3.2198 \(\int \frac{1}{x^3 \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=306 \[ -\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac{3 \log (x) \left (2 b^2-a c\right )}{a^5}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 x \left (b^2-4 a c\right )^2}+\frac{24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{2 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

(-3*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (3*b*(2*b^2
- 9*a*c)*(b^2 - 3*a*c))/(a^4*(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)/(2*a*(b
^2 - 4*a*c)*x^2*(a + b*x + c*x^2)^2) + (4*b^4 - 25*a*b^2*c + 24*a^2*c^2 + 2*b*c*
(2*b^2 - 11*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x + c*x^2)) + (3*b*(2*b^6
- 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/(a^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^2 - a*c)*Log[x])/a^5 - (3*(2*b^2 - a*c)*
Log[a + b*x + c*x^2])/(2*a^5)

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Rubi [A]  time = 0.98046, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac{3 \log (x) \left (2 b^2-a c\right )}{a^5}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 x \left (b^2-4 a c\right )^2}+\frac{24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{2 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x + c*x^2)^3),x]

[Out]

(-3*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (3*b*(2*b^2
- 9*a*c)*(b^2 - 3*a*c))/(a^4*(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)/(2*a*(b
^2 - 4*a*c)*x^2*(a + b*x + c*x^2)^2) + (4*b^4 - 25*a*b^2*c + 24*a^2*c^2 + 2*b*c*
(2*b^2 - 11*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x + c*x^2)) + (3*b*(2*b^6
- 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/(a^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^2 - a*c)*Log[x])/a^5 - (3*(2*b^2 - a*c)*
Log[a + b*x + c*x^2])/(2*a^5)

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Rubi in Sympy [A]  time = 176.022, size = 301, normalized size = 0.98 \[ \frac{- 2 a c + b^{2} + b c x}{2 a x^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{24 a^{2} c^{2} - 25 a b^{2} c + 4 b^{4} + 2 b c x \left (- 11 a c + 2 b^{2}\right )}{2 a^{2} x^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{3 \left (8 a^{2} c^{2} - \frac{13 a b^{2} c}{2} + b^{4}\right )}{a^{3} x^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{3 b \left (- 9 a c + 2 b^{2}\right ) \left (- 3 a c + b^{2}\right )}{a^{4} x \left (- 4 a c + b^{2}\right )^{2}} + \frac{3 b \left (- 70 a^{3} c^{3} + 70 a^{2} b^{2} c^{2} - 21 a b^{4} c + 2 b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{5} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{6 \left (- \frac{a c}{2} + b^{2}\right ) \log{\left (x \right )}}{a^{5}} - \frac{3 \left (- \frac{a c}{2} + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(c*x**2+b*x+a)**3,x)

[Out]

(-2*a*c + b**2 + b*c*x)/(2*a*x**2*(-4*a*c + b**2)*(a + b*x + c*x**2)**2) + (24*a
**2*c**2 - 25*a*b**2*c + 4*b**4 + 2*b*c*x*(-11*a*c + 2*b**2))/(2*a**2*x**2*(-4*a
*c + b**2)**2*(a + b*x + c*x**2)) - 3*(8*a**2*c**2 - 13*a*b**2*c/2 + b**4)/(a**3
*x**2*(-4*a*c + b**2)**2) + 3*b*(-9*a*c + 2*b**2)*(-3*a*c + b**2)/(a**4*x*(-4*a*
c + b**2)**2) + 3*b*(-70*a**3*c**3 + 70*a**2*b**2*c**2 - 21*a*b**4*c + 2*b**6)*a
tanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(a**5*(-4*a*c + b**2)**(5/2)) + 6*(-a*c/2
+ b**2)*log(x)/a**5 - 3*(-a*c/2 + b**2)*log(a + b*x + c*x**2)/a**5

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Mathematica [A]  time = 0.932615, size = 269, normalized size = 0.88 \[ \frac{\frac{a^2 \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac{a^2}{x^2}-\frac{6 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{a \left (-32 a^3 c^3+97 a^2 b^2 c^2+66 a^2 b c^3 x-47 a b^4 c-42 a b^3 c^2 x+6 b^6+6 b^5 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+6 \log (x) \left (2 b^2-a c\right )+3 \left (a c-2 b^2\right ) \log (a+x (b+c x))+\frac{6 a b}{x}}{2 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x + c*x^2)^3),x]

