\(\int \frac {(d+e x)^5}{(a+b x+c x^2)^5} \, dx\) [2221]

Optimal result

Integrand size = 20, antiderivative size = 388 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

-1/4*(2*c*x+b)*(e*x+d)^5/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+1/12*(e*x+d)^4*(14*b*c*d-5*b^2*e-8*a*c*e+14*c*(-b*e+2*c*
d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3+1/12*(e*x+d)^3*(63*b^2*c*d*e+28*a*c^2*d*e-10*b^3*e^2-10*b*c*(3*a*e^2+7*c*
d^2)-c*(140*c^2*d^2+27*b^2*e^2-4*c*e*(-8*a*e+35*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^2+5/2*(-b*e+2*c*d)*(7*c^
2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-10*(-b*e+2*c
*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(9/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {750, 834, 818, 736, 632, 212} \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=-\frac {10 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac {5 (d+e x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (-c x \left (-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right )-10 b c \left (3 a e^2+7 c d^2\right )+28 a c^2 d e-10 b^3 e^2+63 b^2 c d e\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

[In]

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

[Out]

-1/4*((b + 2*c*x)*(d + e*x)^5)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^4*(14*b*c*d - 5*b^2*e - 8*a*c*
e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^3*(63*b^2*c*d*e + 28*a*c^2*d*
e - 10*b^3*e^2 - 10*b*c*(7*c*d^2 + 3*a*e^2) - c*(140*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(35*b*d - 8*a*e))*x))/(12*(b
^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)*(b
*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^
2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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 736

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

Rule 750

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

Rule 818

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

Rule 834

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {\int \frac {(d+e x)^4 (-14 c d+5 b e-4 c e x)}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )} \\ & = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {\int \frac {(d+e x)^3 \left (-2 \left (70 c^2 d^2+10 b^2 e^2-c e (63 b d-16 a e)\right )-14 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2} \\ & = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {\left (5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )^3} \\ & = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4} \\ & = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (10 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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 )^4} \\ & = -\frac {(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(985\) vs. \(2(388)=776\).

Time = 1.47 (sec) , antiderivative size = 985, normalized size of antiderivative = 2.54 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\frac {1}{12} \left (\frac {30 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac {5 b^5 c d e^4+3 b^6 e^5+4 c^3 \left (-48 a^3 e^5+35 c^3 d^5 x+50 a c^2 d^3 e^2 x+15 a^2 c d e^4 x\right )+b^2 c^2 e \left (129 a^2 e^4-25 c^2 d^3 (7 d-12 e x)-30 a c d e^2 (5 d-4 e x)\right )+10 b c^3 \left (7 c^2 d^4 (d-5 e x)+10 a c d^2 e^2 (d-3 e x)+3 a^2 e^4 (d-e x)\right )+10 b^3 c^2 e^2 \left (5 c d^2 (3 d-2 e x)+a e^2 (6 d-e x)\right )+b^4 c e^3 \left (-41 a e^2+10 c d (-5 d+e x)\right )}{c^3 \left (-b^2+4 a c\right )^3 (a+x (b+c x))^2}-\frac {3 \left (b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )\right )}{c^4 \left (-b^2+4 a c\right ) (a+x (b+c x))^4}+\frac {-3 b^6 e^5+3 b^5 c e^4 (5 d+2 e x)+b^4 c e^3 \left (27 a e^2-10 c d (3 d+e x)\right )-10 b^3 c^2 e^2 \left (5 a e^2 (2 d+e x)+c d^2 (-3 d+2 e x)\right )+4 c^3 \left (16 a^3 e^5+7 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (16 d+9 e x)\right )+2 b c^3 \left (7 c^2 d^4 (d-5 e x)+10 a c d^2 e^2 (d-3 e x)+5 a^2 e^4 (23 d+9 e x)\right )+b^2 c^2 e \left (-83 a^2 e^4+5 c^2 d^3 (-7 d+12 e x)+10 a c d e^2 (13 d+12 e x)\right )}{c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac {120 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{9/2}}\right ) \]

[In]

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

[Out]

