3.1513 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=227 \[ -\frac{4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac{2 c^3 x}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.585563, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac{2 c^3 x}{e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 78.0472, size = 224, normalized size = 0.99 \[ \frac{2 c^{3} x}{e^{5}} + \frac{5 c^{2} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{4 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \left (d + e x\right )} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{2 e^{6} \left (d + e x\right )^{2}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{6} \left (d + e x\right )^{3}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{4 e^{6} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d)**5,x)

[Out]

2*c**3*x/e**5 + 5*c**2*(b*e - 2*c*d)*log(d + e*x)/e**6 - 4*c*(a*c*e**2 + b**2*e*
*2 - 5*b*c*d*e + 5*c**2*d**2)/(e**6*(d + e*x)) - (b*e - 2*c*d)*(6*a*c*e**2 + b**
2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/(2*e**6*(d + e*x)**2) - 2*(a*e**2 - b*d*e +
c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(3*e**6*(d + e*x)**3) -
(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(4*e**6*(d + e*x)**4)

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Mathematica [A]  time = 0.998831, size = 292, normalized size = 1.29 \[ -\frac{2 c e^2 \left (a^2 e^2 (d+4 e x)+3 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+6 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+b e^3 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+c^2 e \left (12 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+60 c^2 (d+e x)^4 (2 c d-b e) \log (d+e x)+2 c^3 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(2*c^3*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4
- 12*e^5*x^5) + b*e^3*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*
e^2*x^2)) + 2*c*e^2*(a^2*e^2*(d + 4*e*x) + 3*a*b*e*(d^2 + 4*d*e*x + 6*e^2*x^2) +
6*b^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + c^2*e*(12*a*e*(d^3 + 4*d^2
*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*b*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 4
8*e^3*x^3)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^4*Log[d + e*x])/(12*e^6*(d + e*x)^4
)

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Maple [B]  time = 0.016, size = 507, normalized size = 2.2 \[{\frac{{a}^{2}cd}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+{\frac{a{b}^{2}d}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+2\,{\frac{{c}^{3}x}{{e}^{5}}}-20\,{\frac{{c}^{3}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+5\,{\frac{{c}^{2}\ln \left ( ex+d \right ) b}{{e}^{5}}}-10\,{\frac{{c}^{3}\ln \left ( ex+d \right ) d}{{e}^{6}}}-15\,{\frac{b{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}cd}{{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,{a}^{2}c}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,a{b}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{2\,d{b}^{3}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{10\,{c}^{3}{d}^{4}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{3\,abc{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+4\,{\frac{cabd}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{a{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{2}{d}^{3}c}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{5\,b{d}^{4}{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-4\,{\frac{a{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+20\,{\frac{{c}^{2}bd}{{e}^{5} \left ( ex+d \right ) }}+{\frac{20\,b{d}^{3}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-3\,{\frac{abc}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{a{c}^{2}d}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-4\,{\frac{c{b}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+10\,{\frac{{c}^{3}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-4\,{\frac{a{c}^{2}}{{e}^{4} \left ( ex+d \right ) }}-4\,{\frac{{b}^{2}c}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{a}^{2}b}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{3}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^5,x)

[Out]

1/2/e^2/(e*x+d)^4*a^2*c*d+1/2/e^2/(e*x+d)^4*d*a*b^2+2*c^3*x/e^5-20/e^6*c^3/(e*x+
d)*d^2+5*c^2/e^5*ln(e*x+d)*b-10*c^3/e^6*ln(e*x+d)*d-15/e^5/(e*x+d)^2*b*c^2*d^2+6
/e^4/(e*x+d)^2*b^2*c*d+1/2/e^6/(e*x+d)^4*c^3*d^5-2/3/e^2/(e*x+d)^3*a^2*c-2/3/e^2
/(e*x+d)^3*a*b^2+2/3/e^3/(e*x+d)^3*b^3*d-10/3/e^6/(e*x+d)^3*c^3*d^4-3/2/e^3/(e*x
+d)^4*d^2*a*b*c+4/e^3/(e*x+d)^3*a*b*c*d+1/e^4/(e*x+d)^4*a*c^2*d^3+1/e^4/(e*x+d)^
4*d^3*b^2*c-5/4/e^5/(e*x+d)^4*d^4*b*c^2-4/e^4/(e*x+d)^3*a*c^2*d^2+20/e^5*c^2/(e*
x+d)*b*d+20/3/e^5/(e*x+d)^3*d^3*b*c^2-3/e^3/(e*x+d)^2*a*b*c+6/e^4/(e*x+d)^2*a*c^
2*d-1/2/e^3/(e*x+d)^2*b^3-4/e^4/(e*x+d)^3*b^2*c*d^2+10/e^6/(e*x+d)^2*c^3*d^3-4/e
^4*c^2/(e*x+d)*a-4/e^4*c/(e*x+d)*b^2-1/4/e/(e*x+d)^4*a^2*b-1/4/e^3/(e*x+d)^4*d^2
*b^3

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Maxima [A]  time = 0.735645, size = 456, normalized size = 2.01 \[ -\frac{154 \, c^{3} d^{5} - 125 \, b c^{2} d^{4} e + 3 \, a^{2} b e^{5} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} + 48 \,{\left (5 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} +{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} + 6 \,{\left (100 \, c^{3} d^{3} e^{2} - 90 \, b c^{2} d^{2} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} +{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 4 \,{\left (130 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{2 \, c^{3} x}{e^{5}} - \frac{5 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^5,x, algorithm="maxima")

[Out]