[Out]

(-(a^2/x^2) + (6*a*b)/x + (a^2*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^
2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (a*(6*b^6 - 47*a*b^4*c + 97*a^2*b^2*
c^2 - 32*a^3*c^3 + 6*b^5*c*x - 42*a*b^3*c^2*x + 66*a^2*b*c^3*x))/((b^2 - 4*a*c)^
2*(a + x*(b + c*x))) - (6*b*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*A
rcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + 6*(2*b^2 - a*c)*Lo
g[x] + 3*(-2*b^2 + a*c)*Log[a + x*(b + c*x)])/(2*a^5)

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Maple [B]  time = 0.031, size = 1614, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^2+b*x+a)^3,x)

[Out]

-3/a^5/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b
^6-6/a^5/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8
*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^
4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-
b^10)^(1/2))*b^7+7/2/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^6+24/a^2*c
^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))-20/(c
*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3+6/a^5*b^2*ln(x)+3/a^4*b/x-3*c/a^4*l
n(x)+210/a^2/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a
*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*
c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^
8*c-b^10)^(1/2))*b*c^3-210/a^3/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-16
0*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(1
6*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a
^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^3*c^2+63/a^4/(1024*a^5*c^5-1280*a^4*b^2*c^4
+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-
8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+64
0*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^5*c-16/a/(c*x^2+b*x+a)^2
*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+3/a^4/(c*x^2+b*x+a)^2*b^7/(16*a^2*c^2-8*a*b^
2*c+b^4)*x+115/2/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c^2-55/2/a^2/(
c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4*c+23/a/(c*x^2+b*x+a)^2*b/(16*a^2*c
^2-8*a*b^2*c+b^4)*x*c^3+24/a^2/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*
c^2-20/a^3/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c+51/2/a^4*c/(16*a^2
*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^4+33/a^2/(c*x
^2+b*x+a)^2*b*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-21/a^3/(c*x^2+b*x+a)^2*b^3*c^3/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+3/a^4/(c*x^2+b*x+a)^2*b^5*c^2/(16*a^2*c^2-8*a*b^2
*c+b^4)*x^3+163/2/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2-89/
2/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^4+6/a^4/(c*x^2+b*x+a)
^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^6-1/2/a^3/x^2-60/a^3*c^2/(16*a^2*c^2-8*a*b
^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^2

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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(1/((c*x^2 + b*x + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03639, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^3*x^3),x, algorithm="fricas")

[Out]