((30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a
*b*d*e + 3*a^2*e^2))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (5*b^5*c*d*e^4 + 3*b^6*e^5 + 4*c^3*(
-48*a^3*e^5 + 35*c^3*d^5*x + 50*a*c^2*d^3*e^2*x + 15*a^2*c*d*e^4*x) + b^2*c^2*e*(129*a^2*e^4 - 25*c^2*d^3*(7*d
- 12*e*x) - 30*a*c*d*e^2*(5*d - 4*e*x)) + 10*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 3*a^
2*e^4*(d - e*x)) + 10*b^3*c^2*e^2*(5*c*d^2*(3*d - 2*e*x) + a*e^2*(6*d - e*x)) + b^4*c*e^3*(-41*a*e^2 + 10*c*d*
(-5*d + e*x)))/(c^3*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) - (3*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*
e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*
e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a
^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x))))/(c^4*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (-3*b^6*e^5 + 3*
b^5*c*e^4*(5*d + 2*e*x) + b^4*c*e^3*(27*a*e^2 - 10*c*d*(3*d + e*x)) - 10*b^3*c^2*e^2*(5*a*e^2*(2*d + e*x) + c*
d^2*(-3*d + 2*e*x)) + 4*c^3*(16*a^3*e^5 + 7*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(16*d + 9*e*x)) + 2
*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(23*d + 9*e*x)) + b^2*c^2*e*(-83*a^2*e^
4 + 5*c^2*d^3*(-7*d + 12*e*x) + 10*a*c*d*e^2*(13*d + 12*e*x)))/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (12
0*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*
d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2343\) vs. \(2(376)=752\).

Time = 17.48 (sec) , antiderivative size = 2344, normalized size of antiderivative = 6.04

method	result	size
default	\(\text {Expression too large to display}\)	\(2344\)
risch	\(\text {Expression too large to display}\)	\(4895\)

[In]

