3.234 \(\int \frac{x^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=316 \[ \frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}-\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^5}+\frac{x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}+\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{17/3}}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{17/3}}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]

[Out]

-(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(2*b^5) + ((b^3*c - a*b^2*d + a^2*b
*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^8)/(8*b^3) + ((b*e - a*f)*
x^11)/(11*b^2) + (f*x^14)/(14*b) - (a^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(17/3)) - (a^(5/3)
*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(17/3)) + (a
^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(
2/3)*x^2])/(6*b^(17/3))

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Rubi [A]  time = 0.697341, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}-\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^5}+\frac{x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}+\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{17/3}}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{17/3}}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

-(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(2*b^5) + ((b^3*c - a*b^2*d + a^2*b
*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^8)/(8*b^3) + ((b*e - a*f)*
x^11)/(11*b^2) + (f*x^14)/(14*b) - (a^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(17/3)) - (a^(5/3)
*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(17/3)) + (a
^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(
2/3)*x^2])/(6*b^(17/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{17}{3}}} - \frac{a^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{17}{3}}} + \frac{\sqrt{3} a^{\frac{5}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{17}{3}}} + \frac{a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int x\, dx}{b^{5}} + \frac{f x^{14}}{14 b} - \frac{x^{11} \left (a f - b e\right )}{11 b^{2}} + \frac{x^{8} \left (a^{2} f - a b e + b^{2} d\right )}{8 b^{3}} - \frac{x^{5} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

a**(5/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*b
**(17/3)) - a**(5/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(2/3) - a**(
1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(17/3)) + sqrt(3)*a**(5/3)*(a**3*f - a**2
*b*e + a*b**2*d - b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(
3*b**(17/3)) + a*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*Integral(x, x)/b**5 + f
*x**14/(14*b) - x**11*(a*f - b*e)/(11*b**2) + x**8*(a**2*f - a*b*e + b**2*d)/(8*
b**3) - x**5*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(5*b**4)

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Mathematica [A]  time = 0.182511, size = 311, normalized size = 0.98 \[ \frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}+\frac{a x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 b^5}+\frac{x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}-\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^{17/3}}+\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^{17/3}}+\frac{a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(2*b^5) + ((b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^8)/(8*b^3) + ((b*e - a*f
)*x^11)/(11*b^2) + (f*x^14)/(14*b) + (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^
3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(17/3)) + (a^(5/3)*
(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(17/3)) -
(a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(6*b^(17/3))

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Maple [B]  time = 0.006, size = 554, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

-1/3*a^3/b^4*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/5/b
*x^5*c+1/11/b*x^11*e+1/14*f*x^14/b-1/5/b^4*x^5*f*a^3+1/5/b^3*x^5*a^2*e-1/5/b^2*x
^5*a*d+1/2/b^5*x^2*a^4*f-1/2/b^4*x^2*a^3*e+1/2/b^3*x^2*a^2*d-1/2/b^2*x^2*a*c+1/3
*a^5/b^6/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-1/3*a^4/b^5/(a/b)^(1/3)*ln(x+(a/b)^(1/3
))*e-1/3*a^2/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-1/6*a^5/b^6/(a/b)^(1/3)*ln(x^2-
x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6*a^4/b^5/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^
(2/3))*e-1/6*a^3/b^4/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/6*a^2/b^3
/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-1/11/b^2*x^11*a*f+1/8/b^3*x^8*a
^2*f-1/8/b^2*x^8*a*e+1/8/b*x^8*d+1/3*a^3/b^4/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d+1/3
*a^2/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c+1/3*a^4/b
^5*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-1/3*a^5/b^6*3^(
1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221211, size = 462, normalized size = 1.46 \[ \frac{\sqrt{3}{\left (1540 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 3080 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 9240 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (220 \, b^{4} f x^{14} + 280 \,{\left (b^{4} e - a b^{3} f\right )} x^{11} + 385 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{8} + 616 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{5} - 1540 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2}\right )}\right )}}{27720 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/27720*sqrt(3)*(1540*sqrt(3)*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(a^2/b^2)^
(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2/3) + a*(a^2/b^2)^(1/3)) - 3080*sqrt(3)*(a*b^3
*c - a^2*b^2*d + a^3*b*e - a^4*f)*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)) -
9240*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(a^2/b^2)^(1/3)*arctan(-1/3*(2*sqr
t(3)*a*x - sqrt(3)*b*(a^2/b^2)^(2/3))/(b*(a^2/b^2)^(2/3))) + 3*sqrt(3)*(220*b^4*
f*x^14 + 280*(b^4*e - a*b^3*f)*x^11 + 385*(b^4*d - a*b^3*e + a^2*b^2*f)*x^8 + 61
6*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^5 - 1540*(a*b^3*c - a^2*b^2*d + a^3*
b*e - a^4*f)*x^2))/b^5

