3.1593 \(\int \frac{b+2 c x}{(d+e x) (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{2 e \left (-c x \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )-2 b c \left (c d^2-5 a e^2\right )-12 a c^2 d e+5 b^2 c d e-3 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

[Out]

(-2*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x
^2)^(3/2)) - (2*e*(5*b^2*c*d*e - 12*a*c^2*d*e - 3*b^3*e^2 - 2*b*c*(c*d^2 - 5*a*e^2) - c*(4*c^2*d^2 + 3*b^2*e^2
- 4*c*e*(b*d + 2*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x + c*x^2]) - (e^3*(2*c*d -
b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 -
b*d*e + a*e^2)^(5/2)

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Rubi [A]  time = 0.293255, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {822, 12, 724, 206} \[ -\frac{2 e \left (-c x \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )-2 b c \left (c d^2-5 a e^2\right )-12 a c^2 d e+5 b^2 c d e-3 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x
^2)^(3/2)) - (2*e*(5*b^2*c*d*e - 12*a*c^2*d*e - 3*b^3*e^2 - 2*b*c*(c*d^2 - 5*a*e^2) - c*(4*c^2*d^2 + 3*b^2*e^2
- 4*c*e*(b*d + 2*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x + c*x^2]) - (e^3*(2*c*d -
b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 -
b*d*e + a*e^2)^(5/2)

Rule 822

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (b^2-4 a c\right ) e (2 c d-3 b e)-2 c \left (b^2-4 a c\right ) e^2 x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e \left (5 b^2 c d e-12 a c^2 d e-3 b^3 e^2-2 b c \left (c d^2-5 a e^2\right )-c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{a+b x+c x^2}}+\frac{4 \int -\frac{3 \left (b^2-4 a c\right )^2 e^3 (2 c d-b e)}{4 (d+e x) \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e \left (5 b^2 c d e-12 a c^2 d e-3 b^3 e^2-2 b c \left (c d^2-5 a e^2\right )-c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{\left (e^3 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e \left (5 b^2 c d e-12 a c^2 d e-3 b^3 e^2-2 b c \left (c d^2-5 a e^2\right )-c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{a+b x+c x^2}}+\frac{\left (2 e^3 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e \left (5 b^2 c d e-12 a c^2 d e-3 b^3 e^2-2 b c \left (c d^2-5 a e^2\right )-c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.435503, size = 249, normalized size = 0.86 \[ \frac{2 e \left (2 b c \left (c d (d-2 e x)-5 a e^2\right )+4 c^2 \left (a e (3 d-2 e x)+c d^2 x\right )+b^2 c e (3 e x-5 d)+3 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}+\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{2 (b e-c d+c e x)}{3 (a+x (b+c x))^{3/2} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x))/(3*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2)) + (2*e*(3*b^3*e^2 + b^2*c*e*