int((e*x+d)^5/(c*x^2+b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

(-5*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10
*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*
b^6*c+b^8)*c^3*x^7-35/2*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*
d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+
96*a^2*b^4*c^2-16*a*b^6*c+b^8)*c^2*b*x^6-5/3*c*(11*a*c+13*b^2)*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b
^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14
*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5-1/12*(768*a^4*c^4*e^5+882*a^3*b^2*c^
3*e^5-3300*a^3*b*c^4*d*e^4+1213*a^2*b^4*c^2*e^5-7350*a^2*b^3*c^3*d*e^4+16500*a^2*b^2*c^4*d^2*e^3-11000*a^2*b*c
^5*d^3*e^2+77*a*b^6*c*e^5-2050*a*b^5*c^2*d*e^4+9250*a*b^4*c^3*d^2*e^3-19000*a*b^3*c^4*d^3*e^2+19250*a*b^2*c^5*
d^4*e-7700*a*b*c^6*d^5+3*b^8*e^5-125*b^7*c*d*e^4+1250*b^6*c^2*d^2*e^3-3750*b^5*c^3*d^3*e^2+4375*b^4*c^4*d^4*e-
1750*b^3*c^5*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(219*a^4*b*c^3*e^5+330
*a^4*c^4*d*e^4+376*a^3*b^3*c^2*e^5-2250*a^3*b^2*c^3*d*e^4+2190*a^3*b*c^4*d^2*e^3-1460*a^3*c^5*d^3*e^2+110*a^2*
b^5*c*e^5-1015*a^2*b^4*c^2*d*e^4+3760*a^2*b^3*c^3*d^2*e^3-4210*a^2*b^2*c^4*d^3*e^2+2555*a^2*b*c^5*d^4*e-1022*a
^2*c^6*d^5+3*a*b^7*e^5-185*a*b^6*c*d*e^4+1100*a*b^5*c^2*d^2*e^3-3090*a*b^4*c^3*d^3*e^2+3535*a*b^3*c^4*d^4*e-14
14*a*b^2*c^5*d^5+30*b^7*c*d^2*e^3-90*b^6*c^2*d^3*e^2+105*b^5*c^3*d^4*e-42*b^4*c^4*d^5)/c/(256*a^4*c^4-256*a^3*
b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3-1/6*(256*a^5*c^3*e^5+401*a^4*b^2*c^2*e^5-1570*a^4*b*c^3*d*e^4+2560*
a^4*c^4*d^2*e^3+399*a^3*b^4*c*e^5-2540*a^3*b^3*c^2*d*e^4+4010*a^3*b^2*c^3*d^2*e^3-4380*a^3*b*c^4*d^3*e^2+9*a^2
*b^6*e^5-645*a^2*b^5*c*d*e^4+3990*a^2*b^4*c^2*d^2*e^3-7130*a^2*b^3*c^3*d^3*e^2+7665*a^2*b^2*c^4*d^4*e-3066*a^2
*b*c^5*d^5+90*a*b^6*c*d^2*e^3-820*a*b^5*c^2*d^3*e^2+980*a*b^4*c^3*d^4*e-392*a*b^3*c^4*d^5+30*b^7*c*d^3*e^2-35*
b^6*c^2*d^4*e+14*b^5*c^3*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^2-1/3*(83*a^5*b*
c^2*e^5+90*a^5*c^3*d*e^4+151*a^4*b^3*c*e^5-920*a^4*b^2*c^2*d*e^4+830*a^4*b*c^3*d^2*e^3+300*a^4*c^4*d^3*e^2+3*a
^3*b^5*e^5-235*a^3*b^4*c*d*e^4+1510*a^3*b^3*c^2*d^2*e^3-2790*a^3*b^2*c^3*d^3*e^2+1395*a^3*b*c^4*d^4*e-558*a^3*
c^5*d^5+30*a^2*b^5*c*d^2*e^3-280*a^2*b^4*c^2*d^3*e^2+870*a^2*b^3*c^3*d^4*e-348*a^2*b^2*c^4*d^5+10*a*b^6*c*d^3*
e^2-95*a*b^5*c^2*d^4*e+38*a*b^4*c^3*d^5+5*b^7*c*d^4*e-2*b^6*c^2*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4
*c^2-16*a*b^6*c+b^8)*x-1/12*(128*a^6*c^2*e^5+166*a^5*b^2*c*e^5-1100*a^5*b*c^2*d*e^4+1280*a^5*c^3*d^2*e^3+3*a^4
*b^4*e^5-250*a^4*b^3*c*d*e^4+1660*a^4*b^2*c^2*d^2*e^3-3240*a^4*b*c^3*d^3*e^2+1920*a^4*c^4*d^4*e+30*a^3*b^4*c*d
^2*e^3-280*a^3*b^3*c^2*d^3*e^2+870*a^3*b^2*c^3*d^4*e-1116*a^3*b*c^4*d^5+10*a^2*b^5*c*d^3*e^2-95*a^2*b^4*c^2*d^
4*e+326*a^2*b^3*c^3*d^5+5*a*b^6*c*d^4*e-50*a*b^5*c^2*d^5+3*b^7*c*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^
4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4-10*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2
*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^
4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4419 vs. \(2 (376) = 752\).

Time = 0.54 (sec) , antiderivative size = 8858, normalized size of antiderivative = 22.83 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^5/(c*x^2+b*x+a)^5,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. 2431 vs. \(2 (376) = 752\).

Time = 0.30 (sec) , antiderivative size = 2431, normalized size of antiderivative = 6.27 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

