3.1161 \(\int \frac{A+B x}{(d+e x)^2 (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=331 \[ -\frac{c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}+\frac{\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac{c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac{c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac{A}{2 b^3 d^2 x^2}+\frac{e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}-\frac{e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4} \]

[Out]

-A/(2*b^3*d^2*x^2) - (b*B*d - 3*A*c*d - 2*A*b*e)/(b^4*d^3*x) - (c^3*(b*B - A*c))/(2*b^3*(c*d - b*e)^2*(b + c*x
)^2) - (c^3*(2*b*B*c*d - 3*A*c^2*d - 4*b^2*B*e + 5*A*b*c*e))/(b^4*(c*d - b*e)^3*(b + c*x)) + (e^4*(B*d - A*e))
/(d^3*(c*d - b*e)^3*(d + e*x)) + ((6*A*c^2*d^2 - b^2*e*(2*B*d - 3*A*e) - 3*b*c*d*(B*d - 2*A*e))*Log[x])/(b^5*d
^4) - (c^3*(6*A*c^3*d^2 - 10*b^3*B*e^2 + 5*b^2*c*e*(2*B*d + 3*A*e) - 3*b*c^2*d*(B*d + 6*A*e))*Log[b + c*x])/(b
^5*(c*d - b*e)^4) - (e^4*(B*d*(5*c*d - 2*b*e) - 3*A*e*(2*c*d - b*e))*Log[d + e*x])/(d^4*(c*d - b*e)^4)

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Rubi [A]  time = 0.672028, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}+\frac{\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac{c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac{c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac{A}{2 b^3 d^2 x^2}+\frac{e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}-\frac{e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

-A/(2*b^3*d^2*x^2) - (b*B*d - 3*A*c*d - 2*A*b*e)/(b^4*d^3*x) - (c^3*(b*B - A*c))/(2*b^3*(c*d - b*e)^2*(b + c*x
)^2) - (c^3*(2*b*B*c*d - 3*A*c^2*d - 4*b^2*B*e + 5*A*b*c*e))/(b^4*(c*d - b*e)^3*(b + c*x)) + (e^4*(B*d - A*e))
/(d^3*(c*d - b*e)^3*(d + e*x)) + ((6*A*c^2*d^2 - b^2*e*(2*B*d - 3*A*e) - 3*b*c*d*(B*d - 2*A*e))*Log[x])/(b^5*d
^4) - (c^3*(6*A*c^3*d^2 - 10*b^3*B*e^2 + 5*b^2*c*e*(2*B*d + 3*A*e) - 3*b*c^2*d*(B*d + 6*A*e))*Log[b + c*x])/(b
^5*(c*d - b*e)^4) - (e^4*(B*d*(5*c*d - 2*b*e) - 3*A*e*(2*c*d - b*e))*Log[d + e*x])/(d^4*(c*d - b*e)^4)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{A}{b^3 d^2 x^3}+\frac{b B d-3 A c d-2 A b e}{b^4 d^3 x^2}+\frac{6 A c^2 d^2-b^2 e (2 B d-3 A e)-3 b c d (B d-2 A e)}{b^5 d^4 x}+\frac{c^4 (b B-A c)}{b^3 (-c d+b e)^2 (b+c x)^3}+\frac{c^4 \left (-2 b B c d+3 A c^2 d+4 b^2 B e-5 A b c e\right )}{b^4 (-c d+b e)^3 (b+c x)^2}+\frac{c^4 \left (-6 A c^3 d^2+10 b^3 B e^2-5 b^2 c e (2 B d+3 A e)+3 b c^2 d (B d+6 A e)\right )}{b^5 (c d-b e)^4 (b+c x)}-\frac{e^5 (B d-A e)}{d^3 (c d-b e)^3 (d+e x)^2}+\frac{e^5 (-B d (5 c d-2 b e)+3 A e (2 c d-b e))}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{A}{2 b^3 d^2 x^2}-\frac{b B d-3 A c d-2 A b e}{b^4 d^3 x}-\frac{c^3 (b B-A c)}{2 b^3 (c d-b e)^2 (b+c x)^2}-\frac{c^3 \left (2 b B c d-3 A c^2 d-4 b^2 B e+5 A b c e\right )}{b^4 (c d-b e)^3 (b+c x)}+\frac{e^4 (B d-A e)}{d^3 (c d-b e)^3 (d+e x)}+\frac{\left (6 A c^2 d^2-b^2 e (2 B d-3 A e)-3 b c d (B d-2 A e)\right ) \log (x)}{b^5 d^4}-\frac{c^3 \left (6 A c^3 d^2-10 b^3 B e^2+5 b^2 c e (2 B d+3 A e)-3 b c^2 d (B d+6 A e)\right ) \log (b+c x)}{b^5 (c d-b e)^4}-\frac{e^4 (B d (5 c d-2 b e)-3 A e (2 c d-b e)) \log (d+e x)}{d^4 (c d-b e)^4}\\ \end{align*}

