3.2223 \(\int \frac{(d+e x)^5 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=364 \[ -\frac{2 (d+e x)^3 (-7 b e g+10 c d g+4 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{6 c^3 e^2 (2 c d-b e)}-\frac{5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{4 c^4 e^2}+\frac{5 (2 c d-b e) (-7 b e g+10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{9/2} e^2}+\frac{2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^5)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) -
(2*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)^3)/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2]) - (5*(4*c*e*f + 10*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c^4*e^2) - (5*(4*c*e*f
+ 10*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(6*c^3*e^2*(2*c*d - b*e)) + (5*(2*c
*d - b*e)*(4*c*e*f + 10*c*d*g - 7*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2])])/(8*c^(9/2)*e^2)

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Rubi [A]  time = 0.555418, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {788, 668, 670, 640, 621, 204} \[ -\frac{2 (d+e x)^3 (-7 b e g+10 c d g+4 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{6 c^3 e^2 (2 c d-b e)}-\frac{5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{4 c^4 e^2}+\frac{5 (2 c d-b e) (-7 b e g+10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{9/2} e^2}+\frac{2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^5)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) -
(2*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)^3)/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2]) - (5*(4*c*e*f + 10*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c^4*e^2) - (5*(4*c*e*f
+ 10*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(6*c^3*e^2*(2*c*d - b*e)) + (5*(2*c
*d - b*e)*(4*c*e*f + 10*c*d*g - 7*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2])])/(8*c^(9/2)*e^2)

Rule 788

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

Rule 668

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

Rule 670

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))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

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

Rule 621

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

Rule 204

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(4 c e f+10 c d g-7 b e g) \int \frac{(d+e x)^4}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (4 c e f+10 c d g-7 b e g) (d+e x)^3}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(5 (4 c e f+10 c d g-7 b e g)) \int \frac{(d+e x)^2}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{3 c^2 e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (4 c e f+10 c d g-7 b e g) (d+e x)^3}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (4 c e f+10 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{6 c^3 e^2 (2 c d-b e)}+\frac{(5 (4 c e f+10 c d g-7 b e g)) \int \frac{d+e x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (4 c e f+10 c d g-7 b e g) (d+e x)^3}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (4 c e f+10 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^4 e^2}-\frac{5 (4 c e f+10 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{6 c^3 e^2 (2 c d-b e)}+\frac{(5 (2 c d-b e) (4 c e f+10 c d g-7 b e g)) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 c^4 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (4 c e f+10 c d g-7 b e g) (d+e x)^3}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (4 c e f+10 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^4 e^2}-\frac{5 (4 c e f+10 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{6 c^3 e^2 (2 c d-b e)}+\frac{(5 (2 c d-b e) (4 c e f+10 c d g-7 b e g)) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{4 c^4 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^5}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (4 c e f+10 c d g-7 b e g) (d+e x)^3}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (4 c e f+10 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^4 e^2}-\frac{5 (4 c e f+10 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{6 c^3 e^2 (2 c d-b e)}+\frac{5 (2 c d-b e) (4 c e f+10 c d g-7 b e g) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{9/2} e^2}\\ \end{align*}

Mathematica [C]  time = 0.313255, size = 139, normalized size = 0.38 \[ \frac{2 (d+e x)^5 \left (\left (\frac{b e-c d+c e x}{b e-2 c d}\right )^{3/2} (-7 b e g+10 c d g+4 c e f) \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{c (d+e x)}{2 c d-b e}\right )-7 (-b e g+c d g+c e f)\right )}{21 c e^2 (b e-2 c d) ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^5*(-7*(c*e*f + c*d*g - b*e*g) + (4*c*e*f + 10*c*d*g - 7*b*e*g)*((-(c*d) + b*e + c*e*x)/(-2*c*d +
b*e))^(3/2)*Hypergeometric2F1[3/2, 7/2, 9/2, (c*(d + e*x))/(2*c*d - b*e)]))/(21*c*e^2*(-2*c*d + b*e)*((d + e*x
)*(-(b*e) + c*(d - e*x)))^(3/2))

