∖int 1________________ ∖left(c+ a_ x2 + b x∖right)2x6 ∖,dx

3.429 \(\int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^2 x^6} \, dx\)

Optimal. Leaf size=148 \[ \frac{b \log \left (a+b x+c x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{2 \left (b^2-3 a c\right )}{a^2 x \left (b^2-4 a c\right )}-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

(-2*(b^2 - 3*a*c))/(a^2*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c

)*x*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sq

rt[b^2 - 4*a*c]])/(a^3*(b^2 - 4*a*c)^(3/2)) - (2*b*Log[x])/a^3 + (b*Log[a + b*x

+ c*x^2])/a^3

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Rubi [A] time = 0.400381, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \log \left (a+b x+c x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{2 \left (b^2-3 a c\right )}{a^2 x \left (b^2-4 a c\right )}-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully veriﬁed.

[In] Int[1/((c + a/x^2 + b/x)^2*x^6),x]

[Out]

(-2*(b^2 - 3*a*c))/(a^2*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c

)*x*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sq

rt[b^2 - 4*a*c]])/(a^3*(b^2 - 4*a*c)^(3/2)) - (2*b*Log[x])/a^3 + (b*Log[a + b*x

+ c*x^2])/a^3

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Rubi in Sympy [A] time = 74.9223, size = 143, normalized size = 0.97 \[ \frac{- 2 a c + b^{2} + b c x}{a x \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{2 \left (- 3 a c + b^{2}\right )}{a^{2} x \left (- 4 a c + b^{2}\right )} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (a + b x + c x^{2} \right )}}{a^{3}} - \frac{2 \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(1/(c+a/x**2+b/x)**2/x**6,x)

[Out]

(-2*a*c + b**2 + b*c*x)/(a*x*(-4*a*c + b**2)*(a + b*x + c*x**2)) - 2*(-3*a*c + b

**2)/(a**2*x*(-4*a*c + b**2)) - 2*b*log(x)/a**3 + b*log(a + b*x + c*x**2)/a**3 -

2*(6*a**2*c**2 - 6*a*b**2*c + b**4)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(a**

3*(-4*a*c + b**2)**(3/2))

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Mathematica [A] time = 0.462612, size = 131, normalized size = 0.89 \[ -\frac{\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a \left (-3 a b c-2 a c^2 x+b^3+b^2 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-b \log (a+x (b+c x))+\frac{a}{x}+2 b \log (x)}{a^3} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((c + a/x^2 + b/x)^2*x^6),x]

[Out]

-((a/x + (a*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*

x))) + (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/

(-b^2 + 4*a*c)^(3/2) + 2*b*Log[x] - b*Log[a + x*(b + c*x)])/a^3)

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Maple [B] time = 0.021, size = 545, normalized size = 3.7 \[ -{\frac{1}{{a}^{2}x}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{{c}^{2}x}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{cx{b}^{2}}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{bc}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{3}}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{c\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) b}{ \left ( 4\,ac-{b}^{2} \right ){a}^{2}}}-{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{3}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{c}^{2}}{a\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+12\,{\frac{{b}^{2}c}{{a}^{2}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }-2\,{\frac{{b}^{4}}{{a}^{3}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) } \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(1/(c+a/x^2+b/x)^2/x^6,x)

[Out]

-1/a^2/x-2*b*ln(x)/a^3-2/a/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x+1/a^2/(c*x^2+b*x+a)*c

/(4*a*c-b^2)*x*b^2-3/a/(c*x^2+b*x+a)*b/(4*a*c-b^2)*c+1/a^2/(c*x^2+b*x+a)*b^3/(4*

a*c-b^2)+4/a^2/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b-1/a^3/(4*a*c-b^2)*l

n((4*a*c-b^2)*(c*x^2+b*x+a))*b^3-12/a/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

^(1/2)*arctan((2*(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*

b^4*c-b^6)^(1/2))*c^2+12/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*ar

ctan((2*(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6

)^(1/2))*b^2*c-2/a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*

(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))

*b^4

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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 + b/x + a/x^2)^2*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.421351, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((c + b/x + a/x^2)^2*x^6),x, algorithm="fricas")

[Out]

[-(((b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2

+ (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*x)*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^2*b^2 - 4*a^3*c + 2*(a*b^2*c - 3*a^2*c^2)*x^2 + (2*a*b^3 - 7*a^2*b*c)*x - ((b

^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2*b*c)*x)*log(c*x^2

+ b*x + a) + 2*((b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^

2*b*c)*x)*log(x))*sqrt(b^2 - 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 -

4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(b^2 - 4*a*c)), (2*((b^4*c - 6*a*b^2

*c^2 + 6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2 + (a*b^4 - 6*a^2*b^2