Mathematica [A]  time = 0.637884, size = 328, normalized size = 0.99 \[ \frac{c^3 \log (b+c x) \left (-5 b^2 c e (3 A e+2 B d)+3 b c^2 d (6 A e+B d)-6 A c^3 d^2+10 b^3 B e^2\right )}{b^5 (c d-b e)^4}-\frac{\log (x) \left (b^2 e (2 B d-3 A e)+3 b c d (B d-2 A e)-6 A c^2 d^2\right )}{b^5 d^4}+\frac{c^3 \left (b c (5 A e+2 B d)-3 A c^2 d-4 b^2 B e\right )}{b^4 (b+c x) (b e-c d)^3}+\frac{c^3 (A c-b B)}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 A b e+3 A c d-b B d}{b^4 d^3 x}-\frac{A}{2 b^3 d^2 x^2}+\frac{e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}-\frac{e^4 \log (d+e x) (3 A e (b e-2 c d)+B d (5 c d-2 b e))}{d^4 (c d-b e)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

-A/(2*b^3*d^2*x^2) + (-(b*B*d) + 3*A*c*d + 2*A*b*e)/(b^4*d^3*x) + (c^3*(-(b*B) + A*c))/(2*b^3*(c*d - b*e)^2*(b
+ c*x)^2) + (c^3*(-3*A*c^2*d - 4*b^2*B*e + b*c*(2*B*d + 5*A*e)))/(b^4*(-(c*d) + b*e)^3*(b + c*x)) + (e^4*(B*d
- A*e))/(d^3*(c*d - b*e)^3*(d + e*x)) - ((-6*A*c^2*d^2 + b^2*e*(2*B*d - 3*A*e) + 3*b*c*d*(B*d - 2*A*e))*Log[x
])/(b^5*d^4) + (c^3*(-6*A*c^3*d^2 + 10*b^3*B*e^2 - 5*b^2*c*e*(2*B*d + 3*A*e) + 3*b*c^2*d*(B*d + 6*A*e))*Log[b
+ c*x])/(b^5*(c*d - b*e)^4) - (e^4*(B*d*(5*c*d - 2*b*e) + 3*A*e*(-2*c*d + b*e))*Log[d + e*x])/(d^4*(c*d - b*e)
^4)

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Maple [A]  time = 0.022, size = 598, normalized size = 1.8 \begin{align*} 18\,{\frac{{c}^{5}\ln \left ( cx+b \right ) Ade}{{b}^{4} \left ( be-cd \right ) ^{4}}}-10\,{\frac{{c}^{4}\ln \left ( cx+b \right ) Bde}{{b}^{3} \left ( be-cd \right ) ^{4}}}-{\frac{A}{2\,{d}^{2}{b}^{3}{x}^{2}}}+6\,{\frac{{e}^{5}\ln \left ( ex+d \right ) Ac}{{d}^{3} \left ( be-cd \right ) ^{4}}}+6\,{\frac{Ac\ln \left ( x \right ) e}{{d}^{3}{b}^{4}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3} \left ( be-cd \right ) ^{4}}}-6\,{\frac{{c}^{6}\ln \left ( cx+b \right ) A{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{4}}}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ) B{e}^{2}}{{b}^{2} \left ( be-cd \right ) ^{4}}}+3\,{\frac{{c}^{5}\ln \left ( cx+b \right ) B{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{4}}}+2\,{\frac{{e}^{5}\ln \left ( ex+d \right ) Bb}{{d}^{3} \left ( be-cd \right ) ^{4}}}-5\,{\frac{{e}^{4}\ln \left ( ex+d \right ) Bc}{{d}^{2} \left ( be-cd \right ) ^{4}}}+5\,{\frac{{c}^{4}Ae}{{b}^{3} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}-3\,{\frac{{c}^{5}Ad}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}-4\,{\frac{B{c}^{3}e}{{b}^{2} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}+2\,{\frac{B{c}^{4}d}{{b}^{3} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}-3\,{\frac{{e}^{6}\ln \left ( ex+d \right ) Ab}{{d}^{4} \left ( be-cd \right ) ^{4}}}-{\frac{B}{{d}^{2}{b}^{3}x}}+2\,{\frac{Ae}{{d}^{3}{b}^{3}x}}+{\frac{{e}^{5}A}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{{e}^{4}B}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}+3\,{\frac{Ac}{{d}^{2}{b}^{4}x}}+3\,{\frac{\ln \left ( x \right ) A{e}^{2}}{{d}^{4}{b}^{3}}}+6\,{\frac{\ln \left ( x \right ) A{c}^{2}}{{d}^{2}{b}^{5}}}-2\,{\frac{\ln \left ( x \right ) Be}{{d}^{3}{b}^{3}}}-3\,{\frac{Bc\ln \left ( x \right ) }{{d}^{2}{b}^{4}}}+{\frac{{c}^{4}A}{2\,{b}^{3} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) ^{2}}}-{\frac{B{c}^{3}}{2\,{b}^{2} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^3,x)