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Sympy [A]  time = 2.93698, size = 496, normalized size = 1.57 \[ \operatorname{RootSum}{\left (27 t^{3} b^{17} - a^{14} f^{3} + 3 a^{13} b e f^{2} - 3 a^{12} b^{2} d f^{2} - 3 a^{12} b^{2} e^{2} f + 3 a^{11} b^{3} c f^{2} + 6 a^{11} b^{3} d e f + a^{11} b^{3} e^{3} - 6 a^{10} b^{4} c e f - 3 a^{10} b^{4} d^{2} f - 3 a^{10} b^{4} d e^{2} + 6 a^{9} b^{5} c d f + 3 a^{9} b^{5} c e^{2} + 3 a^{9} b^{5} d^{2} e - 3 a^{8} b^{6} c^{2} f - 6 a^{8} b^{6} c d e - a^{8} b^{6} d^{3} + 3 a^{7} b^{7} c^{2} e + 3 a^{7} b^{7} c d^{2} - 3 a^{6} b^{8} c^{2} d + a^{5} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{11}}{a^{9} f^{2} - 2 a^{8} b e f + 2 a^{7} b^{2} d f + a^{7} b^{2} e^{2} - 2 a^{6} b^{3} c f - 2 a^{6} b^{3} d e + 2 a^{5} b^{4} c e + a^{5} b^{4} d^{2} - 2 a^{4} b^{5} c d + a^{3} b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{14}}{14 b} - \frac{x^{11} \left (a f - b e\right )}{11 b^{2}} + \frac{x^{8} \left (a^{2} f - a b e + b^{2} d\right )}{8 b^{3}} - \frac{x^{5} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{5 b^{4}} + \frac{x^{2} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**17 - a**14*f**3 + 3*a**13*b*e*f**2 - 3*a**12*b**2*d*f**2 - 3
*a**12*b**2*e**2*f + 3*a**11*b**3*c*f**2 + 6*a**11*b**3*d*e*f + a**11*b**3*e**3
- 6*a**10*b**4*c*e*f - 3*a**10*b**4*d**2*f - 3*a**10*b**4*d*e**2 + 6*a**9*b**5*c
*d*f + 3*a**9*b**5*c*e**2 + 3*a**9*b**5*d**2*e - 3*a**8*b**6*c**2*f - 6*a**8*b**
6*c*d*e - a**8*b**6*d**3 + 3*a**7*b**7*c**2*e + 3*a**7*b**7*c*d**2 - 3*a**6*b**8
*c**2*d + a**5*b**9*c**3, Lambda(_t, _t*log(9*_t**2*b**11/(a**9*f**2 - 2*a**8*b*
e*f + 2*a**7*b**2*d*f + a**7*b**2*e**2 - 2*a**6*b**3*c*f - 2*a**6*b**3*d*e + 2*a
**5*b**4*c*e + a**5*b**4*d**2 - 2*a**4*b**5*c*d + a**3*b**6*c**2) + x))) + f*x**
14/(14*b) - x**11*(a*f - b*e)/(11*b**2) + x**8*(a**2*f - a*b*e + b**2*d)/(8*b**3
) - x**5*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(5*b**4) + x**2*(a**4*f - a**3*
b*e + a**2*b**2*d - a*b**3*c)/(2*b**5)

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GIAC/XCAS [A]  time = 0.219766, size = 595, normalized size = 1.88 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{7}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{7}} - \frac{{\left (a^{2} b^{12} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{11} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{5} b^{9} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{4} b^{10} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{14}} + \frac{220 \, b^{13} f x^{14} - 280 \, a b^{12} f x^{11} + 280 \, b^{13} x^{11} e + 385 \, b^{13} d x^{8} + 385 \, a^{2} b^{11} f x^{8} - 385 \, a b^{12} x^{8} e + 616 \, b^{13} c x^{5} - 616 \, a b^{12} d x^{5} - 616 \, a^{3} b^{10} f x^{5} + 616 \, a^{2} b^{11} x^{5} e - 1540 \, a b^{12} c x^{2} + 1540 \, a^{2} b^{11} d x^{2} + 1540 \, a^{4} b^{9} f x^{2} - 1540 \, a^{3} b^{10} x^{2} e}{3080 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(2/3)*a*b^3*c - (-a*b^2)^(2/3)*a^2*b^2*d - (-a*b^2)^(2/3)
*a^4*f + (-a*b^2)^(2/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^
(1/3))/b^7 + 1/6*((-a*b^2)^(2/3)*a*b^3*c - (-a*b^2)^(2/3)*a^2*b^2*d - (-a*b^2)^(
2/3)*a^4*f + (-a*b^2)^(2/3)*a^3*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^7
- 1/3*(a^2*b^12*c*(-a/b)^(1/3) - a^3*b^11*d*(-a/b)^(1/3) - a^5*b^9*f*(-a/b)^(1/
3) + a^4*b^10*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^14) +
1/3080*(220*b^13*f*x^14 - 280*a*b^12*f*x^11 + 280*b^13*x^11*e + 385*b^13*d*x^8 +
385*a^2*b^11*f*x^8 - 385*a*b^12*x^8*e + 616*b^13*c*x^5 - 616*a*b^12*d*x^5 - 616
*a^3*b^10*f*x^5 + 616*a^2*b^11*x^5*e - 1540*a*b^12*c*x^2 + 1540*a^2*b^11*d*x^2 +
1540*a^4*b^9*f*x^2 - 1540*a^3*b^10*x^2*e)/b^14