*c + 6*a^3*c^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (a^2*

b^2 - 4*a^3*c + 2*(a*b^2*c - 3*a^2*c^2)*x^2 + (2*a*b^3 - 7*a^2*b*c)*x - ((b^3*c

- 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2*b*c)*x)*log(c*x^2 + b*

x + a) + 2*((b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2*b*c

)*x)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^

4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A] time = 38.6586, size = 2672, normalized size = 18.05 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/(c+a/x**2+b/x)**2/x**6,x)

[Out]

(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a

**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-1728*a**11*b*c**5

*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*

a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 2256*a**10*b**3*c**4*

(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a

**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 1172*a**9*b**5*c**3*(b

/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**

3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 288*a**9*c**6*(b/a**3 -

sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 -

48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 303*a**8*b**7*c**2*(b/a**3 - sqrt(-

(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a*

*2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 432*a**8*b**2*c**5*(b/a**3 - sqrt(-(4*

a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*

b**2*c**2 + 12*a*b**4*c - b**6))) - 39*a**7*b**9*c*(b/a**3 - sqrt(-(4*a*c - b**2

)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2

+ 12*a*b**4*c - b**6)))**2 + 558*a**7*b**4*c**4*(b/a**3 - sqrt(-(4*a*c - b**2)**

3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1

2*a*b**4*c - b**6))) + 2*a**6*b**11*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c

**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -

b**6)))**2 - 212*a**6*b**6*c**3*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2

- 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b*

*6))) - 576*a**6*b*c**6 + 34*a**5*b**8*c**2*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(

6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*

b**4*c - b**6))) + 6048*a**5*b**3*c**5 - 2*a**4*b**10*c*(b/a**3 - sqrt(-(4*a*c -

b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*

c**2 + 12*a*b**4*c - b**6))) - 7908*a**4*b**5*c**4 + 4264*a**3*b**7*c**3 - 1144*

a**2*b**9*c**2 + 152*a*b**11*c - 8*b**13)/(216*a**6*c**7 + 2808*a**5*b**2*c**6 -

5292*a**4*b**4*c**5 + 3384*a**3*b**6*c**4 - 1008*a**2*b**8*c**3 + 144*a*b**10*c

**2 - 8*b**12*c)) + (b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c

+ b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x +

(-1728*a**11*b*c**5*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*

c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 2

256*a**10*b**3*c**4*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c

+ b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 11

72*a**9*b**5*c**3*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c +

b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 288*

a**9*c**6*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(

a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 303*a**8*b**7*c

**2*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(

64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 432*a**8*b**2*c**5

*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*

a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 39*a**7*b**9*c*(b/a**3 +

sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3

- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 558*a**7*b**4*c**4*(b/a**3 + sq

rt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 4

8*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 2*a**6*b**11*(b/a**3 + sqrt(-(4*a*c -

b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*

c**2 + 12*a*b**4*c - b**6)))**2 - 212*a**6*b**6*c**3*(b/a**3 + sqrt(-(4*a*c - b*

*2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**

2 + 12*a*b**4*c - b**6))) - 576*a**6*b*c**6 + 34*a**5*b**8*c**2*(b/a**3 + sqrt(-

(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a*

*2*b**2*c**2 + 12*a*b**4*c - b**6))) + 6048*a**5*b**3*c**5 - 2*a**4*b**10*c*(b/a

**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*

c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 7908*a**4*b**5*c**4 + 4264*a*

*3*b**7*c**3 - 1144*a**2*b**9*c**2 + 152*a*b**11*c - 8*b**13)/(216*a**6*c**7 + 2

808*a**5*b**2*c**6 - 5292*a**4*b**4*c**5 + 3384*a**3*b**6*c**4 - 1008*a**2*b**8*

c**3 + 144*a*b**10*c**2 - 8*b**12*c)) - (4*a**2*c - a*b**2 + x**2*(6*a*c**2 - 2*

b**2*c) + x*(7*a*b*c - 2*b**3))/(x**3*(4*a**3*c**2 - a**2*b**2*c) + x**2*(4*a**3

*b*c - a**2*b**3) + x*(4*a**4*c - a**3*b**2)) - 2*b*log(x)/a**3

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GIAC/XCAS [A] time = 0.286524, size = 231, normalized size = 1.56 \[ \frac{2 \,{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{2} c x^{2} - 6 \, a c^{2} x^{2} + 2 \, b^{3} x - 7 \, a b c x + a b^{2} - 4 \, a^{2} c}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )}{\left (c x^{3} + b x^{2} + a x\right )}} + \frac{b{\rm ln}\left (c x^{2} + b x + a\right )}{a^{3}} - \frac{2 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((c + b/x + a/x^2)^2*x^6),x, algorithm="giac")

[Out]

2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2

- 4*a^4*c)*sqrt(-b^2 + 4*a*c)) - (2*b^2*c*x^2 - 6*a*c^2*x^2 + 2*b^3*x - 7*a*b*c

*x + a*b^2 - 4*a^2*c)/((a^2*b^2 - 4*a^3*c)*(c*x^3 + b*x^2 + a*x)) + b*ln(c*x^2 +

b*x + a)/a^3 - 2*b*ln(abs(x))/a^3

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