### 3.253 $$\int \frac{(b x+c x^2)^3}{(d+e x)^5} \, dx$$

Optimal. Leaf size=213 $\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{c^3 x^2}{2 e^5}$

[Out]

-((c^2*(5*c*d - 3*b*e)*x)/e^6) + (c^3*x^2)/(2*e^5) - (d^3*(c*d - b*e)^3)/(4*e^7*(d + e*x)^4) + (d^2*(c*d - b*e
)^2*(2*c*d - b*e))/(e^7*(d + e*x)^3) - (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(2*e^7*(d + e*x)^2)
+ ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2
)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.197327, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {698} $\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{c^3 x^2}{2 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

-((c^2*(5*c*d - 3*b*e)*x)/e^6) + (c^3*x^2)/(2*e^5) - (d^3*(c*d - b*e)^3)/(4*e^7*(d + e*x)^4) + (d^2*(c*d - b*e
)^2*(2*c*d - b*e))/(e^7*(d + e*x)^3) - (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(2*e^7*(d + e*x)^2)
+ ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2
)*Log[d + e*x])/e^7

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, 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] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^3}{(d+e x)^5} \, dx &=\int \left (-\frac{c^2 (5 c d-3 b e)}{e^6}+\frac{c^3 x}{e^5}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^5}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^4}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^2}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 (5 c d-3 b e) x}{e^6}+\frac{c^3 x^2}{2 e^5}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{e^7 (d+e x)}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.114337, size = 210, normalized size = 0.99 $\frac{\frac{48 b^2 c d e^2-4 b^3 e^3-120 b c^2 d^2 e+80 c^3 d^3}{d+e x}+\frac{6 d \left (-6 b^2 c d e^2+b^3 e^3+10 b c^2 d^2 e-5 c^3 d^3\right )}{(d+e x)^2}+12 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)-4 c^2 e x (5 c d-3 b e)+\frac{4 d^2 (c d-b e)^2 (2 c d-b e)}{(d+e x)^3}-\frac{d^3 (c d-b e)^3}{(d+e x)^4}+2 c^3 e^2 x^2}{4 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

(-4*c^2*e*(5*c*d - 3*b*e)*x + 2*c^3*e^2*x^2 - (d^3*(c*d - b*e)^3)/(d + e*x)^4 + (4*d^2*(c*d - b*e)^2*(2*c*d -
b*e))/(d + e*x)^3 + (6*d*(-5*c^3*d^3 + 10*b*c^2*d^2*e - 6*b^2*c*d*e^2 + b^3*e^3))/(d + e*x)^2 + (80*c^3*d^3 -
120*b*c^2*d^2*e + 48*b^2*c*d*e^2 - 4*b^3*e^3)/(d + e*x) + 12*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Log[d + e*x])
/(4*e^7)

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Maple [A]  time = 0.054, size = 370, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{x}^{2}}{2\,{e}^{5}}}+3\,{\frac{{c}^{2}xb}{{e}^{5}}}-5\,{\frac{{c}^{3}dx}{{e}^{6}}}+{\frac{3\,d{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-9\,{\frac{{b}^{2}{d}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{2}}}+15\,{\frac{b{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{4}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+3\,{\frac{c\ln \left ( ex+d \right ){b}^{2}}{{e}^{5}}}-15\,{\frac{{c}^{2}\ln \left ( ex+d \right ) bd}{{e}^{6}}}+15\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}-{\frac{{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{d}^{3}{b}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{3}}}-5\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}cd}{{e}^{5} \left ( ex+d \right ) }}-30\,{\frac{b{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{{d}^{3}{b}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{d}^{4}{b}^{2}c}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{d}^{5}b{c}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}{d}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^5,x)

[Out]

1/2*c^3*x^2/e^5+3*c^2/e^5*x*b-5*c^3*d*x/e^6+3/2*d/e^4/(e*x+d)^2*b^3-9*d^2/e^5/(e*x+d)^2*b^2*c+15*d^3/e^6/(e*x+
d)^2*b*c^2-15/2*d^4/e^7/(e*x+d)^2*c^3+3*c/e^5*ln(e*x+d)*b^2-15*c^2/e^6*ln(e*x+d)*b*d+15*c^3/e^7*ln(e*x+d)*d^2-
d^2/e^4/(e*x+d)^3*b^3+4*d^3/e^5/(e*x+d)^3*b^2*c-5*d^4/e^6/(e*x+d)^3*b*c^2+2*d^5/e^7/(e*x+d)^3*c^3-1/e^4/(e*x+d
)*b^3+12/e^5/(e*x+d)*b^2*c*d-30/e^6/(e*x+d)*b*c^2*d^2+20/e^7/(e*x+d)*c^3*d^3+1/4*d^3/e^4/(e*x+d)^4*b^3-3/4*d^4
/e^5/(e*x+d)^4*b^2*c+3/4*d^5/e^6/(e*x+d)^4*b*c^2-1/4*d^6/e^7/(e*x+d)^4*c^3

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Maxima [A]  time = 1.2026, size = 409, normalized size = 1.92 \begin{align*} \frac{57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e + 25 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 4 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} + 22 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^5,x, algorithm="maxima")

[Out]