10*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*
c^8*d^5*x^7 - 2100*b*c^7*d^4*e*x^7 + 1800*b^2*c^6*d^3*e^2*x^7 + 1200*a*c^7*d^3*e^2*x^7 - 600*b^3*c^5*d^2*e^3*x
^7 - 1800*a*b*c^6*d^2*e^3*x^7 + 60*b^4*c^4*d*e^4*x^7 + 720*a*b^2*c^5*d*e^4*x^7 + 360*a^2*c^6*d*e^4*x^7 - 60*a*
b^3*c^4*e^5*x^7 - 180*a^2*b*c^5*e^5*x^7 + 2940*b*c^7*d^5*x^6 - 7350*b^2*c^6*d^4*e*x^6 + 6300*b^3*c^5*d^3*e^2*x
^6 + 4200*a*b*c^6*d^3*e^2*x^6 - 2100*b^4*c^4*d^2*e^3*x^6 - 6300*a*b^2*c^5*d^2*e^3*x^6 + 210*b^5*c^3*d*e^4*x^6
+ 2520*a*b^3*c^4*d*e^4*x^6 + 1260*a^2*b*c^5*d*e^4*x^6 - 210*a*b^4*c^3*e^5*x^6 - 630*a^2*b^2*c^4*e^5*x^6 + 3640
*b^2*c^6*d^5*x^5 + 3080*a*c^7*d^5*x^5 - 9100*b^3*c^5*d^4*e*x^5 - 7700*a*b*c^6*d^4*e*x^5 + 7800*b^4*c^4*d^3*e^2
*x^5 + 11800*a*b^2*c^5*d^3*e^2*x^5 + 4400*a^2*c^6*d^3*e^2*x^5 - 2600*b^5*c^3*d^2*e^3*x^5 - 10000*a*b^3*c^4*d^2
*e^3*x^5 - 6600*a^2*b*c^5*d^2*e^3*x^5 + 260*b^6*c^2*d*e^4*x^5 + 3340*a*b^4*c^3*d*e^4*x^5 + 4200*a^2*b^2*c^4*d*
e^4*x^5 + 1320*a^3*c^5*d*e^4*x^5 - 260*a*b^5*c^2*e^5*x^5 - 1000*a^2*b^3*c^3*e^5*x^5 - 660*a^3*b*c^4*e^5*x^5 +
1750*b^3*c^5*d^5*x^4 + 7700*a*b*c^6*d^5*x^4 - 4375*b^4*c^4*d^4*e*x^4 - 19250*a*b^2*c^5*d^4*e*x^4 + 3750*b^5*c^
3*d^3*e^2*x^4 + 19000*a*b^3*c^4*d^3*e^2*x^4 + 11000*a^2*b*c^5*d^3*e^2*x^4 - 1250*b^6*c^2*d^2*e^3*x^4 - 9250*a*
b^4*c^3*d^2*e^3*x^4 - 16500*a^2*b^2*c^4*d^2*e^3*x^4 + 125*b^7*c*d*e^4*x^4 + 2050*a*b^5*c^2*d*e^4*x^4 + 7350*a^
2*b^3*c^3*d*e^4*x^4 + 3300*a^3*b*c^4*d*e^4*x^4 - 3*b^8*e^5*x^4 - 77*a*b^6*c*e^5*x^4 - 1213*a^2*b^4*c^2*e^5*x^4
- 882*a^3*b^2*c^3*e^5*x^4 - 768*a^4*c^4*e^5*x^4 + 168*b^4*c^4*d^5*x^3 + 5656*a*b^2*c^5*d^5*x^3 + 4088*a^2*c^6
*d^5*x^3 - 420*b^5*c^3*d^4*e*x^3 - 14140*a*b^3*c^4*d^4*e*x^3 - 10220*a^2*b*c^5*d^4*e*x^3 + 360*b^6*c^2*d^3*e^2
*x^3 + 12360*a*b^4*c^3*d^3*e^2*x^3 + 16840*a^2*b^2*c^4*d^3*e^2*x^3 + 5840*a^3*c^5*d^3*e^2*x^3 - 120*b^7*c*d^2*