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Maple [B]  time = 0.043, size = 6704, normalized size = 18.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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((e*x+d)^5*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 54.1475, size = 2591, normalized size = 7.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(15*((4*(2*c^4*d*e^3 - b*c^3*e^4)*f + (20*c^4*d^2*e^2 - 24*b*c^3*d*e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c
^4*d^3*e - 5*b*c^3*d^2*e^2 + 4*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e + 75*b^2*c^2*d^2*e^
2 - 38*b^3*c*d*e^3 + 7*b^4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + b^2*c^2*e^4)*f + (20*c^4*d^3*e - 44*
b*c^3*d^2*e^2 + 31*b^2*c^2*d*e^3 - 7*b^3*c*e^4)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4
*b*c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*(6*c^4*e^3*g*x
^3 + 3*(4*c^4*e^3*f + (16*c^4*d*e^2 - 7*b*c^3*e^3)*g)*x^2 + 4*(23*c^4*d^2*e - 40*b*c^3*d*e^2 + 15*b^2*c^2*e^3)
*f + (236*c^4*d^3 - 561*b*c^3*d^2*e + 430*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g - 2*(4*(17*c^4*d*e^2 - 10*b*c^3*e^3
)*f + (161*c^4*d^2*e - 219*b*c^3*d*e^2 + 70*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^
7*e^4*x^2 + c^7*d^2*e^2 - 2*b*c^6*d*e^3 + b^2*c^5*e^4 - 2*(c^7*d*e^3 - b*c^6*e^4)*x), -1/24*(15*((4*(2*c^4*d*e
^3 - b*c^3*e^4)*f + (20*c^4*d^2*e^2 - 24*b*c^3*d*e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c^4*d^3*e - 5*b*c^3*d^2*e^
2 + 4*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e + 75*b^2*c^2*d^2*e^2 - 38*b^3*c*d*e^3 + 7*b^
4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + b^2*c^2*e^4)*f + (20*c^4*d^3*e - 44*b*c^3*d^2*e^2 + 31*b^2*c^
2*d*e^3 - 7*b^3*c*e^4)*g)*x)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqr
t(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*(6*c^4*e^3*g*x^3 + 3*(4*c^4*e^3*f + (16*c^4*d*e^2 - 7*
b*c^3*e^3)*g)*x^2 + 4*(23*c^4*d^2*e - 40*b*c^3*d*e^2 + 15*b^2*c^2*e^3)*f + (236*c^4*d^3 - 561*b*c^3*d^2*e + 43
0*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g - 2*(4*(17*c^4*d*e^2 - 10*b*c^3*e^3)*f + (161*c^4*d^2*e - 219*b*c^3*d*e^2 +
70*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^7*e^4*x^2 + c^7*d^2*e^2 - 2*b*c^6*d*e^3
+ b^2*c^5*e^4 - 2*(c^7*d*e^3 - b*c^6*e^4)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Integral((d + e*x)**5*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)