[-1/2*(3*((2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5)*x^6 + 2*(2*
b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)*x^5 + (2*b^9 - 17*a*b^7*
c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^4 + 2*(2*a*b^8 - 21*a^2*b
^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^3)*x^3 + (2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4
*b^3*c^2 - 70*a^5*b*c^3)*x^2)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x - (2*c
^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) + (a^4*b^4
- 8*a^5*b^2*c + 16*a^6*c^2 - 6*(2*a*b^5*c^2 - 15*a^2*b^3*c^3 + 27*a^3*b*c^4)*x^
5 - 3*(8*a*b^6*c - 62*a^2*b^4*c^2 + 121*a^3*b^2*c^3 - 16*a^4*c^4)*x^4 - 2*(6*a*b
^7 - 39*a^2*b^5*c + 32*a^3*b^3*c^2 + 103*a^4*b*c^3)*x^3 - (18*a^2*b^6 - 145*a^3*
b^4*c + 307*a^4*b^2*c^2 - 72*a^5*c^3)*x^2 - 4*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*
c^2)*x + 3*((2*b^6*c^2 - 17*a*b^4*c^3 + 40*a^2*b^2*c^4 - 16*a^3*c^5)*x^6 + 2*(2*
b^7*c - 17*a*b^5*c^2 + 40*a^2*b^3*c^3 - 16*a^3*b*c^4)*x^5 + (2*b^8 - 13*a*b^6*c
+ 6*a^2*b^4*c^2 + 64*a^3*b^2*c^3 - 32*a^4*c^4)*x^4 + 2*(2*a*b^7 - 17*a^2*b^5*c +
40*a^3*b^3*c^2 - 16*a^4*b*c^3)*x^3 + (2*a^2*b^6 - 17*a^3*b^4*c + 40*a^4*b^2*c^2
- 16*a^5*c^3)*x^2)*log(c*x^2 + b*x + a) - 6*((2*b^6*c^2 - 17*a*b^4*c^3 + 40*a^2
*b^2*c^4 - 16*a^3*c^5)*x^6 + 2*(2*b^7*c - 17*a*b^5*c^2 + 40*a^2*b^3*c^3 - 16*a^3
*b*c^4)*x^5 + (2*b^8 - 13*a*b^6*c + 6*a^2*b^4*c^2 + 64*a^3*b^2*c^3 - 32*a^4*c^4)
*x^4 + 2*(2*a*b^7 - 17*a^2*b^5*c + 40*a^3*b^3*c^2 - 16*a^4*b*c^3)*x^3 + (2*a^2*b
^6 - 17*a^3*b^4*c + 40*a^4*b^2*c^2 - 16*a^5*c^3)*x^2)*log(x))*sqrt(b^2 - 4*a*c))
/(((a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*x^6 + 2*(a^5*b^5*c - 8*a^6*b^3*c^2
+ 16*a^7*b*c^3)*x^5 + (a^5*b^6 - 6*a^6*b^4*c + 32*a^8*c^3)*x^4 + 2*(a^6*b^5 - 8
*a^7*b^3*c + 16*a^8*b*c^2)*x^3 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*x^2)*sqrt(
b^2 - 4*a*c)), -1/2*(6*((2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^
5)*x^6 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)*x^5 + (2*b
^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^4 + 2*(2*a*
b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^3)*x^3 + (2*a^2*b^7 - 21*a^3*
b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x +
b)/(b^2 - 4*a*c)) + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 - 6*(2*a*b^5*c^2 - 15*a^
2*b^3*c^3 + 27*a^3*b*c^4)*x^5 - 3*(8*a*b^6*c - 62*a^2*b^4*c^2 + 121*a^3*b^2*c^3
- 16*a^4*c^4)*x^4 - 2*(6*a*b^7 - 39*a^2*b^5*c + 32*a^3*b^3*c^2 + 103*a^4*b*c^3)*
x^3 - (18*a^2*b^6 - 145*a^3*b^4*c + 307*a^4*b^2*c^2 - 72*a^5*c^3)*x^2 - 4*(a^3*b
^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x + 3*((2*b^6*c^2 - 17*a*b^4*c^3 + 40*a^2*b^2*c
^4 - 16*a^3*c^5)*x^6 + 2*(2*b^7*c - 17*a*b^5*c^2 + 40*a^2*b^3*c^3 - 16*a^3*b*c^4
)*x^5 + (2*b^8 - 13*a*b^6*c + 6*a^2*b^4*c^2 + 64*a^3*b^2*c^3 - 32*a^4*c^4)*x^4 +
2*(2*a*b^7 - 17*a^2*b^5*c + 40*a^3*b^3*c^2 - 16*a^4*b*c^3)*x^3 + (2*a^2*b^6 - 1
7*a^3*b^4*c + 40*a^4*b^2*c^2 - 16*a^5*c^3)*x^2)*log(c*x^2 + b*x + a) - 6*((2*b^6
*c^2 - 17*a*b^4*c^3 + 40*a^2*b^2*c^4 - 16*a^3*c^5)*x^6 + 2*(2*b^7*c - 17*a*b^5*c
^2 + 40*a^2*b^3*c^3 - 16*a^3*b*c^4)*x^5 + (2*b^8 - 13*a*b^6*c + 6*a^2*b^4*c^2 +
64*a^3*b^2*c^3 - 32*a^4*c^4)*x^4 + 2*(2*a*b^7 - 17*a^2*b^5*c + 40*a^3*b^3*c^2 -
16*a^4*b*c^3)*x^3 + (2*a^2*b^6 - 17*a^3*b^4*c + 40*a^4*b^2*c^2 - 16*a^5*c^3)*x^2
)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*x^6 +
2*(a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*x^5 + (a^5*b^6 - 6*a^6*b^4*c + 32*
a^8*c^3)*x^4 + 2*(a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*x^3 + (a^7*b^4 - 8*a^8*b
^2*c + 16*a^9*c^2)*x^2)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 151.281, size = 7465, normalized size = 24.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**2+b*x+a)**3,x)

