### 3.292 $$\int \frac{\sqrt{b x+c x^2}}{(d+e x)^4} \, dx$$

Optimal. Leaf size=183 $-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)}$

[Out]

((2*c*d - b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(8*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(b*x + c*x^2)
^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d
- b*e]*Sqrt[b*x + c*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))

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Rubi [A]  time = 0.138381, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.19, Rules used = {730, 720, 724, 206} $-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

((2*c*d - b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(8*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(b*x + c*x^2)
^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d
- b*e]*Sqrt[b*x + c*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))

Rule 730

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[(2*c*d - b*e)/(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, 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] && EqQ[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{b x+c x^2}}{(d+e x)^4} \, dx &=-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac{(2 c d-b e) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)}\\ &=\frac{(2 c d-b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac{\left (b^2 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{16 d^2 (c d-b e)^2}\\ &=\frac{(2 c d-b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac{\left (b^2 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{8 d^2 (c d-b e)^2}\\ &=\frac{(2 c d-b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.293452, size = 191, normalized size = 1.04 $\frac{\sqrt{x (b+c x)} \left (\frac{3 (2 c d-b e) \left (\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d} (b (d-e x)+2 c d x)-b^2 (d+e x)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{8 d^{3/2} \sqrt{b+c x} (d+e x)^2 (b e-c d)^{3/2}}+\frac{e x^{3/2} (b+c x)}{(d+e x)^3}\right )}{3 d \sqrt{x} (b e-c d)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[x*(b + c*x)]*((e*x^(3/2)*(b + c*x))/(d + e*x)^3 + (3*(2*c*d - b*e)*(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[x]*S
qrt[b + c*x]*(2*c*d*x + b*(d - e*x)) - b^2*(d + e*x)^2*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c
*x])]))/(8*d^(3/2)*(-(c*d) + b*e)^(3/2)*Sqrt[b + c*x]*(d + e*x)^2)))/(3*d*(-(c*d) + b*e)*Sqrt[x])

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Maple [B]  time = 0.214, size = 2891, normalized size = 15.8 \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)^(1/2)/(e*x+d)^4,x)

[Out]

5/4/e^2*d/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^
2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c^3-1/2*e/d^2/(b*e-c*d)^3/(d/e+
x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*c-1/8*e^2/d^3/(b*e-c*d)^3*c*(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b^2+1/2*e/d^2/(b*e-c*d)^3*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b
*e-c*d)/e^2)^(1/2)*x*b-1/4/e/d/(b*e-c*d)^2*c/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/
e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2-1/2/d/
(b*e-c*d)^3*c^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-5/8/d/(b*e-c*d)^3*ln((1/2*(b*e-2*c
*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b^2-1/4/d^2/(b*e-c
*d)^2*c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b-1/8*e^2/d^3/(b*e-c*d)^3*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^3+1/2/e/d/(b*e-c*d)^2*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(1/2)-1/2/e^2*d/(b*e-c*d)^3*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(7/2)+1/3/e^2/d/(b*e-c*d)/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(3/2)-1/8/d^2/(b*e-c*d)^2*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^2+1/4/d^2/(b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(3/2)*b+1/2/d/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2-1/d/(b
*e-c*d)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c^2+1/e/(b*e-c*d)^3*ln((1/2*(b*e-2*c*d)/
e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*b+1/2/e/(b*e-c*d)^3*(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^3-1/2/e^2/(b*e-c*d)^2*c^(5/2)*ln((1/2*(b*e-2*c*d)/e+
(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))-9/8/e/(b*e-c*d)^3/(-d*(b*e-c*d)/
e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c^2+3/4/e^2/(b*e-c*d)^2*c^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*
e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2
)^(1/2))/(d/e+x))*b-1/2/e/d/(b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c+
1/8*e^2/d^3/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2+5/8*e/d^2/(b*e-c
*d)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*c+1/8*e/d^2/(b*e-c*d)^3*ln((1/2*(b*e-2*c*d
)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^3-1/16*e/d^2/(b*e-
c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^4+1/2/e/d/(b*e-c*d)^2*c^(3/2)*ln((1/2*(b*e-2*c
*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b+7/16/d/(b*e-c*d)^3/(-d*(
b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*c-1/2/e^3*d/(b*e-c*d)^2*c^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln(
(-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e
-c*d)/e^2)^(1/2))/(d/e+x))-1/2/e^3*d^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)
/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^4

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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)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.41711, size = 1925, normalized size = 10.52 \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)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/48*(3*(2*b^2*c*d^4 - b^3*d^3*e + (2*b^2*c*d*e^3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d^2*e^2 - b^3*d*e^3)*x^2 + 3*(
2*b^2*c*d^3*e - b^3*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*
x^2 + b*x))/(e*x + d)) - 2*(6*b*c^2*d^5 - 9*b^2*c*d^4*e + 3*b^3*d^3*e^2 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + 7*b
^2*c*d^2*e^3 - 3*b^3*d*e^4)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11*b^2*c*d^3*e^2 - 4*b^3*d^2*e^3)*x)*sqrt(c*
x^2 + b*x))/(c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*
c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*
d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x), -1/24*(3*(2*b^2*c*d^4 - b^3*d^3*e + (2*b^2*c*d*e^
3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d^2*e^2 - b^3*d*e^3)*x^2 + 3*(2*b^2*c*d^3*e - b^3*d^2*e^2)*x)*sqrt(-c*d^2 + b*d*
e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (6*b*c^2*d^5 - 9*b^2*c*d^4*e + 3*b^3*d^3*
e^2 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + 7*b^2*c*d^2*e^3 - 3*b^3*d*e^4)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11
*b^2*c*d^3*e^2 - 4*b^3*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3
+ (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*
b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**4, x)

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Giac [B]  time = 1.48826, size = 1114, normalized size = 6.09 \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)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/8*(2*b^2*c*d - b^3*e)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4
- 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^(7/2)*d^4*e
+ 32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^4*d^5 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c^3*d^4*e + 48*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b*c^(7/2)*d^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b*c^(5/2)*d^3*e^2 - 36*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b^2*c^(5/2)*d^4*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c^3*d^5 - 84*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*b^2*c^2*d^3*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*c^2*d^4*e + 4*b^3*c^(5/2)*d^5
+ 78*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^2*c^(3/2)*d^2*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c^(3/2)*
d^3*e^2 - 4*b^4*c^(3/2)*d^4*e + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^2*c*d*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*b^3*c*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*c*d^3*e^2 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^4*b^3*sqrt(c)*d*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^4*sqrt(c)*d^2*e^3 + 3*b^5*sqrt(c)*d^3*e^2 - 3*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*e^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^4*d*e^4 + 3*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*b^5*d^2*e^3)/((c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e +
2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^3)