3.2331 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=304 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^6 (d+e x)^6}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{7 e^6 (d+e x)^7}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{8 e^6 (d+e x)^8}+\frac{c (-A c e-2 b B e+5 B c d)}{4 e^6 (d+e x)^4}-\frac{B c^2}{3 e^6 (d+e x)^3} \]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(8*e^6*(d + e*x)^8) + ((c*d^2 - b*d*e +
a*e^2)*(2*A*e*(2*c*d - b*e) - B*(5*c*d^2 - e*(3*b*d - a*e))))/(7*e^6*(d + e*x)^7
) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2
*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(6*e^6*(d + e*x)^6) + (2*A*c*e*(2*c*d - b
*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(5*e^6*(d + e*x)^5) + (c*(
5*B*c*d - 2*b*B*e - A*c*e))/(4*e^6*(d + e*x)^4) - (B*c^2)/(3*e^6*(d + e*x)^3)

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Rubi [A]  time = 1.01496, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^6 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6 (d+e x)^7}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{8 e^6 (d+e x)^8}+\frac{c (-A c e-2 b B e+5 B c d)}{4 e^6 (d+e x)^4}-\frac{B c^2}{3 e^6 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^9,x]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(8*e^6*(d + e*x)^8) - ((c*d^2 - b*d*e +
a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(7*e^6*(d + e*x)^7
) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2
*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(6*e^6*(d + e*x)^6) + (2*A*c*e*(2*c*d - b
*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(5*e^6*(d + e*x)^5) + (c*(
5*B*c*d - 2*b*B*e - A*c*e))/(4*e^6*(d + e*x)^4) - (B*c^2)/(3*e^6*(d + e*x)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2/(e*x+d)**9,x)

[Out]

Timed out

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Mathematica [A]  time = 0.75728, size = 375, normalized size = 1.23 \[ -\frac{A e \left (5 e^2 \left (21 a^2 e^2+6 a b e (d+8 e x)+b^2 \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 c e \left (5 a e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+3 c^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+B \left (e^2 \left (15 a^2 e^2 (d+8 e x)+10 a b e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 c e \left (a e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+5 c^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^9,x]

[Out]

-(A*e*(3*c^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*
e^2*(21*a^2*e^2 + 6*a*b*e*(d + 8*e*x) + b^2*(d^2 + 8*d*e*x + 28*e^2*x^2)) + 2*c*
e*(5*a*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*b*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56
*e^3*x^3))) + B*(5*c^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d
*e^4*x^4 + 56*e^5*x^5) + e^2*(15*a^2*e^2*(d + 8*e*x) + 10*a*b*e*(d^2 + 8*d*e*x +
28*e^2*x^2) + 3*b^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 6*c*e*(a*e
*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b*(d^4 + 8*d^3*e*x + 28*d^2*e^2
*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4))))/(840*e^6*(d + e*x)^8)

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Maple [A]  time = 0.01, size = 453, normalized size = 1.5 \[ -{\frac{B{c}^{2}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{A{a}^{2}{e}^{5}-2\,Adab{e}^{4}+2\,A{d}^{2}ac{e}^{3}+A{d}^{2}{b}^{2}{e}^{3}-2\,A{d}^{3}bc{e}^{2}+A{d}^{4}{c}^{2}e-Bd{a}^{2}{e}^{4}+2\,B{d}^{2}ab{e}^{3}-2\,B{d}^{3}ac{e}^{2}-B{d}^{3}{b}^{2}{e}^{2}+2\,B{d}^{4}bce-B{c}^{2}{d}^{5}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+2\,aBc{e}^{2}+{b}^{2}B{e}^{2}-8\,Bdbce+10\,B{c}^{2}{d}^{2}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{2\,aAc{e}^{3}+A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e+2\,B{e}^{3}ab-6\,Bdac{e}^{2}-3\,Bd{b}^{2}{e}^{2}+12\,B{d}^{2}bce-10\,B{c}^{2}{d}^{3}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{2\,Aab{e}^{4}-4\,Adac{e}^{3}-2\,Ad{b}^{2}{e}^{3}+6\,A{d}^{2}bc{e}^{2}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}-4\,Bdab{e}^{3}+6\,B{d}^{2}ac{e}^{2}+3\,B{d}^{2}{b}^{2}{e}^{2}-8\,B{d}^{3}bce+5\,B{c}^{2}{d}^{4}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^9,x)

[Out]

-1/3*B*c^2/e^6/(e*x+d)^3-1/8*(A*a^2*e^5-2*A*a*b*d*e^4+2*A*a*c*d^2*e^3+A*b^2*d^2*
e^3-2*A*b*c*d^3*e^2+A*c^2*d^4*e-B*a^2*d*e^4+2*B*a*b*d^2*e^3-2*B*a*c*d^3*e^2-B*b^
2*d^3*e^2+2*B*b*c*d^4*e-B*c^2*d^5)/e^6/(e*x+d)^8-1/4*c*(A*c*e+2*B*b*e-5*B*c*d)/e
^6/(e*x+d)^4-1/5*(2*A*b*c*e^2-4*A*c^2*d*e+2*B*a*c*e^2+B*b^2*e^2-8*B*b*c*d*e+10*B
*c^2*d^2)/e^6/(e*x+d)^5-1/6*(2*A*a*c*e^3+A*b^2*e^3-6*A*b*c*d*e^2+6*A*c^2*d^2*e+2
*B*a*b*e^3-6*B*a*c*d*e^2-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)/e^6/(e*x+d)^
6-1/7*(2*A*a*b*e^4-4*A*a*c*d*e^3-2*A*b^2*d*e^3+6*A*b*c*d^2*e^2-4*A*c^2*d^3*e+B*a
^2*e^4-4*B*a*b*d*e^3+6*B*a*c*d^2*e^2+3*B*b^2*d^2*e^2-8*B*b*c*d^3*e+5*B*c^2*d^4)/
e^6/(e*x+d)^7