[Out]

(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c -
2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 16
0*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))*log(x + (9
8304*a**19*c**9*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2
+ 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**
3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a
**5))**2 - 429056*a**18*b**2*c**8*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 -
70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b
**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(
a*c - 2*b**2)/(2*a**5))**2 + 645888*a**17*b**4*c**7*(-3*b*sqrt(-(4*a*c - b**2)**
5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*
c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8
*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 508032*a**16*b**6*c**6*(-3*b*sqrt
(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(
2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**
6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 241376*a**15*b*
*8*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*
b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c
**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2
+ 147456*a**15*c**10*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**
2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2
)/(2*a**5)) - 73436*a**14*b**10*c**4*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**
3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**
4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) +
3*(a*c - 2*b**2)/(2*a**5))**2 - 542016*a**14*b**2*c**9*(-3*b*sqrt(-(4*a*c - b**2
)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a*
*5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b
**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 14479*a**13*b**12*c**3*(-3*b*sqrt
(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(
2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**
6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 923760*a**13*b*
*4*c**8*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*
b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c
**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) -
1797*a**12*b**14*c**2*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**
2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2
)/(2*a**5))**2 - 849948*a**12*b**6*c**7*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*
c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*
a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))
+ 3*(a*c - 2*b**2)/(2*a**5)) + 128*a**11*b**16*c*(-3*b*sqrt(-(4*a*c - b**2)**5)
*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c
- b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 464829*a**11*b**8*c**6*(-3*b*sqrt(-
(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*
a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*
c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 442368*a**11*c**11 -
4*a**10*b**18*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2
+ 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3
*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a*
*5))**2 - 159318*a**10*b**10*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 -
70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b
**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(
a*c - 2*b**2)/(2*a**5)) + 4889664*a**10*b**2*c**10 + 34731*a**9*b**12*c**4*(-3*b
*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b*
*6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**
2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 18774576*a**9
*b**4*c**9 - 4695*a**8*b**14*c**3*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 -
70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b
**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(
a*c - 2*b**2)/(2*a**5)) + 35177868*a**8*b**6*c**8 + 360*a**7*b**16*c**2*(-3*b*sq
rt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)
/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b
**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 37219329*a**7*b*
*8*c**7 - 12*a**6*b**18*c*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2
*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*
b**2)/(2*a**5)) + 24372684*a**6*b**10*c**6 - 10403442*a**5*b**12*c**5 + 2958642*
a**4*b**14*c**4 - 557838*a**3*b**16*c**3 + 67140*a**2*b**18*c**2 - 4680*a*b**20*
c + 144*b**22)/(1451520*a**10*b*c**11 - 8300250*a**9*b**3*c**10 + 19711566*a**8*
b**5*c**9 - 24401871*a**7*b**7*c**8 + 17859492*a**6*b**9*c**7 - 8284248*a**5*b**
11*c**6 + 2513700*a**4*b**13*c**5 - 499338*a**3*b**15*c**4 + 62748*a**2*b**17*c*
*3 - 4536*a*b**19*c**2 + 144*b**21*c)) + (3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*
c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*
a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))
+ 3*(a*c - 2*b**2)/(2*a**5))*log(x + (98304*a**19*c**9*(3*b*sqrt(-(4*a*c - b**2
)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a*
*5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b
**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 429056*a**18*b**2*c**8*(3*b*sq
rt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)
/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b
**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 645888*a**17*
b**4*c**7*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a
*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*
c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**
2 - 508032*a**16*b**6*c**6*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2
*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*