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Giac [B]  time = 1.30187, size = 1971, normalized size = 5.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*((((3*(2*(16*c^7*d^4*g*e^11 - 32*b*c^6*d^3*g*e^12 + 24*b^2*c^
5*d^2*g*e^13 - 8*b^3*c^4*d*g*e^14 + b^4*c^3*g*e^15)*x/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d^2*e^10
- 8*b^3*c^5*d*e^11 + b^4*c^4*e^12) + (320*c^7*d^5*g*e^10 + 64*c^7*d^4*f*e^11 - 752*b*c^6*d^4*g*e^11 - 128*b*c
^6*d^3*f*e^12 + 704*b^2*c^5*d^3*g*e^12 + 96*b^2*c^5*d^2*f*e^13 - 328*b^3*c^4*d^2*g*e^13 - 32*b^3*c^4*d*f*e^14
+ 76*b^4*c^3*d*g*e^14 + 4*b^4*c^3*f*e^15 - 7*b^5*c^2*g*e^15)/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d
^2*e^10 - 8*b^3*c^5*d*e^11 + b^4*c^4*e^12))*x - 4*(880*c^7*d^6*g*e^9 + 448*c^7*d^5*f*e^10 - 3344*b*c^6*d^5*g*e
^10 - 1216*b*c^6*d^4*f*e^11 + 5048*b^2*c^5*d^4*g*e^11 + 1312*b^2*c^5*d^3*f*e^12 - 3936*b^3*c^4*d^3*g*e^12 - 70
4*b^3*c^4*d^2*f*e^13 + 1687*b^4*c^3*d^2*g*e^13 + 188*b^4*c^3*d*f*e^14 - 379*b^5*c^2*d*g*e^14 - 20*b^5*c^2*f*e^
15 + 35*b^6*c*g*e^15)/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d^2*e^10 - 8*b^3*c^5*d*e^11 + b^4*c^4*e^
12))*x - 3*(1920*c^7*d^7*g*e^8 + 896*c^7*d^6*f*e^9 - 5408*b*c^6*d^6*g*e^9 - 1792*b*c^6*d^5*f*e^10 + 5216*b^2*c
^5*d^5*g*e^10 + 1024*b^2*c^5*d^4*f*e^11 - 1152*b^3*c^4*d^4*g*e^11 + 192*b^3*c^4*d^3*f*e^12 - 1416*b^4*c^3*d^3*
g*e^12 - 424*b^4*c^3*d^2*f*e^13 + 1142*b^5*c^2*d^2*g*e^13 + 160*b^5*c^2*d*f*e^14 - 330*b^6*c*d*g*e^14 - 20*b^6
*c*f*e^15 + 35*b^7*g*e^15)/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d^2*e^10 - 8*b^3*c^5*d*e^11 + b^4*c
^4*e^12))*x + 6*(400*c^7*d^8*g*e^7 + 128*c^7*d^7*f*e^8 - 2624*b*c^6*d^7*g*e^8 - 896*b*c^6*d^6*f*e^9 + 6168*b^2
*c^5*d^6*g*e^9 + 1792*b^2*c^5*d^5*f*e^10 - 7336*b^3*c^4*d^5*g*e^10 - 1664*b^3*c^4*d^4*f*e^11 + 4937*b^4*c^3*d^
4*g*e^11 + 808*b^4*c^3*d^3*f*e^12 - 1914*b^5*c^2*d^3*g*e^12 - 200*b^5*c^2*d^2*f*e^13 + 400*b^6*c*d^2*g*e^13 +
20*b^6*c*d*f*e^14 - 35*b^7*d*g*e^14)/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d^2*e^10 - 8*b^3*c^5*d*e^
11 + b^4*c^4*e^12))*x + (3776*c^7*d^9*g*e^6 + 1472*c^7*d^8*f*e^7 - 16528*b*c^6*d^8*g*e^7 - 5504*b*c^6*d^7*f*e^
8 + 30496*b^2*c^5*d^7*g*e^8 + 8288*b^2*c^5*d^6*f*e^9 - 30792*b^3*c^4*d^6*g*e^9 - 6496*b^3*c^4*d^5*f*e^10 + 184
04*b^4*c^3*d^5*g*e^10 + 2812*b^4*c^3*d^4*f*e^11 - 6521*b^5*c^2*d^4*g*e^11 - 640*b^5*c^2*d^3*f*e^12 + 1270*b^6*
c*d^3*g*e^12 + 60*b^6*c*d^2*f*e^13 - 105*b^7*d^2*g*e^13)/(16*c^8*d^4*e^8 - 32*b*c^7*d^3*e^9 + 24*b^2*c^6*d^2*e
^10 - 8*b^3*c^5*d*e^11 + b^4*c^4*e^12))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2 + 5/8*(20*c^2*d^2*g + 8*c^2*d*
f*e - 24*b*c*d*g*e - 4*b*c*f*e^2 + 7*b^2*g*e^2)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e
^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2)*b))/c^5