e^3*x^3 - 4400*a*b^5*c^2*d^2*e^3*x^3 - 15040*a^2*b^3*c^3*d^2*e^3*x^3 - 8760*a^3*b*c^4*d^2*e^3*x^3 + 740*a*b^6*
c*d*e^4*x^3 + 4060*a^2*b^4*c^2*d*e^4*x^3 + 9000*a^3*b^2*c^3*d*e^4*x^3 - 1320*a^4*c^4*d*e^4*x^3 - 12*a*b^7*e^5*
x^3 - 440*a^2*b^5*c*e^5*x^3 - 1504*a^3*b^3*c^2*e^5*x^3 - 876*a^4*b*c^3*e^5*x^3 - 28*b^5*c^3*d^5*x^2 + 784*a*b^
3*c^4*d^5*x^2 + 6132*a^2*b*c^5*d^5*x^2 + 70*b^6*c^2*d^4*e*x^2 - 1960*a*b^4*c^3*d^4*e*x^2 - 15330*a^2*b^2*c^4*d
^4*e*x^2 - 60*b^7*c*d^3*e^2*x^2 + 1640*a*b^5*c^2*d^3*e^2*x^2 + 14260*a^2*b^3*c^3*d^3*e^2*x^2 + 8760*a^3*b*c^4*
d^3*e^2*x^2 - 180*a*b^6*c*d^2*e^3*x^2 - 7980*a^2*b^4*c^2*d^2*e^3*x^2 - 8020*a^3*b^2*c^3*d^2*e^3*x^2 - 5120*a^4
*c^4*d^2*e^3*x^2 + 1290*a^2*b^5*c*d*e^4*x^2 + 5080*a^3*b^3*c^2*d*e^4*x^2 + 3140*a^4*b*c^3*d*e^4*x^2 - 18*a^2*b
^6*e^5*x^2 - 798*a^3*b^4*c*e^5*x^2 - 802*a^4*b^2*c^2*e^5*x^2 - 512*a^5*c^3*e^5*x^2 + 8*b^6*c^2*d^5*x - 152*a*b
^4*c^3*d^5*x + 1392*a^2*b^2*c^4*d^5*x + 2232*a^3*c^5*d^5*x - 20*b^7*c*d^4*e*x + 380*a*b^5*c^2*d^4*e*x - 3480*a
^2*b^3*c^3*d^4*e*x - 5580*a^3*b*c^4*d^4*e*x - 40*a*b^6*c*d^3*e^2*x + 1120*a^2*b^4*c^2*d^3*e^2*x + 11160*a^3*b^
2*c^3*d^3*e^2*x - 1200*a^4*c^4*d^3*e^2*x - 120*a^2*b^5*c*d^2*e^3*x - 6040*a^3*b^3*c^2*d^2*e^3*x - 3320*a^4*b*c
^3*d^2*e^3*x + 940*a^3*b^4*c*d*e^4*x + 3680*a^4*b^2*c^2*d*e^4*x - 360*a^5*c^3*d*e^4*x - 12*a^3*b^5*e^5*x - 604
*a^4*b^3*c*e^5*x - 332*a^5*b*c^2*e^5*x - 3*b^7*c*d^5 + 50*a*b^5*c^2*d^5 - 326*a^2*b^3*c^3*d^5 + 1116*a^3*b*c^4
*d^5 - 5*a*b^6*c*d^4*e + 95*a^2*b^4*c^2*d^4*e - 870*a^3*b^2*c^3*d^4*e - 1920*a^4*c^4*d^4*e - 10*a^2*b^5*c*d^3*
e^2 + 280*a^3*b^3*c^2*d^3*e^2 + 3240*a^4*b*c^3*d^3*e^2 - 30*a^3*b^4*c*d^2*e^3 - 1660*a^4*b^2*c^2*d^2*e^3 - 128
0*a^5*c^3*d^2*e^3 + 250*a^4*b^3*c*d*e^4 + 1100*a^5*b*c^2*d*e^4 - 3*a^4*b^4*e^5 - 166*a^5*b^2*c*e^5 - 128*a^6*c
^2*e^5)/((b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*(c*x^2 + b*x + a)^4)