[Out]

18*c^5/b^4/(b*e-c*d)^4*ln(c*x+b)*A*d*e-10*c^4/b^3/(b*e-c*d)^4*ln(c*x+b)*B*d*e-1/2*A/b^3/d^2/x^2+6*e^5/d^3/(b*e
-c*d)^4*ln(e*x+d)*A*c+6/d^3/b^4*ln(x)*A*c*e-15*c^4/b^3/(b*e-c*d)^4*ln(c*x+b)*A*e^2-6*c^6/b^5/(b*e-c*d)^4*ln(c*
x+b)*A*d^2+10*c^3/b^2/(b*e-c*d)^4*ln(c*x+b)*B*e^2+3*c^5/b^4/(b*e-c*d)^4*ln(c*x+b)*B*d^2+2*e^5/d^3/(b*e-c*d)^4*
ln(e*x+d)*B*b-5*e^4/d^2/(b*e-c*d)^4*ln(e*x+d)*B*c+5*c^4/b^3/(b*e-c*d)^3/(c*x+b)*A*e-3*c^5/b^4/(b*e-c*d)^3/(c*x
+b)*A*d-4*c^3/b^2/(b*e-c*d)^3/(c*x+b)*B*e+2*c^4/b^3/(b*e-c*d)^3/(c*x+b)*B*d-3*e^6/d^4/(b*e-c*d)^4*ln(e*x+d)*A*
b-1/d^2/b^3/x*B+2/d^3/b^3/x*A*e+e^5/d^3/(b*e-c*d)^3/(e*x+d)*A-e^4/d^2/(b*e-c*d)^3/(e*x+d)*B+3/d^2/b^4/x*A*c+3/
d^4/b^3*ln(x)*A*e^2+6/d^2/b^5*ln(x)*A*c^2-2/d^3/b^3*ln(x)*B*e-3/d^2/b^4*ln(x)*B*c+1/2*c^4/b^3/(b*e-c*d)^2/(c*x
+b)^2*A-1/2*c^3/b^2/(b*e-c*d)^2/(c*x+b)^2*B

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Maxima [B]  time = 1.37682, size = 1408, normalized size = 4.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

(3*(B*b*c^5 - 2*A*c^6)*d^2 - 2*(5*B*b^2*c^4 - 9*A*b*c^5)*d*e + 5*(2*B*b^3*c^3 - 3*A*b^2*c^4)*e^2)*log(c*x + b)
/(b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4) - (5*B*c*d^2*e^4 + 3*A*b*e^6 -
2*(B*b + 3*A*c)*d*e^5)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e
^4) - 1/2*(A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + 2*(3*A*b^4*c^2*e^5 + 3*(B*b
*c^5 - 2*A*c^6)*d^4*e - (7*B*b^2*c^4 - 12*A*b*c^5)*d^3*e^2 + 3*(B*b^3*c^3 - A*b^2*c^4)*d^2*e^3 - (2*B*b^4*c^2
+ 3*A*b^3*c^3)*d*e^4)*x^4 + (12*A*b^5*c*e^5 + 6*(B*b*c^5 - 2*A*c^6)*d^5 - (5*B*b^2*c^4 - 6*A*b*c^5)*d^4*e - 15
*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3*e^2 + 5*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^3 - (8*B*b^5*c + 9*A*b^4*c^2)*d*e^4)*
x^3 - (4*B*b^6*d*e^4 - 6*A*b^6*e^5 - 9*(B*b^2*c^4 - 2*A*b*c^5)*d^5 + (19*B*b^3*c^3 - 32*A*b^2*c^4)*d^4*e - (6*
B*b^4*c^2 - A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c - 13*A*b^4*c^2)*d^2*e^3)*x^2 + (3*A*b^6*d*e^4 + 2*(B*b^3*c^3 - 2*A
*b^2*c^4)*d^5 - 3*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^4*e + 3*(2*B*b^5*c - A*b^4*c^2)*d^3*e^2 - (2*B*b^6 + 5*A*b^5*c
)*d^2*e^3)*x)/((b^4*c^5*d^6*e - 3*b^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3*e^4)*x^5 + (b^4*c^5*d^7 -
b^5*c^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5*c^4*d^7 - 5*b^6*c^3*d^6*
e + 3*b^7*c^2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^3 + (b^6*c^3*d^7 - 3*b^7*c^2*d^6*e + 3*b^8*c*d^5*e^2 -
b^9*d^4*e^3)*x^2) + (3*A*b^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^2 - 2*(B*b^2 - 3*A*b*c)*d*e)*log(x)/(b^5*d^4)