(-5*d + 3*e*x) + 2*b*c*(-5*a*e^2 + c*d*(d - 2*e*x)) + 4*c^2*(c*d^2*x + a*e*(3*d - 2*e*x))))/(3*(b^2 - 4*a*c)*(
c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) + (e^3*(2*c*d - b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*
e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)

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Maple [B]  time = 0.014, size = 2451, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^3-e
^3/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2
*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b
+8*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b
*c^2*d+4/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b
^2*c*d+32/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*b^2*d-8/3*e/(a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*b^3+e^3/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b+1/
3*e/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b-2/3/(a*e^2-b*d*e+c
*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d+2*e^2/(a*e^2-b*d*e+c*d^2)^2/((a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d+32/3*c^2/e/(4*a*c-b^2)^2/(c*
x^2+b*x+a)^(1/2)*b-e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*b^3+8/3*c^2/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+4/3*c/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b+64/3*c^
3/e/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-2*e^2/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*c*d+64/3/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*d+8/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(3/2)*x*c^2*b*d-2*e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c-8*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^3*d^2-64/3/e/(a*e^2-b*d*e+c*d^2)*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^2-32/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^2-2/3*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c*b^2-8/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^3*d^2-4*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^2*d^2-4/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c^2*d^2-16/3*e/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b
^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2+4*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a
*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c*d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 24.5995, size = 7698, normalized size = 26.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*(2*(a^2*b^2*c - 4*a^3*c^2)*d*e^3 - (a^2*b^3 - 4*a^3*b*c)*e^4 + (2*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^