Mupad [B] (verification not implemented)

Time = 12.20 (sec) , antiderivative size = 2722, normalized size of antiderivative = 7.02 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(10*atan((((5*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e)*(b^9 + 256*a
^4*b*c^4 + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 - 16*a*b^7*c))/((4*a*c - b^2)^(9/2)*(b^8 + 256*a^4*c^4 + 96*a^2*b^
4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (10*c*x*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)*(b^2*e^2 + 7*c^2*d^2 +
3*a*c*e^2 - 7*b*c*d*e))/(4*a*c - b^2)^(9/2))*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*
c))/(70*c^4*d^5 - 5*a*b^3*e^5 + 5*b^4*d*e^4 + 100*a*c^3*d^3*e^2 + 30*a^2*c^2*d*e^4 - 50*b^3*c*d^2*e^3 + 150*b^
2*c^2*d^3*e^2 - 15*a^2*b*c*e^5 - 175*b*c^3*d^4*e + 60*a*b^2*c*d*e^4 - 150*a*b*c^2*d^2*e^3))*(b*e - 2*c*d)*(a*e
^2 + c*d^2 - b*d*e)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(4*a*c - b^2)^(9/2) - ((3*b^7*c*d^5 + 3*a^4
*b^4*e^5 + 128*a^6*c^2*e^5 - 50*a*b^5*c^2*d^5 - 1116*a^3*b*c^4*d^5 + 166*a^5*b^2*c*e^5 + 1920*a^4*c^4*d^4*e +
326*a^2*b^3*c^3*d^5 + 1280*a^5*c^3*d^2*e^3 + 5*a*b^6*c*d^4*e - 280*a^3*b^3*c^2*d^3*e^2 + 1660*a^4*b^2*c^2*d^2*
e^3 - 250*a^4*b^3*c*d*e^4 - 1100*a^5*b*c^2*d*e^4 - 95*a^2*b^4*c^2*d^4*e + 10*a^2*b^5*c*d^3*e^2 + 870*a^3*b^2*c
^3*d^4*e + 30*a^3*b^4*c*d^2*e^3 - 3240*a^4*b*c^3*d^3*e^2)/(12*c*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*
b^2*c^3 - 16*a*b^6*c)) + (x^3*(3*a*b^7*e^5 - 1022*a^2*c^6*d^5 - 42*b^4*c^4*d^5 - 1414*a*b^2*c^5*d^5 + 110*a^2*
b^5*c*e^5 + 219*a^4*b*c^3*e^5 + 330*a^4*c^4*d*e^4 + 105*b^5*c^3*d^4*e + 30*b^7*c*d^2*e^3 + 376*a^3*b^3*c^2*e^5
- 1460*a^3*c^5*d^3*e^2 - 90*b^6*c^2*d^3*e^2 - 185*a*b^6*c*d*e^4 - 4210*a^2*b^2*c^4*d^3*e^2 + 3760*a^2*b^3*c^3
*d^2*e^3 + 3535*a*b^3*c^4*d^4*e + 2555*a^2*b*c^5*d^4*e - 3090*a*b^4*c^3*d^3*e^2 + 1100*a*b^5*c^2*d^2*e^3 - 101
5*a^2*b^4*c^2*d*e^4 + 2190*a^3*b*c^4*d^2*e^3 - 2250*a^3*b^2*c^3*d*e^4))/(3*c*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c
^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x^2*(9*a^2*b^6*e^5 + 256*a^5*c^3*e^5 + 14*b^5*c^3*d^5 - 392*a*b^3*c^4*d
^5 - 3066*a^2*b*c^5*d^5 + 399*a^3*b^4*c*e^5 - 35*b^6*c^2*d^4*e + 30*b^7*c*d^3*e^2 + 401*a^4*b^2*c^2*e^5 + 2560
*a^4*c^4*d^2*e^3 - 7130*a^2*b^3*c^3*d^3*e^2 + 3990*a^2*b^4*c^2*d^2*e^3 + 4010*a^3*b^2*c^3*d^2*e^3 + 980*a*b^4*