-1/12*(154*c^3*d^5 - 125*b*c^2*d^4*e + 3*a^2*b*e^5 + 12*(b^2*c + a*c^2)*d^3*e^2
+ (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4 + 48*(5*c^3*d^2*e^3 - 5*b*c^
2*d*e^4 + (b^2*c + a*c^2)*e^5)*x^3 + 6*(100*c^3*d^3*e^2 - 90*b*c^2*d^2*e^3 + 12*
(b^2*c + a*c^2)*d*e^4 + (b^3 + 6*a*b*c)*e^5)*x^2 + 4*(130*c^3*d^4*e - 110*b*c^2*
d^3*e^2 + 12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)
*e^5)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + 2*c^
3*x/e^5 - 5*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^6

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Fricas [A]  time = 0.264825, size = 618, normalized size = 2.72 \[ \frac{24 \, c^{3} e^{5} x^{5} + 96 \, c^{3} d e^{4} x^{4} - 154 \, c^{3} d^{5} + 125 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} - 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 48 \,{\left (2 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} +{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 6 \,{\left (84 \, c^{3} d^{3} e^{2} - 90 \, b c^{2} d^{2} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} +{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 4 \,{\left (124 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{5} - b c^{2} d^{4} e +{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (2 \, c^{3} d^{2} e^{3} - b c^{2} d e^{4}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{3} e^{2} - b c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (2 \, c^{3} d^{4} e - b c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/12*(24*c^3*e^5*x^5 + 96*c^3*d*e^4*x^4 - 154*c^3*d^5 + 125*b*c^2*d^4*e - 3*a^2*
b*e^5 - 12*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 - 2*(a*b^2 + a^2*c)
*d*e^4 - 48*(2*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)*e^5)*x^3 - 6*(84*c^
3*d^3*e^2 - 90*b*c^2*d^2*e^3 + 12*(b^2*c + a*c^2)*d*e^4 + (b^3 + 6*a*b*c)*e^5)*x
^2 - 4*(124*c^3*d^4*e - 110*b*c^2*d^3*e^2 + 12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 +
6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*x - 60*(2*c^3*d^5 - b*c^2*d^4*e + (2*c^3
*d*e^4 - b*c^2*e^5)*x^4 + 4*(2*c^3*d^2*e^3 - b*c^2*d*e^4)*x^3 + 6*(2*c^3*d^3*e^2
- b*c^2*d^2*e^3)*x^2 + 4*(2*c^3*d^4*e - b*c^2*d^3*e^2)*x)*log(e*x + d))/(e^10*x
^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279703, size = 709, normalized size = 3.12 \[ 2 \,{\left (x e + d\right )} c^{3} e^{\left (-6\right )} + 5 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-6\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{1}{12} \,{\left (\frac{240 \, c^{3} d^{2} e^{22}}{x e + d} - \frac{120 \, c^{3} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac{40 \, c^{3} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac{6 \, c^{3} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac{240 \, b c^{2} d e^{23}}{x e + d} + \frac{180 \, b c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac{80 \, b c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac{15 \, b c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{48 \, b^{2} c e^{24}}{x e + d} + \frac{48 \, a c^{2} e^{24}}{x e + d} - \frac{72 \, b^{2} c d e^{24}}{{\left (x e + d\right )}^{2}} - \frac{72 \, a c^{2} d e^{24}}{{\left (x e + d\right )}^{2}} + \frac{48 \, b^{2} c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac{48 \, a c^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac{12 \, b^{2} c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac{12 \, a c^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac{6 \, b^{3} e^{25}}{{\left (x e + d\right )}^{2}} + \frac{36 \, a b c e^{25}}{{\left (x e + d\right )}^{2}} - \frac{8 \, b^{3} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac{48 \, a b c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b^{3} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac{18 \, a b c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac{8 \, a b^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac{8 \, a^{2} c e^{26}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a b^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a^{2} c d e^{26}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a^{2} b e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^5,x, algorithm="giac")

[Out]

2*(x*e + d)*c^3*e^(-6) + 5*(2*c^3*d - b*c^2*e)*e^(-6)*ln(abs(x*e + d)*e^(-1)/(x*
e + d)^2) - 1/12*(240*c^3*d^2*e^22/(x*e + d) - 120*c^3*d^3*e^22/(x*e + d)^2 + 40
*c^3*d^4*e^22/(x*e + d)^3 - 6*c^3*d^5*e^22/(x*e + d)^4 - 240*b*c^2*d*e^23/(x*e +
d) + 180*b*c^2*d^2*e^23/(x*e + d)^2 - 80*b*c^2*d^3*e^23/(x*e + d)^3 + 15*b*c^2*
d^4*e^23/(x*e + d)^4 + 48*b^2*c*e^24/(x*e + d) + 48*a*c^2*e^24/(x*e + d) - 72*b^
2*c*d*e^24/(x*e + d)^2 - 72*a*c^2*d*e^24/(x*e + d)^2 + 48*b^2*c*d^2*e^24/(x*e +
d)^3 + 48*a*c^2*d^2*e^24/(x*e + d)^3 - 12*b^2*c*d^3*e^24/(x*e + d)^4 - 12*a*c^2*
d^3*e^24/(x*e + d)^4 + 6*b^3*e^25/(x*e + d)^2 + 36*a*b*c*e^25/(x*e + d)^2 - 8*b^
3*d*e^25/(x*e + d)^3 - 48*a*b*c*d*e^25/(x*e + d)^3 + 3*b^3*d^2*e^25/(x*e + d)^4
+ 18*a*b*c*d^2*e^25/(x*e + d)^4 + 8*a*b^2*e^26/(x*e + d)^3 + 8*a^2*c*e^26/(x*e +
d)^3 - 6*a*b^2*d*e^26/(x*e + d)^4 - 6*a^2*c*d*e^26/(x*e + d)^4 + 3*a^2*b*e^27/(
x*e + d)^4)*e^(-28)