2 - 4*a*b*c^3)*e^4)*x^4 + 2*(2*(b^3*c^2 - 4*a*b*c^3)*d*e^3 - (b^4*c - 4*a*b^2*c^2)*e^4)*x^3 + (2*(b^4*c - 2*a*
b^2*c^2 - 8*a^2*c^3)*d*e^3 - (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^4)*x^2 + 2*(2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 - (
a*b^4 - 4*a^2*b^2*c)*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c
^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e
+ (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*((b^2*
c^3 - 4*a*c^4)*d^5 - (3*b^3*c^2 - 10*a*b*c^3)*d^4*e + (3*b^4*c - 3*a*b^2*c^2 - 20*a^2*c^3)*d^3*e^2 - (b^5 + 8*
a*b^3*c - 36*a^2*b*c^2)*d^2*e^3 + (5*a*b^4 - 12*a^2*b^2*c - 16*a^3*c^2)*d*e^4 - 2*(2*a^2*b^3 - 7*a^3*b*c)*e^5
- (4*c^5*d^4*e - 8*b*c^4*d^3*e^2 + (7*b^2*c^3 - 4*a*c^4)*d^2*e^3 - (3*b^3*c^2 - 4*a*b*c^3)*d*e^4 + (3*a*b^2*c^
2 - 8*a^2*c^3)*e^5)*x^3 - 3*(2*b*c^4*d^4*e - (5*b^2*c^3 - 4*a*c^4)*d^3*e^2 + (5*b^3*c^2 - 8*a*b*c^3)*d^2*e^3 -
(2*b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d*e^4 + 2*(a*b^3*c - 3*a^2*b*c^2)*e^5)*x^2 - 3*(b^2*c^3*d^4*e - (3*b^3*c^
2 - 4*a*b*c^3)*d^3*e^2 + (3*b^4*c - 5*a*b^2*c^2 - 4*a^2*c^3)*d^2*e^3 - (b^5 - 8*a^2*b*c^2)*d*e^4 + (a*b^4 - 2*
a^2*b^2*c - 4*a^3*c^2)*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^2*c^3 - 4*a^3*c^4)*d^6 - 3*(a^2*b^3*c^2 - 4*a^3*
b*c^3)*d^5*e + 3*(a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^4*e^2 - (a^2*b^5 + 2*a^3*b^3*c - 24*a^4*b*c^2)*d^3*
e^3 + 3*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^2*e^4 - 3*(a^4*b^3 - 4*a^5*b*c)*d*e^5 + (a^5*b^2 - 4*a^6*c)*e^6
+ ((b^2*c^5 - 4*a*c^6)*d^6 - 3*(b^3*c^4 - 4*a*b*c^5)*d^5*e + 3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^4*e^2 - (
b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*e^3 + 3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^3*c^4)*d^2*e^4 - 3*(a^2*b^3
*c^2 - 4*a^3*b*c^3)*d*e^5 + (a^3*b^2*c^2 - 4*a^4*c^3)*e^6)*x^4 + 2*((b^3*c^4 - 4*a*b*c^5)*d^6 - 3*(b^4*c^3 - 4
*a*b^2*c^4)*d^5*e + 3*(b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e^2 - (b^6*c + 2*a*b^4*c^2 - 24*a^2*b^2*c^3)*d
^3*e^3 + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^4 - 3*(a^2*b^4*c - 4*a^3*b^2*c^2)*d*e^5 + (a^3*b^3*c
- 4*a^4*b*c^2)*e^6)*x^3 + ((b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^6 - 3*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d
^5*e + 3*(b^6*c - a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^4*e^2 - (b^7 + 4*a*b^5*c - 20*a^2*b^3*c^2 - 48*a^3
*b*c^3)*d^3*e^3 + 3*(a*b^6 - a^2*b^4*c - 10*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^4 - 3*(a^2*b^5 - 2*a^3*b^3*c - 8*a^
