### 3.334 $$\int \frac{1}{(d+e x) (b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=230 $\frac{2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{e^4 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}}$

[Out]

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

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Rubi [A]  time = 0.17944, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {740, 822, 12, 724, 206} $\frac{2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{e^4 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]

[Out]

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

Rule 740

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

Mathematica [A]  time = 0.297749, size = 238, normalized size = 1.03 $\frac{2 \sqrt{d} \sqrt{b e-c d} \left (-b^3 c^2 \left (9 d^2 e x+d^3-3 d e^2 x^2-3 e^3 x^3\right )+2 b^2 c^3 d x \left (3 d^2-18 d e x+e^2 x^2\right )+2 b^4 c e \left (d^2+3 e^2 x^2\right )+b^5 e^2 (3 e x-d)+24 b c^4 d^2 x^2 (d-e x)+16 c^5 d^3 x^3\right )+6 b^4 e^4 x^{3/2} (b+c x)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{3 b^4 d^{5/2} (x (b+c x))^{3/2} (b e-c d)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]

[Out]

(2*Sqrt[d]*Sqrt[-(c*d) + b*e]*(16*c^5*d^3*x^3 + 24*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-d + 3*e*x) + 2*b^2*c^3*
d*x*(3*d^2 - 18*d*e*x + e^2*x^2) + 2*b^4*c*e*(d^2 + 3*e^2*x^2) - b^3*c^2*(d^3 + 9*d^2*e*x - 3*d*e^2*x^2 - 3*e^
3*x^3)) + 6*b^4*e^4*x^(3/2)*(b + c*x)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(3*b
^4*d^(5/2)*(-(c*d) + b*e)^(5/2)*(x*(b + c*x))^(3/2))

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Maple [B]  time = 0.219, size = 950, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.11545, size = 1993, normalized size = 8.67 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(b^4*c^2*e^4*x^4 + 2*b^5*c*e^4*x^3 + b^6*e^4*x^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*s
qrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*d^
2*e^3 - (16*c^6*d^5 - 40*b*c^5*d^4*e + 26*b^2*c^4*d^3*e^2 + b^3*c^3*d^2*e^3 - 3*b^4*c^2*d*e^4)*x^3 - 3*(8*b*c^
5*d^5 - 20*b^2*c^4*d^4*e + 13*b^3*c^3*d^3*e^2 + b^4*c^2*d^2*e^3 - 2*b^5*c*d*e^4)*x^2 - 3*(2*b^2*c^4*d^5 - 5*b^
3*c^3*d^4*e + 3*b^4*c^2*d^3*e^2 + b^5*c*d^2*e^3 - b^6*d*e^4)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d
^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*d^3*e^3)*x^4 + 2*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c
*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^7*c^2*d^5*e + 3*b^8*c*d^4*e^2 - b^9*d^3*e^3)*x^2), 2/3*(3*(b^4*c^2*e^4*x^4
+ 2*b^5*c*e^4*x^3 + b^6*e^4*x^2)*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)) - (b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*d^2*e^3 - (16*c^6*d^5 - 40*b*c^5*d^4*e + 26*b
^2*c^4*d^3*e^2 + b^3*c^3*d^2*e^3 - 3*b^4*c^2*d*e^4)*x^3 - 3*(8*b*c^5*d^5 - 20*b^2*c^4*d^4*e + 13*b^3*c^3*d^3*e
^2 + b^4*c^2*d^2*e^3 - 2*b^5*c*d*e^4)*x^2 - 3*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e^2 + b^5*c*d^2
*e^3 - b^6*d*e^4)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*d^3*e^3)
*x^4 + 2*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^7*c^2*d^
5*e + 3*b^8*c*d^4*e^2 - b^9*d^3*e^3)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.41056, size = 562, normalized size = 2.44 \begin{align*} -\frac{2 \, \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right ) e^{4}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt{-c d^{2} + b d e}} - \frac{{\left (x{\left (\frac{{\left (16 \, c^{7} d^{10} - 56 \, b c^{6} d^{9} e + 66 \, b^{2} c^{5} d^{8} e^{2} - 25 \, b^{3} c^{4} d^{7} e^{3} - 4 \, b^{4} c^{3} d^{6} e^{4} + 3 \, b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{6} d^{10} - 28 \, b^{2} c^{5} d^{9} e + 33 \, b^{3} c^{4} d^{8} e^{2} - 12 \, b^{4} c^{3} d^{7} e^{3} - 3 \, b^{5} c^{2} d^{6} e^{4} + 2 \, b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (2 \, b^{2} c^{5} d^{10} - 7 \, b^{3} c^{4} d^{9} e + 8 \, b^{4} c^{3} d^{8} e^{2} - 2 \, b^{5} c^{2} d^{7} e^{3} - 2 \, b^{6} c d^{6} e^{4} + b^{7} d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} x - \frac{b^{3} c^{4} d^{10} - 4 \, b^{4} c^{3} d^{9} e + 6 \, b^{5} c^{2} d^{8} e^{2} - 4 \, b^{6} c d^{7} e^{3} + b^{7} d^{6} e^{4}}{b^{4} c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^4/((c^2*d^4 - 2*b*c*d^3*e +
b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 1/3*((x*((16*c^7*d^10 - 56*b*c^6*d^9*e + 66*b^2*c^5*d^8*e^2 - 25*b^3*c^4*
d^7*e^3 - 4*b^4*c^3*d^6*e^4 + 3*b^5*c^2*d^5*e^5)*x/(b^4*c^2) + 3*(8*b*c^6*d^10 - 28*b^2*c^5*d^9*e + 33*b^3*c^4
*d^8*e^2 - 12*b^4*c^3*d^7*e^3 - 3*b^5*c^2*d^6*e^4 + 2*b^6*c*d^5*e^5)/(b^4*c^2)) + 3*(2*b^2*c^5*d^10 - 7*b^3*c^
4*d^9*e + 8*b^4*c^3*d^8*e^2 - 2*b^5*c^2*d^7*e^3 - 2*b^6*c*d^6*e^4 + b^7*d^5*e^5)/(b^4*c^2))*x - (b^3*c^4*d^10
- 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)/(b^4*c^2))/(c*x^2 + b*x)^(3/2)