b**2)/(2*a**5))**2 + 241376*a**15*b**8*c**5*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a*
*3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 12
80*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**1
0)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 147456*a**15*c**10*(3*b*sqrt(-(4*a*c - b**
2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a
**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*
b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 73436*a**14*b**10*c**4*(3*b*sqrt
(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(
2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**
6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 542016*a**14*b*
*2*c**9*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b
**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c*
*3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 1
4479*a**13*b**12*c**3*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2
*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)
/(2*a**5))**2 + 923760*a**13*b**4*c**8*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c*
*3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a*
*4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) +
3*(a*c - 2*b**2)/(2*a**5)) - 1797*a**12*b**14*c**2*(3*b*sqrt(-(4*a*c - b**2)**5
)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c
**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*
c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 849948*a**12*b**6*c**7*(3*b*sqrt(-
(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*
a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*
c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 128*a**11*b**16*c*(3
*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*
b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 464829*a
**11*b**8*c**6*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**
5)) - 442368*a**11*c**11 - 4*a**10*b**18*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*
c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*
a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))
+ 3*(a*c - 2*b**2)/(2*a**5))**2 - 159318*a**10*b**10*c**5*(3*b*sqrt(-(4*a*c - b
**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024
*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*
a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 4889664*a**10*b**2*c**10 + 347
31*a**9*b**12*c**4*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c*
*2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2
*a**5)) - 18774576*a**9*b**4*c**9 - 4695*a**8*b**14*c**3*(3*b*sqrt(-(4*a*c - b**
2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a
**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*
b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 35177868*a**8*b**6*c**8 + 360*a*
*7*b**16*c**2*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b
**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5
)) - 37219329*a**7*b**8*c**7 - 12*a**6*b**18*c*(3*b*sqrt(-(4*a*c - b**2)**5)*(70
*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b
**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 24372684*a**6*b**10*c**6 - 10403442*a**5*b
**12*c**5 + 2958642*a**4*b**14*c**4 - 557838*a**3*b**16*c**3 + 67140*a**2*b**18*
c**2 - 4680*a*b**20*c + 144*b**22)/(1451520*a**10*b*c**11 - 8300250*a**9*b**3*c*
*10 + 19711566*a**8*b**5*c**9 - 24401871*a**7*b**7*c**8 + 17859492*a**6*b**9*c**
7 - 8284248*a**5*b**11*c**6 + 2513700*a**4*b**13*c**5 - 499338*a**3*b**15*c**4 +
62748*a**2*b**17*c**3 - 4536*a*b**19*c**2 + 144*b**21*c)) + (-16*a**5*c**2 + 8*
a**4*b**2*c - a**3*b**4 + x**5*(162*a**2*b*c**4 - 90*a*b**3*c**3 + 12*b**5*c**2)
+ x**4*(-48*a**3*c**4 + 363*a**2*b**2*c**3 - 186*a*b**4*c**2 + 24*b**6*c) + x**
3*(206*a**3*b*c**3 + 64*a**2*b**3*c**2 - 78*a*b**5*c + 12*b**7) + x**2*(-72*a**4
*c**3 + 307*a**3*b**2*c**2 - 145*a**2*b**4*c + 18*a*b**6) + x*(64*a**4*b*c**2 -
32*a**3*b**3*c + 4*a**2*b**5))/(x**6*(32*a**6*c**4 - 16*a**5*b**2*c**3 + 2*a**4*
b**4*c**2) + x**5*(64*a**6*b*c**3 - 32*a**5*b**3*c**2 + 4*a**4*b**5*c) + x**4*(6
4*a**7*c**3 - 12*a**5*b**4*c + 2*a**4*b**6) + x**3*(64*a**7*b*c**2 - 32*a**6*b**
3*c + 4*a**5*b**5) + x**2*(32*a**8*c**2 - 16*a**7*b**2*c + 2*a**6*b**4)) - 3*(a*
c - 2*b**2)*log(x + (-442368*a**11*c**11 + 4889664*a**10*b**2*c**10 - 442368*a**
10*c**10*(a*c - 2*b**2) - 18774576*a**9*b**4*c**9 + 1626048*a**9*b**2*c**9*(a*c
- 2*b**2) + 884736*a**9*c**9*(a*c - 2*b**2)**2 + 35177868*a**8*b**6*c**8 - 27712
80*a**8*b**4*c**8*(a*c - 2*b**2) - 3861504*a**8*b**2*c**8*(a*c - 2*b**2)**2 - 37
219329*a**7*b**8*c**7 + 2549844*a**7*b**6*c**7*(a*c - 2*b**2) + 5812992*a**7*b**
4*c**7*(a*c - 2*b**2)**2 + 24372684*a**6*b**10*c**6 - 1394487*a**6*b**8*c**6*(a*
c - 2*b**2) - 4572288*a**6*b**6*c**6*(a*c - 2*b**2)**2 - 10403442*a**5*b**12*c**
5 + 477954*a**5*b**10*c**5*(a*c - 2*b**2) + 2172384*a**5*b**8*c**5*(a*c - 2*b**2
)**2 + 2958642*a**4*b**14*c**4 - 104193*a**4*b**12*c**4*(a*c - 2*b**2) - 660924*
a**4*b**10*c**4*(a*c - 2*b**2)**2 - 557838*a**3*b**16*c**3 + 14085*a**3*b**14*c*
*3*(a*c - 2*b**2) + 130311*a**3*b**12*c**3*(a*c - 2*b**2)**2 + 67140*a**2*b**18*
c**2 - 1080*a**2*b**16*c**2*(a*c - 2*b**2) - 16173*a**2*b**14*c**2*(a*c - 2*b**2
)**2 - 4680*a*b**20*c + 36*a*b**18*c*(a*c - 2*b**2) + 1152*a*b**16*c*(a*c - 2*b*
*2)**2 + 144*b**22 - 36*b**18*(a*c - 2*b**2)**2)/(1451520*a**10*b*c**11 - 830025
0*a**9*b**3*c**10 + 19711566*a**8*b**5*c**9 - 24401871*a**7*b**7*c**8 + 17859492
*a**6*b**9*c**7 - 8284248*a**5*b**11*c**6 + 2513700*a**4*b**13*c**5 - 499338*a**
3*b**15*c**4 + 62748*a**2*b**17*c**3 - 4536*a*b**19*c**2 + 144*b**21*c))/a**5

