2*b^3*c*d^3*e^5 + 48*a^3*b*c^2*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 20*a^3*b^2*c*d^2*e^6 - 16*a^4*c^2*d^2*e^6 -
4*a^3*b^3*d*e^7 + 16*a^4*b*c*d*e^7 + a^4*b^2*e^8 - 4*a^5*c*e^8)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*b*c^4*d^7 - 8*b^
2*c^3*d^6*e + 12*a*c^4*d^6*e + 12*b^3*c^2*d^5*e^2 - 28*a*b*c^3*d^5*e^2 - 8*b^4*c*d^4*e^3 + 29*a*b^2*c^2*d^4*e^
3 - 28*a^2*c^3*d^4*e^3 + 2*b^5*d^3*e^4 - 16*a*b^3*c*d^3*e^4 + 42*a^2*b*c^2*d^3*e^4 + 3*a*b^4*d^2*e^5 - 2*a^2*b
^2*c*d^2*e^5 - 44*a^3*c^2*d^2*e^5 - 6*a^2*b^3*d*e^6 + 24*a^3*b*c*d*e^6 + a^3*b^2*e^7 - 4*a^4*c*e^7 + 2*(2*c^5*
d^5*e^2 - 5*b*c^4*d^4*e^3 + 10*b^2*c^3*d^3*e^4 - 20*a*c^4*d^3*e^4 - 10*b^3*c^2*d^2*e^5 + 30*a*b*c^3*d^2*e^5 +
3*b^4*c*d*e^6 - 4*a*b^2*c^2*d*e^6 - 22*a^2*c^3*d*e^6 - 3*a*b^3*c*e^7 + 11*a^2*b*c^2*e^7)*x^3 + (8*c^5*d^6*e -
18*b*c^4*d^5*e^2 + 25*b^2*c^3*d^4*e^3 - 40*a*c^4*d^4*e^3 - 10*b^3*c^2*d^3*e^4 + 20*a*b*c^3*d^3*e^4 - 11*b^4*c*
d^2*e^5 + 58*a*b^2*c^2*d^2*e^5 - 56*a^2*c^3*d^2*e^5 + 6*b^5*d*e^6 - 20*a*b^3*c*d*e^6 - 10*a^2*b*c^2*d*e^6 - 6*
a*b^4*e^7 + 25*a^2*b^2*c*e^7 - 8*a^3*c^2*e^7)*x^2 + (4*c^5*d^7 - 6*b*c^4*d^6*e - 4*b^2*c^3*d^5*e^2 + 16*a*c^4*
d^5*e^2 + 25*b^3*c^2*d^4*e^3 - 80*a*b*c^3*d^4*e^3 - 28*b^4*c*d^3*e^4 + 104*a*b^2*c^2*d^3*e^4 - 28*a^2*c^3*d^3*
e^4 + 9*b^5*d^2*e^5 - 28*a*b^3*c*d^2*e^5 - 14*a^2*b*c^2*d^2*e^5 - 6*a*b^4*d*e^6 + 32*a^2*b^2*c*d*e^6 - 40*a^3*
c^2*d*e^6 - 3*a^2*b^3*e^7 + 12*a^3*b*c*e^7)*x)/((c*d^2 - b*d*e + a*e^2)^4*(c*x^2 + b*x + a)*(b^2 - 4*a*c)*(e*x
+ d)^2)

Mupad [B] (verification not implemented)

Time = 17.87 (sec) , antiderivative size = 3748, normalized size of antiderivative = 7.73 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^2),x)

[Out]

(log(b^3 + (-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c - 8*a*c^2*x + 2*b^2*c*x)*((3*b^8*e^5)/2 + 64*a^4*c^4*e^5 - (3*b^
5*e^5*(-(4*a*c - b^2)^3)^(1/2))/2 - 2*c^5*d^5*(-(4*a*c - b^2)^3)^(1/2) + 84*a^2*b^4*c^2*e^5 - 144*a^3*b^2*c^3*
e^5 - 320*a^3*c^5*d^2*e^3 + 5*b^6*c^2*d^2*e^3 - 19*a*b^6*c*e^5 - 5*b^7*c*d*e^4 + 240*a^2*b^2*c^4*d^2*e^3 - 5*b
^3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^3*c*e^5*(-(4*a*c - b^2)^3)^(1/2) + 60*a*b^5*c^2*d*e^4 + 320*a
^3*b*c^4*d*e^4 + 5*b*c^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 5*b^4*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 15*a^2*b*c^
2*e^5*(-(4*a*c - b^2)^3)^(1/2) - 60*a*b^4*c^3*d^2*e^3 - 240*a^2*b^3*c^3*d*e^4 - 20*a*c^4*d^3*e^2*(-(4*a*c - b^
2)^3)^(1/2) + 30*a^2*c^3*d*e^4*(-(4*a*c - b^2)^3)^(1/2) + 30*a*b*c^3*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 30*a*b