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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((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^3,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((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.48116, size = 1744, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/2*(6*B*b*c^5*d^6*e^2 - 12*A*c^6*d^6*e^2 - 20*B*b^2*c^4*d^5*e^3 + 36*A*b*c^5*d^5*e^3 + 20*B*b^3*c^3*d^4*e^4
- 30*A*b^2*c^4*d^4*e^4 - 5*B*b^5*c*d^2*e^6 + 2*B*b^6*d*e^7 + 6*A*b^5*c*d*e^7 - 3*A*b^6*e^8)*e^(-2)*log(abs(2*c
*d*e - 2*c*d^2*e/(x*e + d) - b*e^2 + 2*b*d*e^2/(x*e + d) - abs(b)*e^2)/abs(2*c*d*e - 2*c*d^2*e/(x*e + d) - b*e
^2 + 2*b*d*e^2/(x*e + d) + abs(b)*e^2))/((b^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3
+ b^8*d^4*e^4)*abs(b)) + 1/2*(5*B*c*d^2*e^4 - 2*B*b*d*e^5 - 6*A*c*d*e^5 + 3*A*b*e^6)*log(abs(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))/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*
b^3*c*d^5*e^3 + b^4*d^4*e^4) + (B*d*e^10/(x*e + d) - A*e^11/(x*e + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*
c*d^4*e^8 - b^3*d^3*e^9) - 1/2*(6*B*b*c^6*d^5*e - 12*A*c^7*d^5*e - 17*B*b^2*c^5*d^4*e^2 + 30*A*b*c^6*d^4*e^2 +
12*B*b^3*c^4*d^3*e^3 - 16*A*b^2*c^5*d^3*e^3 - 8*B*b^4*c^3*d^2*e^4 - 6*A*b^3*c^4*d^2*e^4 + 2*B*b^5*c^2*d*e^5 +
14*A*b^4*c^3*d*e^5 - 5*A*b^5*c^2*e^6 - 2*(9*B*b*c^6*d^6*e^2 - 18*A*c^7*d^6*e^2 - 30*B*b^2*c^5*d^5*e^3 + 54*A*
b*c^6*d^5*e^3 + 31*B*b^3*c^4*d^4*e^4 - 47*A*b^2*c^5*d^4*e^4 - 24*B*b^4*c^3*d^3*e^5 + 4*A*b^3*c^4*d^3*e^5 + 11*
B*b^5*c^2*d^2*e^6 + 29*A*b^4*c^3*d^2*e^6 - 2*B*b^6*c*d*e^7 - 22*A*b^5*c^2*d*e^7 + 5*A*b^6*c*e^8)*e^(-1)/(x*e +
d) + (18*B*b*c^6*d^7*e^3 - 36*A*c^7*d^7*e^3 - 69*B*b^2*c^5*d^6*e^4 + 126*A*b*c^6*d^6*e^4 + 90*B*b^3*c^4*d^5*e
^5 - 144*A*b^2*c^5*d^5*e^5 - 80*B*b^4*c^3*d^4*e^6 + 45*A*b^3*c^4*d^4*e^6 + 50*B*b^5*c^2*d^3*e^7 + 70*A*b^4*c^3
*d^3*e^7 - 16*B*b^6*c*d^2*e^8 - 87*A*b^5*c^2*d^2*e^8 + 2*B*b^7*d*e^9 + 36*A*b^6*c*d*e^9 - 5*A*b^7*e^10)*e^(-2)
/(x*e + d)^2 - 2*(3*B*b*c^6*d^8*e^4 - 6*A*c^7*d^8*e^4 - 13*B*b^2*c^5*d^7*e^5 + 24*A*b*c^6*d^7*e^5 + 20*B*b^3*c
^4*d^6*e^6 - 33*A*b^2*c^5*d^6*e^6 - 20*B*b^4*c^3*d^5*e^7 + 15*A*b^3*c^4*d^5*e^7 + 15*B*b^5*c^2*d^4*e^8 + 15*A*
b^4*c^3*d^4*e^8 - 6*B*b^6*c*d^3*e^9 - 27*A*b^5*c^2*d^3*e^9 + B*b^7*d^2*e^10 + 15*A*b^6*c*d^2*e^10 - 3*A*b^7*d*
e^11)*e^(-3)/(x*e + d)^3)/((c*d - b*e)^4*b^4*(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)^2*d^4)