4*b*c^2)*d*e^5 + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*e^6)*x^2 + 2*((a*b^3*c^3 - 4*a^2*b*c^4)*d^6 - 3*(a*b^4*c^
2 - 4*a^2*b^2*c^3)*d^5*e + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^4*e^2 - (a*b^6 + 2*a^2*b^4*c - 24*a^3*b
^2*c^2)*d^3*e^3 + 3*(a^2*b^5 - 3*a^3*b^3*c - 4*a^4*b*c^2)*d^2*e^4 - 3*(a^3*b^4 - 4*a^4*b^2*c)*d*e^5 + (a^4*b^3
- 4*a^5*b*c)*e^6)*x), -1/3*(3*(2*(a^2*b^2*c - 4*a^3*c^2)*d*e^3 - (a^2*b^3 - 4*a^3*b*c)*e^4 + (2*(b^2*c^3 - 4*
a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^3)*e^4)*x^4 + 2*(2*(b^3*c^2 - 4*a*b*c^3)*d*e^3 - (b^4*c - 4*a*b^2*c^2)*e^4)*
x^3 + (2*(b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^3 - (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^4)*x^2 + 2*(2*(a*b^3*c -
4*a^2*b*c^2)*d*e^3 - (a*b^4 - 4*a^2*b^2*c)*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*
e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c
*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + 2*((b^2*c^3 - 4*a*c^4)*d^5 - (3*b^3*c^2 - 10*a*b*c^3
)*d^4*e + (3*b^4*c - 3*a*b^2*c^2 - 20*a^2*c^3)*d^3*e^2 - (b^5 + 8*a*b^3*c - 36*a^2*b*c^2)*d^2*e^3 + (5*a*b^4 -
12*a^2*b^2*c - 16*a^3*c^2)*d*e^4 - 2*(2*a^2*b^3 - 7*a^3*b*c)*e^5 - (4*c^5*d^4*e - 8*b*c^4*d^3*e^2 + (7*b^2*c^
3 - 4*a*c^4)*d^2*e^3 - (3*b^3*c^2 - 4*a*b*c^3)*d*e^4 + (3*a*b^2*c^2 - 8*a^2*c^3)*e^5)*x^3 - 3*(2*b*c^4*d^4*e -
(5*b^2*c^3 - 4*a*c^4)*d^3*e^2 + (5*b^3*c^2 - 8*a*b*c^3)*d^2*e^3 - (2*b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d*e^4 +
2*(a*b^3*c - 3*a^2*b*c^2)*e^5)*x^2 - 3*(b^2*c^3*d^4*e - (3*b^3*c^2 - 4*a*b*c^3)*d^3*e^2 + (3*b^4*c - 5*a*b^2*
c^2 - 4*a^2*c^3)*d^2*e^3 - (b^5 - 8*a^2*b*c^2)*d*e^4 + (a*b^4 - 2*a^2*b^2*c - 4*a^3*c^2)*e^5)*x)*sqrt(c*x^2 +
b*x + a))/((a^2*b^2*c^3 - 4*a^3*c^4)*d^6 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^5*e + 3*(a^2*b^4*c - 3*a^3*b^2*c^2
- 4*a^4*c^3)*d^4*e^2 - (a^2*b^5 + 2*a^3*b^3*c - 24*a^4*b*c^2)*d^3*e^3 + 3*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*
d^2*e^4 - 3*(a^4*b^3 - 4*a^5*b*c)*d*e^5 + (a^5*b^2 - 4*a^6*c)*e^6 + ((b^2*c^5 - 4*a*c^6)*d^6 - 3*(b^3*c^4 - 4*
a*b*c^5)*d^5*e + 3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^4*e^2 - (b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*e^
3 + 3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^3*c^4)*d^2*e^4 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^5 + (a^3*b^2*c^2 - 4
*a^4*c^3)*e^6)*x^4 + 2*((b^3*c^4 - 4*a*b*c^5)*d^6 - 3*(b^4*c^3 - 4*a*b^2*c^4)*d^5*e + 3*(b^5*c^2 - 3*a*b^3*c^3
- 4*a^2*b*c^4)*d^4*e^2 - (b^6*c + 2*a*b^4*c^2 - 24*a^2*b^2*c^3)*d^3*e^3 + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*