^2*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2)))/(64*a^3*c^7*d^8 - a^4*b^6*e^8 + 64*a^7*c^3*e^8 - b^6*c^4*d^8 - b^10*d^
4*e^4 + 12*a*b^4*c^5*d^8 + 12*a^5*b^4*c*e^8 + 4*a*b^9*d^3*e^5 + 4*a^3*b^7*d*e^7 + 4*b^7*c^3*d^7*e + 4*b^9*c*d^
5*e^3 - 48*a^2*b^2*c^6*d^8 - 48*a^6*b^2*c^2*e^8 - 6*a^2*b^8*d^2*e^6 + 256*a^4*c^6*d^6*e^2 + 384*a^5*c^5*d^4*e^
4 + 256*a^6*c^4*d^2*e^6 - 6*b^8*c^2*d^6*e^2 - 240*a^2*b^4*c^4*d^6*e^2 + 48*a^2*b^5*c^3*d^5*e^3 + 90*a^2*b^6*c^
2*d^4*e^4 + 192*a^3*b^2*c^5*d^6*e^2 + 320*a^3*b^3*c^4*d^5*e^3 - 440*a^3*b^4*c^3*d^4*e^4 + 48*a^3*b^5*c^2*d^3*e
^5 + 480*a^4*b^2*c^4*d^4*e^4 + 320*a^4*b^3*c^3*d^3*e^5 - 240*a^4*b^4*c^2*d^2*e^6 + 192*a^5*b^2*c^3*d^2*e^6 - 4
8*a*b^5*c^4*d^7*e - 256*a^3*b*c^6*d^7*e - 48*a^4*b^5*c*d*e^7 - 256*a^6*b*c^3*d*e^7 + 68*a*b^6*c^3*d^6*e^2 - 36
*a*b^7*c^2*d^5*e^3 + 192*a^2*b^3*c^5*d^7*e - 36*a^2*b^7*c*d^3*e^5 + 68*a^3*b^6*c*d^2*e^6 - 768*a^4*b*c^5*d^5*e
^3 - 768*a^5*b*c^4*d^3*e^5 + 192*a^5*b^3*c^2*d*e^7) - (log(d + e*x)*(e^5*(2*a*c - 3*b^2) - 10*c^2*d^2*e^3 + 10
*b*c*d*e^4))/(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*(3*b^3*c*e^5 - 2*c^4*d^3*e^2 + 3*b*c^3*d^2*e^3 -
7*b^2*c^2*d*e^4 - 11*a*b*c^2*e^5 + 22*a*c^3*d*e^4))/(4*a*c^4*d^6 + 4*a^4*c*e^6 - a^3*b^2*e^6 - b^2*c^3*d^6 +
b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 + 3*b^3*c^2*d^5*e - 3*b^4*c*d^4*e^2 + 12*a^2*c^3*d^4*e^2 + 12*
a^3*c^2*d^2*e^4 - 12*a*b*c^3*d^5*e - 12*a^3*b*c*d*e^5 + 2*a*b^3*c*d^3*e^3 + 9*a*b^2*c^2*d^4*e^2 - 24*a^2*b*c^2
*d^3*e^3 + 9*a^2*b^2*c*d^2*e^4) - (2*b*c^3*d^5 - 4*a^3*c*e^5 + a^2*b^2*e^5 - 2*b^4*d^2*e^3 - 6*b^2*c^2*d^4*e +
6*b^3*c*d^3*e^2 - 40*a^2*c^2*d^2*e^3 - 5*a*b^3*d*e^4 + 12*a*c^3*d^4*e + 20*a^2*b*c*d*e^4 - 18*a*b*c^2*d^3*e^2
+ 17*a*b^2*c*d^2*e^3)/(2*(4*a*c^4*d^6 + 4*a^4*c*e^6 - a^3*b^2*e^6 - b^2*c^3*d^6 + b^5*d^3*e^3 - 3*a*b^4*d^2*e
^4 + 3*a^2*b^3*d*e^5 + 3*b^3*c^2*d^5*e - 3*b^4*c*d^4*e^2 + 12*a^2*c^3*d^4*e^2 + 12*a^3*c^2*d^2*e^4 - 12*a*b*c^
3*d^5*e - 12*a^3*b*c*d*e^5 + 2*a*b^3*c*d^3*e^3 + 9*a*b^2*c^2*d^4*e^2 - 24*a^2*b*c^2*d^3*e^3 + 9*a^2*b^2*c*d^2*
e^4)) + (x*(3*a*b^3*e^5 - 4*c^4*d^5 + 9*b^4*d*e^4 - 12*a*c^3*d^3*e^2 + 40*a^2*c^2*d*e^4 - 19*b^3*c*d^2*e^3 + 6
*b^2*c^2*d^3*e^2 - 12*a^2*b*c*e^5 + 2*b*c^3*d^4*e - 44*a*b^2*c*d*e^4 + 66*a*b*c^2*d^2*e^3))/(2*(4*a*c^4*d^6 +
4*a^4*c*e^6 - a^3*b^2*e^6 - b^2*c^3*d^6 + b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 + 3*b^3*c^2*d^5*e -
3*b^4*c*d^4*e^2 + 12*a^2*c^3*d^4*e^2 + 12*a^3*c^2*d^2*e^4 - 12*a*b*c^3*d^5*e - 12*a^3*b*c*d*e^5 + 2*a*b^3*c*d^