1/4*(57*c^3*d^6 - 77*b*c^2*d^5*e + 25*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 4*(20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*
b^2*c*d*e^5 - b^3*e^6)*x^3 + 6*(35*c^3*d^4*e^2 - 50*b*c^2*d^3*e^3 + 18*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 4*(47*
c^3*d^5*e - 65*b*c^2*d^4*e^2 + 22*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4
*d^3*e^8*x + d^4*e^7) + 1/2*(c^3*e*x^2 - 2*(5*c^3*d - 3*b*c^2*e)*x)/e^6 + 3*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e
^2)*log(e*x + d)/e^7

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Fricas [B]  time = 1.58442, size = 976, normalized size = 4.58 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e + 25 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 12 \,{\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \,{\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \,{\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} + 22 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + b^{2} c d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + b^{2} c d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + b^{2} c d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + b^{2} c d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/4*(2*c^3*e^6*x^6 + 57*c^3*d^6 - 77*b*c^2*d^5*e + 25*b^2*c*d^4*e^2 - b^3*d^3*e^3 - 12*(c^3*d*e^5 - b*c^2*e^6)
*x^5 - 4*(17*c^3*d^2*e^4 - 12*b*c^2*d*e^5)*x^4 - 4*(8*c^3*d^3*e^3 + 12*b*c^2*d^2*e^4 - 12*b^2*c*d*e^5 + b^3*e^
6)*x^3 + 6*(22*c^3*d^4*e^2 - 42*b*c^2*d^3*e^3 + 18*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 4*(42*c^3*d^5*e - 62*b*c^2
*d^4*e^2 + 22*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x + 12*(5*c^3*d^6 - 5*b*c^2*d^5*e + b^2*c*d^4*e^2 + (5*c^3*d^2*e^4
- 5*b*c^2*d*e^5 + b^2*c*e^6)*x^4 + 4*(5*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + b^2*c*d*e^5)*x^3 + 6*(5*c^3*d^4*e^2 -
5*b*c^2*d^3*e^3 + b^2*c*d^2*e^4)*x^2 + 4*(5*c^3*d^5*e - 5*b*c^2*d^4*e^2 + b^2*c*d^3*e^3)*x)*log(e*x + d))/(e^1
1*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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Sympy [A]  time = 14.4251, size = 314, normalized size = 1.47 \begin{align*} \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{b^{3} d^{3} e^{3} - 25 b^{2} c d^{4} e^{2} + 77 b c^{2} d^{5} e - 57 c^{3} d^{6} + x^{3} \left (4 b^{3} e^{6} - 48 b^{2} c d e^{5} + 120 b c^{2} d^{2} e^{4} - 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (6 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 300 b c^{2} d^{3} e^{3} - 210 c^{3} d^{4} e^{2}\right ) + x \left (4 b^{3} d^{2} e^{4} - 88 b^{2} c d^{3} e^{3} + 260 b c^{2} d^{4} e^{2} - 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (3 b c^{2} e - 5 c^{3} d\right )}{e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**5,x)

[Out]

c**3*x**2/(2*e**5) + 3*c*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 - (b**3*d**3*e**3 - 25*b**2*c
*d**4*e**2 + 77*b*c**2*d**5*e - 57*c**3*d**6 + x**3*(4*b**3*e**6 - 48*b**2*c*d*e**5 + 120*b*c**2*d**2*e**4 - 8
0*c**3*d**3*e**3) + x**2*(6*b**3*d*e**5 - 108*b**2*c*d**2*e**4 + 300*b*c**2*d**3*e**3 - 210*c**3*d**4*e**2) +
x*(4*b**3*d**2*e**4 - 88*b**2*c*d**3*e**3 + 260*b*c**2*d**4*e**2 - 188*c**3*d**5*e))/(4*d**4*e**7 + 16*d**3*e*
*8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + x*(3*b*c**2*e - 5*c**3*d)/e**6

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Giac [A]  time = 1.27752, size = 522, normalized size = 2.45 \begin{align*} \frac{1}{2} \,{\left (c^{3} - \frac{6 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac{30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac{120 \, b c^{2} d^{2} e^{30}}{x e + d} + \frac{60 \, b c^{2} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac{20 \, b c^{2} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b c^{2} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac{48 \, b^{2} c d e^{31}}{x e + d} - \frac{36 \, b^{2} c d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{16 \, b^{2} c d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} c d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{4 \, b^{3} e^{32}}{x e + d} + \frac{6 \, b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac{4 \, b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac{b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

1/2*(c^3 - 6*(2*c^3*d*e - b*c^2*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-7) - 3*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c
*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/4*(80*c^3*d^3*e^29/(x*e + d) - 30*c^3*d^4*e^29/(x*e + d)
^2 + 8*c^3*d^5*e^29/(x*e + d)^3 - c^3*d^6*e^29/(x*e + d)^4 - 120*b*c^2*d^2*e^30/(x*e + d) + 60*b*c^2*d^3*e^30/
(x*e + d)^2 - 20*b*c^2*d^4*e^30/(x*e + d)^3 + 3*b*c^2*d^5*e^30/(x*e + d)^4 + 48*b^2*c*d*e^31/(x*e + d) - 36*b^
2*c*d^2*e^31/(x*e + d)^2 + 16*b^2*c*d^3*e^31/(x*e + d)^3 - 3*b^2*c*d^4*e^31/(x*e + d)^4 - 4*b^3*e^32/(x*e + d)
+ 6*b^3*d*e^32/(x*e + d)^2 - 4*b^3*d^2*e^32/(x*e + d)^3 + b^3*d^3*e^32/(x*e + d)^4)*e^(-36)