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Maxima [A]  time = 0.716281, size = 632, normalized size = 2.08 \[ -\frac{280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \,{\left (5 \, B c^{2} d e^{4} + 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B c^{2} d^{2} e^{3} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B c^{2} d^{3} e^{2} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B c^{2} d^{4} e + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 105*A*a^2*e^5 + 3*(2*B*b*c + A*c^2)*d^
4*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^
3 + 15*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 3*(2*B*b*c + A*c^2)*e^5)*x^
4 + 56*(5*B*c^2*d^2*e^3 + 3*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 2*(B*a + A*b)*c
)*e^5)*x^3 + 28*(5*B*c^2*d^3*e^2 + 3*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B
*a + A*b)*c)*d*e^4 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 8*(5*B*c^2*d^4*e +
3*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b
+ A*b^2 + 2*A*a*c)*d*e^4 + 15*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7
+ 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6
*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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Fricas [A]  time = 0.259853, size = 632, normalized size = 2.08 \[ -\frac{280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \,{\left (5 \, B c^{2} d e^{4} + 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B c^{2} d^{2} e^{3} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B c^{2} d^{3} e^{2} + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B c^{2} d^{4} e + 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 105*A*a^2*e^5 + 3*(2*B*b*c + A*c^2)*d^
4*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^
3 + 15*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 3*(2*B*b*c + A*c^2)*e^5)*x^
4 + 56*(5*B*c^2*d^2*e^3 + 3*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 2*(B*a + A*b)*c
)*e^5)*x^3 + 28*(5*B*c^2*d^3*e^2 + 3*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B
*a + A*b)*c)*d*e^4 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 8*(5*B*c^2*d^4*e +
3*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b
+ A*b^2 + 2*A*a*c)*d*e^4 + 15*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7
+ 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6
*e^8*x^2 + 8*d^7*e^7*x + d^8*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((B*x+A)*(c*x**2+b*x+a)**2/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295749, size = 624, normalized size = 2.05 \[ -\frac{{\left (280 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 280 \, B c^{2} d^{2} x^{3} e^{3} + 140 \, B c^{2} d^{3} x^{2} e^{2} + 40 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 420 \, B b c x^{4} e^{5} + 210 \, A c^{2} x^{4} e^{5} + 336 \, B b c d x^{3} e^{4} + 168 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 48 \, B b c d^{3} x e^{2} + 24 \, A c^{2} d^{3} x e^{2} + 6 \, B b c d^{4} e + 3 \, A c^{2} d^{4} e + 168 \, B b^{2} x^{3} e^{5} + 336 \, B a c x^{3} e^{5} + 336 \, A b c x^{3} e^{5} + 84 \, B b^{2} d x^{2} e^{4} + 168 \, B a c d x^{2} e^{4} + 168 \, A b c d x^{2} e^{4} + 24 \, B b^{2} d^{2} x e^{3} + 48 \, B a c d^{2} x e^{3} + 48 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, B a c d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 280 \, B a b x^{2} e^{5} + 140 \, A b^{2} x^{2} e^{5} + 280 \, A a c x^{2} e^{5} + 80 \, B a b d x e^{4} + 40 \, A b^{2} d x e^{4} + 80 \, A a c d x e^{4} + 10 \, B a b d^{2} e^{3} + 5 \, A b^{2} d^{2} e^{3} + 10 \, A a c d^{2} e^{3} + 120 \, B a^{2} x e^{5} + 240 \, A a b x e^{5} + 15 \, B a^{2} d e^{4} + 30 \, A a b d e^{4} + 105 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{840 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/840*(280*B*c^2*x^5*e^5 + 350*B*c^2*d*x^4*e^4 + 280*B*c^2*d^2*x^3*e^3 + 140*B*
c^2*d^3*x^2*e^2 + 40*B*c^2*d^4*x*e + 5*B*c^2*d^5 + 420*B*b*c*x^4*e^5 + 210*A*c^2
*x^4*e^5 + 336*B*b*c*d*x^3*e^4 + 168*A*c^2*d*x^3*e^4 + 168*B*b*c*d^2*x^2*e^3 + 8
4*A*c^2*d^2*x^2*e^3 + 48*B*b*c*d^3*x*e^2 + 24*A*c^2*d^3*x*e^2 + 6*B*b*c*d^4*e +
3*A*c^2*d^4*e + 168*B*b^2*x^3*e^5 + 336*B*a*c*x^3*e^5 + 336*A*b*c*x^3*e^5 + 84*B
*b^2*d*x^2*e^4 + 168*B*a*c*d*x^2*e^4 + 168*A*b*c*d*x^2*e^4 + 24*B*b^2*d^2*x*e^3
+ 48*B*a*c*d^2*x*e^3 + 48*A*b*c*d^2*x*e^3 + 3*B*b^2*d^3*e^2 + 6*B*a*c*d^3*e^2 +
6*A*b*c*d^3*e^2 + 280*B*a*b*x^2*e^5 + 140*A*b^2*x^2*e^5 + 280*A*a*c*x^2*e^5 + 80
*B*a*b*d*x*e^4 + 40*A*b^2*d*x*e^4 + 80*A*a*c*d*x*e^4 + 10*B*a*b*d^2*e^3 + 5*A*b^
2*d^2*e^3 + 10*A*a*c*d^2*e^3 + 120*B*a^2*x*e^5 + 240*A*a*b*x*e^5 + 15*B*a^2*d*e^
4 + 30*A*a*b*d*e^4 + 105*A*a^2*e^5)*e^(-6)/(x*e + d)^8