3*e^3 + 9*a*b^2*c^2*d^4*e^2 - 24*a^2*b*c^2*d^3*e^3 + 9*a^2*b^2*c*d^2*e^4)) + (x^2*(6*b^4*e^5 - 8*c^4*d^4*e + 8
*a^2*c^2*e^5 + 48*a*c^3*d^2*e^3 + 10*b*c^3*d^3*e^2 - 15*b^2*c^2*d^2*e^3 - 25*a*b^2*c*e^5 - 5*b^3*c*d*e^4 + 18*
a*b*c^2*d*e^4))/(2*(4*a*c^4*d^6 + 4*a^4*c*e^6 - a^3*b^2*e^6 - b^2*c^3*d^6 + b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*
a^2*b^3*d*e^5 + 3*b^3*c^2*d^5*e - 3*b^4*c*d^4*e^2 + 12*a^2*c^3*d^4*e^2 + 12*a^3*c^2*d^2*e^4 - 12*a*b*c^3*d^5*e
- 12*a^3*b*c*d*e^5 + 2*a*b^3*c*d^3*e^3 + 9*a*b^2*c^2*d^4*e^2 - 24*a^2*b*c^2*d^3*e^3 + 9*a^2*b^2*c*d^2*e^4)))/
(x^2*(a*e^2 + c*d^2 + 2*b*d*e) + x*(b*d^2 + 2*a*d*e) + a*d^2 + x^3*(b*e^2 + 2*c*d*e) + c*e^2*x^4) + (log((-(4*
a*c - b^2)^3)^(1/2) - b^3 + 4*a*b*c + 8*a*c^2*x - 2*b^2*c*x)*((3*b^8*e^5)/2 + 64*a^4*c^4*e^5 + (3*b^5*e^5*(-(4
*a*c - b^2)^3)^(1/2))/2 + 2*c^5*d^5*(-(4*a*c - b^2)^3)^(1/2) + 84*a^2*b^4*c^2*e^5 - 144*a^3*b^2*c^3*e^5 - 320*
a^3*c^5*d^2*e^3 + 5*b^6*c^2*d^2*e^3 - 19*a*b^6*c*e^5 - 5*b^7*c*d*e^4 + 240*a^2*b^2*c^4*d^2*e^3 + 5*b^3*c^2*d^2
*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^3*c*e^5*(-(4*a*c - b^2)^3)^(1/2) + 60*a*b^5*c^2*d*e^4 + 320*a^3*b*c^4*d
*e^4 - 5*b*c^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) - 5*b^4*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b*c^2*e^5*(-(4
*a*c - b^2)^3)^(1/2) - 60*a*b^4*c^3*d^2*e^3 - 240*a^2*b^3*c^3*d*e^4 + 20*a*c^4*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2
) - 30*a^2*c^3*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 30*a*b*c^3*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 30*a*b^2*c^2*d*e
^4*(-(4*a*c - b^2)^3)^(1/2)))/(64*a^3*c^7*d^8 - a^4*b^6*e^8 + 64*a^7*c^3*e^8 - b^6*c^4*d^8 - b^10*d^4*e^4 + 12
*a*b^4*c^5*d^8 + 12*a^5*b^4*c*e^8 + 4*a*b^9*d^3*e^5 + 4*a^3*b^7*d*e^7 + 4*b^7*c^3*d^7*e + 4*b^9*c*d^5*e^3 - 48
*a^2*b^2*c^6*d^8 - 48*a^6*b^2*c^2*e^8 - 6*a^2*b^8*d^2*e^6 + 256*a^4*c^6*d^6*e^2 + 384*a^5*c^5*d^4*e^4 + 256*a^
6*c^4*d^2*e^6 - 6*b^8*c^2*d^6*e^2 - 240*a^2*b^4*c^4*d^6*e^2 + 48*a^2*b^5*c^3*d^5*e^3 + 90*a^2*b^6*c^2*d^4*e^4
+ 192*a^3*b^2*c^5*d^6*e^2 + 320*a^3*b^3*c^4*d^5*e^3 - 440*a^3*b^4*c^3*d^4*e^4 + 48*a^3*b^5*c^2*d^3*e^5 + 480*a
^4*b^2*c^4*d^4*e^4 + 320*a^4*b^3*c^3*d^3*e^5 - 240*a^4*b^4*c^2*d^2*e^6 + 192*a^5*b^2*c^3*d^2*e^6 - 48*a*b^5*c^
4*d^7*e - 256*a^3*b*c^6*d^7*e - 48*a^4*b^5*c*d*e^7 - 256*a^6*b*c^3*d*e^7 + 68*a*b^6*c^3*d^6*e^2 - 36*a*b^7*c^2
*d^5*e^3 + 192*a^2*b^3*c^5*d^7*e - 36*a^2*b^7*c*d^3*e^5 + 68*a^3*b^6*c*d^2*e^6 - 768*a^4*b*c^5*d^5*e^3 - 768*a
^5*b*c^4*d^3*e^5 + 192*a^5*b^3*c^2*d*e^7)