b*c^3)*d^2*e^4 - 3*(a^2*b^4*c - 4*a^3*b^2*c^2)*d*e^5 + (a^3*b^3*c - 4*a^4*b*c^2)*e^6)*x^3 + ((b^4*c^3 - 2*a*b^
2*c^4 - 8*a^2*c^5)*d^6 - 3*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^5*e + 3*(b^6*c - a*b^4*c^2 - 10*a^2*b^2*c^3
- 8*a^3*c^4)*d^4*e^2 - (b^7 + 4*a*b^5*c - 20*a^2*b^3*c^2 - 48*a^3*b*c^3)*d^3*e^3 + 3*(a*b^6 - a^2*b^4*c - 10*
a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^4 - 3*(a^2*b^5 - 2*a^3*b^3*c - 8*a^4*b*c^2)*d*e^5 + (a^3*b^4 - 2*a^4*b^2*c - 8*
a^5*c^2)*e^6)*x^2 + 2*((a*b^3*c^3 - 4*a^2*b*c^4)*d^6 - 3*(a*b^4*c^2 - 4*a^2*b^2*c^3)*d^5*e + 3*(a*b^5*c - 3*a^
2*b^3*c^2 - 4*a^3*b*c^3)*d^4*e^2 - (a*b^6 + 2*a^2*b^4*c - 24*a^3*b^2*c^2)*d^3*e^3 + 3*(a^2*b^5 - 3*a^3*b^3*c -
4*a^4*b*c^2)*d^2*e^4 - 3*(a^3*b^4 - 4*a^4*b^2*c)*d*e^5 + (a^4*b^3 - 4*a^5*b*c)*e^6)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.24111, size = 5536, normalized size = 19.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*c*d*e^3 - b*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2
))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)
) + 1/3*((((4*b^2*c^10*d^14*e - 16*a*c^11*d^14*e - 28*b^3*c^9*d^13*e^2 + 112*a*b*c^10*d^13*e^2 + 87*b^4*c^8*d^
12*e^3 - 332*a*b^2*c^9*d^12*e^3 - 64*a^2*c^10*d^12*e^3 - 158*b^5*c^7*d^11*e^4 + 536*a*b^3*c^8*d^11*e^4 + 384*a
^2*b*c^9*d^11*e^4 + 185*b^6*c^6*d^10*e^5 - 482*a*b^4*c^7*d^10*e^5 - 1020*a^2*b^2*c^8*d^10*e^5 - 48*a^3*c^9*d^1
0*e^5 - 144*b^7*c^5*d^9*e^6 + 166*a*b^5*c^6*d^9*e^6 + 1580*a^2*b^3*c^7*d^9*e^6 + 240*a^3*b*c^8*d^9*e^6 + 73*b^
8*c^4*d^8*e^7 + 128*a*b^6*c^5*d^8*e^7 - 1515*a^2*b^4*c^6*d^8*e^7 - 700*a^3*b^2*c^7*d^8*e^7 + 160*a^4*c^8*d^8*e
^7 - 22*b^9*c^3*d^7*e^8 - 188*a*b^7*c^4*d^7*e^8 + 804*a^2*b^5*c^5*d^7*e^8 + 1360*a^3*b^3*c^6*d^7*e^8 - 640*a^4
*b*c^7*d^7*e^8 + 3*b^10*c^2*d^6*e^9 + 94*a*b^8*c^3*d^6*e^9 - 94*a^2*b^6*c^4*d^6*e^9 - 1500*a^3*b^4*c^5*d^6*e^9
+ 620*a^4*b^2*c^6*d^6*e^9 + 400*a^5*c^7*d^6*e^9 - 18*a*b^9*c^2*d^5*e^10 - 120*a^2*b^7*c^3*d^5*e^10 + 748*a^3*
b^5*c^4*d^5*e^10 + 380*a^4*b^3*c^5*d^5*e^10 - 1200*a^5*b*c^6*d^5*e^10 + 45*a^2*b^8*c^2*d^4*e^11 - 40*a^3*b^6*c
^3*d^4*e^11 - 815*a^4*b^4*c^4*d^4*e^11 + 924*a^5*b^2*c^5*d^4*e^11 + 384*a^6*c^6*d^4*e^11 - 60*a^3*b^7*c^2*d^3*
e^12 + 250*a^4*b^5*c^3*d^3*e^12 + 152*a^5*b^3*c^4*d^3*e^12 - 768*a^6*b*c^5*d^3*e^12 + 45*a^4*b^6*c^2*d^2*e^13
- 258*a^5*b^4*c^3*d^2*e^13 + 268*a^6*b^2*c^4*d^2*e^13 + 176*a^7*c^5*d^2*e^13 - 18*a^5*b^5*c^2*d*e^14 + 116*a^6
*b^3*c^3*d*e^14 - 176*a^7*b*c^4*d*e^14 + 3*a^6*b^4*c^2*e^15 - 20*a^7*b^2*c^3*e^15 + 32*a^8*c^4*e^15)*x/(b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(2*b^3*c^9*d^14*e - 8*a*b*c^10*d^14*e - 15*b^4*c^8*d^13*e^2 + 64*a*b^2*c^9*d
