### 3.2341 $$\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx$$

Optimal. Leaf size=308 $\frac{\sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}$

[Out]

((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*
d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(a + b*x + c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (5
*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - ((b^2 - 4*a*c)*(16*c^2*
d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*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])])/(128*(c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 0.415327, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {744, 806, 720, 724, 206} $\frac{\sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*
d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(a + b*x + c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (5
*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - ((b^2 - 4*a*c)*(16*c^2*
d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*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])])/(128*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 744

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, 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]

Rule 720

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)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), 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] && GtQ[p, 0]

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{\sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx &=-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{\int \frac{\left (\frac{1}{2} (-8 c d+5 b e)+c e x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx}{4 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac{\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac{\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac{\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\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 )}{64 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac{\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \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 )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.694276, size = 276, normalized size = 0.9 $\frac{3 \left (-2 c e (a e+4 b d)+\frac{5 b^2 e^2}{2}+8 c^2 d^2\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )-\frac{6 e (a+x (b+c x))^{3/2} \left (e (a e-b d)+c d^2\right )}{(d+e x)^4}-\frac{5 e (a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x)^3}}{24 \left (e (a e-b d)+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-6*e*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 - (5*e*(2*c*d - b*e)*(a + x*(b + c*x))^
(3/2))/(d + e*x)^3 + 3*(8*c^2*d^2 + (5*b^2*e^2)/2 - 2*c*e*(4*b*d + a*e))*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c
*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*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)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))
)/(24*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.238, size = 7991, normalized size = 25.9 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x)

[Out]

result too large to display

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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((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)

[Out]

Timed out

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Giac [B]  time = 21.6727, size = 4086, normalized size = 13.27 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