c^3*d^4*e + 90*a*b^6*c*d^2*e^3 - 645*a^2*b^5*c*d*e^4 - 1570*a^4*b*c^3*d*e^4 - 820*a*b^5*c^2*d^3*e^2 + 7665*a^2
*b^2*c^4*d^4*e - 4380*a^3*b*c^4*d^3*e^2 - 2540*a^3*b^3*c^2*d*e^4))/(6*c*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 -
256*a^3*b^2*c^3 - 16*a*b^6*c)) - (5*x^5*(11*a*c^2 + 13*b^2*c)*(14*c^4*d^5 - a*b^3*e^5 + b^4*d*e^4 + 20*a*c^3*d
^3*e^2 + 6*a^2*c^2*d*e^4 - 10*b^3*c*d^2*e^3 + 30*b^2*c^2*d^3*e^2 - 3*a^2*b*c*e^5 - 35*b*c^3*d^4*e + 12*a*b^2*c
*d*e^4 - 30*a*b*c^2*d^2*e^3))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x^4*(
3*b^8*e^5 + 768*a^4*c^4*e^5 - 1750*b^3*c^5*d^5 + 4375*b^4*c^4*d^4*e + 1213*a^2*b^4*c^2*e^5 + 882*a^3*b^2*c^3*e
^5 - 3750*b^5*c^3*d^3*e^2 + 1250*b^6*c^2*d^2*e^3 - 7700*a*b*c^6*d^5 + 77*a*b^6*c*e^5 - 125*b^7*c*d*e^4 + 16500
*a^2*b^2*c^4*d^2*e^3 + 19250*a*b^2*c^5*d^4*e - 2050*a*b^5*c^2*d*e^4 - 3300*a^3*b*c^4*d*e^4 - 19000*a*b^3*c^4*d
^3*e^2 + 9250*a*b^4*c^3*d^2*e^3 - 11000*a^2*b*c^5*d^3*e^2 - 7350*a^2*b^3*c^3*d*e^4))/(12*c*(b^8 + 256*a^4*c^4
+ 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x*(3*a^3*b^5*e^5 - 558*a^3*c^5*d^5 - 2*b^6*c^2*d^5 + 38*a
*b^4*c^3*d^5 + 151*a^4*b^3*c*e^5 + 83*a^5*b*c^2*e^5 + 90*a^5*c^3*d*e^4 - 348*a^2*b^2*c^4*d^5 + 300*a^4*c^4*d^3
*e^2 + 5*b^7*c*d^4*e - 280*a^2*b^4*c^2*d^3*e^2 - 2790*a^3*b^2*c^3*d^3*e^2 + 1510*a^3*b^3*c^2*d^2*e^3 - 95*a*b^
5*c^2*d^4*e + 10*a*b^6*c*d^3*e^2 + 1395*a^3*b*c^4*d^4*e - 235*a^3*b^4*c*d*e^4 + 870*a^2*b^3*c^3*d^4*e + 30*a^2
*b^5*c*d^2*e^3 + 830*a^4*b*c^3*d^2*e^3 - 920*a^4*b^2*c^2*d*e^4))/(3*c*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 25
6*a^3*b^2*c^3 - 16*a*b^6*c)) - (5*c^3*x^7*(14*c^4*d^5 - a*b^3*e^5 + b^4*d*e^4 + 20*a*c^3*d^3*e^2 + 6*a^2*c^2*d
*e^4 - 10*b^3*c*d^2*e^3 + 30*b^2*c^2*d^3*e^2 - 3*a^2*b*c*e^5 - 35*b*c^3*d^4*e + 12*a*b^2*c*d*e^4 - 30*a*b*c^2*
d^2*e^3))/(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c) - (35*b*c^2*x^6*(14*c^4*d^5 - a*
b^3*e^5 + b^4*d*e^4 + 20*a*c^3*d^3*e^2 + 6*a^2*c^2*d*e^4 - 10*b^3*c*d^2*e^3 + 30*b^2*c^2*d^3*e^2 - 3*a^2*b*c*e
^5 - 35*b*c^3*d^4*e + 12*a*b^2*c*d*e^4 - 30*a*b*c^2*d^2*e^3))/(2*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3
*b^2*c^3 - 16*a*b^6*c)))/(x^4*(b^4 + 6*a^2*c^2 + 12*a*b^2*c) + a^4 + c^4*x^8 + x^2*(4*a^3*c + 6*a^2*b^2) + x^6
*(4*a*c^3 + 6*b^2*c^2) + x^3*(4*a*b^3 + 12*a^2*b*c) + x^5*(4*b^3*c + 12*a*b*c^2) + 4*b*c^3*x^7 + 4*a^3*b*x)