^13*e^2 - 16*a^2*c^10*d^13*e^2 + 50*b^5*c^7*d^12*e^3 - 218*a*b^3*c^8*d^12*e^3 + 72*a^2*b*c^9*d^12*e^3 - 97*b^6
*c^6*d^11*e^4 + 406*a*b^4*c^7*d^11*e^4 - 48*a^2*b^2*c^8*d^11*e^4 - 96*a^3*c^9*d^11*e^4 + 120*b^7*c^5*d^10*e^5
- 428*a*b^5*c^6*d^10*e^5 - 334*a^2*b^3*c^7*d^10*e^5 + 504*a^3*b*c^8*d^10*e^5 - 97*b^8*c^4*d^9*e^6 + 208*a*b^6*
c^5*d^9*e^6 + 975*a^2*b^4*c^6*d^9*e^6 - 960*a^3*b^2*c^7*d^9*e^6 - 240*a^4*c^8*d^9*e^6 + 50*b^9*c^3*d^8*e^7 + 4
6*a*b^7*c^4*d^8*e^7 - 1194*a^2*b^5*c^5*d^8*e^7 + 550*a^3*b^3*c^6*d^8*e^7 + 1160*a^4*b*c^7*d^8*e^7 - 15*b^10*c^
2*d^7*e^8 - 122*a*b^8*c^3*d^7*e^8 + 698*a^2*b^6*c^4*d^7*e^8 + 660*a^3*b^4*c^5*d^7*e^8 - 2080*a^4*b^2*c^6*d^7*e
^8 - 320*a^5*c^7*d^7*e^8 + 2*b^11*c*d^6*e^9 + 64*a*b^9*c^2*d^6*e^9 - 102*a^2*b^7*c^3*d^6*e^9 - 1184*a^3*b^5*c^
4*d^6*e^9 + 1430*a^4*b^3*c^5*d^6*e^9 + 1320*a^5*b*c^6*d^6*e^9 - 12*a*b^10*c*d^5*e^10 - 81*a^2*b^8*c^2*d^5*e^10
+ 596*a^3*b^6*c^3*d^5*e^10 + 175*a^4*b^4*c^4*d^5*e^10 - 1920*a^5*b^2*c^5*d^5*e^10 - 240*a^6*c^6*d^5*e^10 + 30
*a^2*b^9*c*d^4*e^11 - 30*a^3*b^7*c^2*d^4*e^11 - 650*a^4*b^5*c^3*d^4*e^11 + 962*a^5*b^3*c^4*d^4*e^11 + 792*a^6*
b*c^5*d^4*e^11 - 40*a^3*b^8*c*d^3*e^12 + 175*a^4*b^6*c^2*d^3*e^12 + 150*a^5*b^4*c^3*d^3*e^12 - 816*a^6*b^2*c^4
*d^3*e^12 - 96*a^7*c^5*d^3*e^12 + 30*a^4*b^7*c*d^2*e^13 - 180*a^5*b^5*c^2*d^2*e^13 + 182*a^6*b^3*c^3*d^2*e^13
+ 232*a^7*b*c^4*d^2*e^13 - 12*a^5*b^6*c*d*e^14 + 81*a^6*b^4*c^2*d*e^14 - 128*a^7*b^2*c^3*d*e^14 - 16*a^8*c^4*d
*e^14 + 2*a^6*b^5*c*e^15 - 14*a^7*b^3*c^2*e^15 + 24*a^8*b*c^3*e^15)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x +
3*(b^4*c^8*d^14*e - 4*a*b^2*c^9*d^14*e - 8*b^5*c^7*d^13*e^2 + 36*a*b^3*c^8*d^13*e^2 - 16*a^2*b*c^9*d^13*e^2 +
28*b^6*c^6*d^12*e^3 - 132*a*b^4*c^7*d^12*e^3 + 76*a^2*b^2*c^8*d^12*e^3 + 16*a^3*c^9*d^12*e^3 - 56*b^7*c^5*d^11
*e^4 + 254*a*b^5*c^6*d^11*e^4 - 72*a^2*b^3*c^7*d^11*e^4 - 192*a^3*b*c^8*d^11*e^4 + 70*b^8*c^4*d^10*e^5 - 264*a
*b^6*c^5*d^10*e^5 - 241*a^2*b^4*c^6*d^10*e^5 + 684*a^3*b^2*c^7*d^10*e^5 + 96*a^4*c^8*d^10*e^5 - 56*b^9*c^3*d^9
*e^6 + 114*a*b^7*c^4*d^9*e^6 + 730*a^2*b^5*c^5*d^9*e^6 - 980*a^3*b^3*c^6*d^9*e^6 - 720*a^4*b*c^7*d^9*e^6 + 28*
b^10*c^2*d^8*e^7 + 44*a*b^8*c^3*d^8*e^7 - 819*a^2*b^6*c^4*d^8*e^7 + 290*a^3*b^4*c^5*d^8*e^7 + 1900*a^4*b^2*c^6
*d^8*e^7 + 240*a^5*c^7*d^8*e^7 - 8*b^11*c*d^7*e^8 - 78*a*b^9*c^2*d^7*e^8 + 404*a^2*b^7*c^3*d^7*e^8 + 764*a^3*b
^5*c^4*d^7*e^8 - 2160*a^4*b^3*c^5*d^7*e^8 - 1280*a^5*b*c^6*d^7*e^8 + b^12*d^6*e^9 + 36*a*b^10*c*d^6*e^9 - 23*a
^2*b^8*c^2*d^6*e^9 - 888*a^3*b^6*c^3*d^6*e^9 + 735*a^4*b^4*c^4*d^6*e^9 + 2420*a^5*b^2*c^5*d^6*e^9 + 320*a^6*c^
6*d^6*e^9 - 6*a*b^11*d^5*e^10 - 54*a^2*b^9*c*d^5*e^10 + 316*a^3*b^7*c^2*d^5*e^10 + 524*a^4*b^5*c^3*d^5*e^10 -