1/384*(2*sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)
*(2*(4*((2*c^3*d^5*e^6*sgn(1/(x*e + d)) - 5*b*c^2*d^4*e^7*sgn(1/(x*e + d)) + 4*b^2*c*d^3*e^8*sgn(1/(x*e + d))
+ 4*a*c^2*d^3*e^8*sgn(1/(x*e + d)) - b^3*d^2*e^9*sgn(1/(x*e + d)) - 6*a*b*c*d^2*e^9*sgn(1/(x*e + d)) + 2*a*b^2
*d*e^10*sgn(1/(x*e + d)) + 2*a^2*c*d*e^10*sgn(1/(x*e + d)) - a^2*b*e^11*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c
^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 1
2*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*
c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16) - 6*(c^3*d^6*e^7*sgn(1/(x*e + d)) - 3*b*c^2*d^5*e^8*sgn(1/(x*e + d)) +
3*b^2*c*d^4*e^9*sgn(1/(x*e + d)) + 3*a*c^2*d^4*e^9*sgn(1/(x*e + d)) - b^3*d^3*e^10*sgn(1/(x*e + d)) - 6*a*b*c
*d^3*e^10*sgn(1/(x*e + d)) + 3*a*b^2*d^2*e^11*sgn(1/(x*e + d)) + 3*a^2*c*d^2*e^11*sgn(1/(x*e + d)) - 3*a^2*b*d
*e^12*sgn(1/(x*e + d)) + a^3*e^13*sgn(1/(x*e + d)))*e^(-1)/((c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^1
0 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2
*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 +
a^4*e^16)*(x*e + d)))*e^(-1)/(x*e + d) + (8*c^3*d^4*e^5*sgn(1/(x*e + d)) - 16*b*c^2*d^3*e^6*sgn(1/(x*e + d)) +
13*b^2*c*d^2*e^7*sgn(1/(x*e + d)) - 4*a*c^2*d^2*e^7*sgn(1/(x*e + d)) - 5*b^3*d*e^8*sgn(1/(x*e + d)) + 4*a*b*c
*d*e^8*sgn(1/(x*e + d)) + 5*a*b^2*e^9*sgn(1/(x*e + d)) - 12*a^2*c*e^9*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3
*d^7*e^9 + 6*b^2*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*
a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*
d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16))*e^(-1)/(x*e + d) + (16*c^3*d^3*e^4*sgn(1/(x*e + d)) - 24*b*c^2*d^2*e^5*
sgn(1/(x*e + d)) + 38*b^2*c*d*e^6*sgn(1/(x*e + d)) - 104*a*c^2*d*e^6*sgn(1/(x*e + d)) - 15*b^3*e^7*sgn(1/(x*e
+ d)) + 52*a*b*c*e^7*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10
- 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d
^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16)) + 3*(16*b^
2*c^2*d^2*sgn(1/(x*e + d)) - 64*a*c^3*d^2*sgn(1/(x*e + d)) - 16*b^3*c*d*e*sgn(1/(x*e + d)) + 64*a*b*c^2*d*e*sg
n(1/(x*e + d)) + 5*b^4*e^2*sgn(1/(x*e + d)) - 24*a*b^2*c*e^2*sgn(1/(x*e + d)) + 16*a^2*c^2*e^2*sgn(1/(x*e + d)
))*sqrt(c*d^2 - b*d*e + a*e^2)*log(abs(2*(c*d^2 - b*d*e + a*e^2)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2
+ b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3 + a*e^4)*e^(-1)/(x*e + d)
) - sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d - b*e)))/(c^5*d^10*e^2 - 5*b*c^4*d^9*e^3 + 10*b^2*c^3*d^8*e^4 + 5*a*c^4
*d^8*e^4 - 10*b^3*c^2*d^7*e^5 - 20*a*b*c^3*d^7*e^5 + 5*b^4*c*d^6*e^6 + 30*a*b^2*c^2*d^6*e^6 + 10*a^2*c^3*d^6*e
^6 - b^5*d^5*e^7 - 20*a*b^3*c*d^5*e^7 - 30*a^2*b*c^2*d^5*e^7 + 5*a*b^4*d^4*e^8 + 30*a^2*b^2*c*d^4*e^8 + 10*a^3
*c^2*d^4*e^8 - 10*a^2*b^3*d^3*e^9 - 20*a^3*b*c*d^3*e^9 + 10*a^3*b^2*d^2*e^10 + 5*a^4*c*d^2*e^10 - 5*a^4*b*d*e^
11 + a^5*e^12) - (32*c^(9/2)*d^5 - 80*b*c^(7/2)*d^4*e + 124*b^2*c^(5/2)*d^3*e^2 - 176*a*c^(7/2)*d^3*e^2 + 48*s
qrt(c*d^2 - b*d*e + a*e^2)*b^2*c^2*d^2*e^2*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*
e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) - 192*sqrt(c*d^2 - b*d*e + a*e^2)*a*c^3*d^2*e^2
*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 -
b*d*e + a*e^2)*b*e)) - 106*b^3*c^(3/2)*d^2*e^3 + 264*a*b*c^(5/2)*d^2*e^3 - 48*sqrt(c*d^2 - b*d*e + a*e^2)*b^3*
c*d*e^3*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c
*d^2 - b*d*e + a*e^2)*b*e)) + 192*sqrt(c*d^2 - b*d*e + a*e^2)*a*b*c^2*d*e^3*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c
)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 30*b^4*sqrt(
c)*d*e^4 - 28*a*b^2*c^(3/2)*d*e^4 - 208*a^2*c^(5/2)*d*e^4 + 15*sqrt(c*d^2 - b*d*e + a*e^2)*b^4*e^4*log(abs(2*c
^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^
2)*b*e)) - 72*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c*e^4*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 -
b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^2
*c^2*e^4*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(
c*d^2 - b*d*e + a*e^2)*b*e)) - 30*a*b^3*sqrt(c)*e^5 + 104*a^2*b*c^(3/2)*e^5)*sgn(1/(x*e + d))/(c^5*d^10*e^4 -
5*b*c^4*d^9*e^5 + 10*b^2*c^3*d^8*e^6 + 5*a*c^4*d^8*e^6 - 10*b^3*c^2*d^7*e^7 - 20*a*b*c^3*d^7*e^7 + 5*b^4*c*d^6
*e^8 + 30*a*b^2*c^2*d^6*e^8 + 10*a^2*c^3*d^6*e^8 - b^5*d^5*e^9 - 20*a*b^3*c*d^5*e^9 - 30*a^2*b*c^2*d^5*e^9 + 5
*a*b^4*d^4*e^10 + 30*a^2*b^2*c*d^4*e^10 + 10*a^3*c^2*d^4*e^10 - 10*a^2*b^3*d^3*e^11 - 20*a^3*b*c*d^3*e^11 + 10
*a^3*b^2*d^2*e^12 + 5*a^4*c*d^2*e^12 - 5*a^4*b*d*e^13 + a^5*e^14))*e^2