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GIAC/XCAS [A]  time = 0.20863, size = 554, normalized size = 1.81 \[ -\frac{3 \,{\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, b^{5} c^{2} x^{5} - 90 \, a b^{3} c^{3} x^{5} + 162 \, a^{2} b c^{4} x^{5} + 24 \, b^{6} c x^{4} - 186 \, a b^{4} c^{2} x^{4} + 363 \, a^{2} b^{2} c^{3} x^{4} - 48 \, a^{3} c^{4} x^{4} + 12 \, b^{7} x^{3} - 78 \, a b^{5} c x^{3} + 64 \, a^{2} b^{3} c^{2} x^{3} + 206 \, a^{3} b c^{3} x^{3} + 18 \, a b^{6} x^{2} - 145 \, a^{2} b^{4} c x^{2} + 307 \, a^{3} b^{2} c^{2} x^{2} - 72 \, a^{4} c^{3} x^{2} + 4 \, a^{2} b^{5} x - 32 \, a^{3} b^{3} c x + 64 \, a^{4} b c^{2} x - a^{3} b^{4} + 8 \, a^{4} b^{2} c - 16 \, a^{5} c^{2}}{2 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )}{\left (c x^{3} + b x^{2} + a x\right )}^{2}} - \frac{3 \,{\left (2 \, b^{2} - a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{5}} + \frac{3 \,{\left (2 \, b^{2} - a c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^3*x^3),x, algorithm="giac")

[Out]

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(
-b^2 + 4*a*c))/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(
12*b^5*c^2*x^5 - 90*a*b^3*c^3*x^5 + 162*a^2*b*c^4*x^5 + 24*b^6*c*x^4 - 186*a*b^4
*c^2*x^4 + 363*a^2*b^2*c^3*x^4 - 48*a^3*c^4*x^4 + 12*b^7*x^3 - 78*a*b^5*c*x^3 +
64*a^2*b^3*c^2*x^3 + 206*a^3*b*c^3*x^3 + 18*a*b^6*x^2 - 145*a^2*b^4*c*x^2 + 307*
a^3*b^2*c^2*x^2 - 72*a^4*c^3*x^2 + 4*a^2*b^5*x - 32*a^3*b^3*c*x + 64*a^4*b*c^2*x
- a^3*b^4 + 8*a^4*b^2*c - 16*a^5*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*
x^3 + b*x^2 + a*x)^2) - 3/2*(2*b^2 - a*c)*ln(c*x^2 + b*x + a)/a^5 + 3*(2*b^2 - a
*c)*ln(abs(x))/a^5