1860*a^5*b^3*c^4*d^5*e^10 - 1200*a^6*b*c^5*d^5*e^10 + 15*a^2*b^10*d^4*e^11 + 10*a^3*b^8*c*d^4*e^11 - 450*a^4*b
^6*c^2*d^4*e^11 + 296*a^5*b^4*c^3*d^4*e^11 + 1476*a^6*b^2*c^4*d^4*e^11 + 240*a^7*c^5*d^4*e^11 - 20*a^3*b^9*d^3
*e^12 + 60*a^4*b^7*c*d^3*e^12 + 262*a^5*b^5*c^2*d^3*e^12 - 584*a^6*b^3*c^3*d^3*e^12 - 576*a^7*b*c^4*d^3*e^12 +
15*a^4*b^8*d^2*e^13 - 72*a^5*b^6*c*d^2*e^13 - 47*a^6*b^4*c^2*d^2*e^13 + 356*a^7*b^2*c^3*d^2*e^13 + 96*a^8*c^4
*d^2*e^13 - 6*a^5*b^7*d*e^14 + 34*a^6*b^5*c*d*e^14 - 12*a^7*b^3*c^2*d*e^14 - 112*a^8*b*c^3*d*e^14 + a^6*b^6*e^
15 - 6*a^7*b^4*c*e^15 + 4*a^8*b^2*c^2*e^15 + 16*a^9*c^3*e^15)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^4*c
^8*d^15 - 8*a*b^2*c^9*d^15 + 16*a^2*c^10*d^15 - 8*b^5*c^7*d^14*e + 62*a*b^3*c^8*d^14*e - 120*a^2*b*c^9*d^14*e
+ 28*b^6*c^6*d^13*e^2 - 200*a*b^4*c^7*d^13*e^2 + 312*a^2*b^2*c^8*d^13*e^2 + 160*a^3*c^9*d^13*e^2 - 56*b^7*c^5*
d^12*e^3 + 336*a*b^5*c^6*d^12*e^3 - 182*a^2*b^3*c^7*d^12*e^3 - 1064*a^3*b*c^8*d^12*e^3 + 70*b^8*c^4*d^11*e^4 -
280*a*b^6*c^5*d^11*e^4 - 717*a^2*b^4*c^6*d^11*e^4 + 2712*a^3*b^2*c^7*d^11*e^4 + 624*a^4*c^8*d^11*e^4 - 56*b^9
*c^3*d^10*e^5 + 28*a*b^7*c^4*d^10*e^5 + 1740*a^2*b^5*c^5*d^10*e^5 - 2930*a^3*b^3*c^6*d^10*e^5 - 3576*a^4*b*c^7
*d^10*e^5 + 28*b^10*c^2*d^9*e^6 + 168*a*b^8*c^3*d^9*e^6 - 1655*a^2*b^6*c^4*d^9*e^6 + 90*a^3*b^4*c^5*d^9*e^6 +
7880*a^4*b^2*c^6*d^9*e^6 + 1280*a^5*c^7*d^9*e^6 - 8*b^11*c*d^8*e^7 - 160*a*b^9*c^2*d^8*e^7 + 618*a^2*b^7*c^3*d
^8*e^7 + 2900*a^3*b^5*c^4*d^8*e^7 - 7670*a^4*b^3*c^5*d^8*e^7 - 6120*a^5*b*c^6*d^8*e^7 + b^12*d^7*e^8 + 64*a*b^
10*c*d^7*e^8 + 117*a^2*b^8*c^2*d^7*e^8 - 2676*a^3*b^6*c^3*d^7*e^8 + 1655*a^4*b^4*c^4*d^7*e^8 + 10920*a^5*b^2*c
^5*d^7*e^8 + 1520*a^6*c^6*d^7*e^8 - 10*a*b^11*d^6*e^9 - 168*a^2*b^9*c*d^6*e^9 + 758*a^3*b^7*c^2*d^6*e^9 + 2696
*a^4*b^5*c^3*d^6*e^9 - 8150*a^5*b^3*c^4*d^6*e^9 - 5800*a^6*b*c^5*d^6*e^9 + 39*a^2*b^10*d^5*e^10 + 130*a^3*b^8*
c*d^5*e^10 - 1894*a^4*b^6*c^2*d^5*e^10 + 996*a^5*b^4*c^3*d^5*e^10 + 7752*a^6*b^2*c^4*d^5*e^10 + 1056*a^7*c^5*d
^5*e^10 - 80*a^3*b^9*d^4*e^11 + 190*a^4*b^7*c*d^4*e^11 + 1632*a^5*b^5*c^2*d^4*e^11 - 3698*a^6*b^3*c^3*d^4*e^11
- 3000*a^7*b*c^4*d^4*e^11 + 95*a^4*b^8*d^3*e^12 - 492*a^5*b^6*c*d^3*e^12 - 235*a^6*b^4*c^2*d^3*e^12 + 2632*a^
7*b^2*c^3*d^3*e^12 + 400*a^8*c^4*d^3*e^12 - 66*a^5*b^7*d^2*e^13 + 436*a^6*b^5*c*d^2*e^13 - 502*a^7*b^3*c^2*d^2
*e^13 - 744*a^8*b*c^3*d^2*e^13 + 25*a^6*b^6*d*e^14 - 182*a^7*b^4*c*d*e^14 + 312*a^8*b^2*c^2*d*e^14 + 64*a^9*c^
3*d*e^14 - 4*a^7*b^5*e^15 + 30*a^8*b^3*c*e^15 - 56*a^9*b*c^2*e^15)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^
2 + b*x + a)^(3/2)