3.430 \(\int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^2 x^7} \, dx\)

Optimal. Leaf size=202 \[ -\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{b \left (3 b^2-11 a c\right )}{a^3 x \left (b^2-4 a c\right )}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(3*b^2 - 8*a*c)/(2*a^2*(b^2 - 4*a*c)*x^2) + (b*(3*b^2 - 11*a*c))/(a^3*(b^2 - 4*
a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + (b*(3*
b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*(b^2
- 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[x])/a^4 - ((3*b^2 - 2*a*c)*Log[a + b*x +
c*x^2])/(2*a^4)

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Rubi [A]  time = 0.507093, antiderivative size = 202, 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 \[ -\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{b \left (3 b^2-11 a c\right )}{a^3 x \left (b^2-4 a c\right )}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((c + a/x^2 + b/x)^2*x^7),x]

[Out]

-(3*b^2 - 8*a*c)/(2*a^2*(b^2 - 4*a*c)*x^2) + (b*(3*b^2 - 11*a*c))/(a^3*(b^2 - 4*
a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + (b*(3*
b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*(b^2
- 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[x])/a^4 - ((3*b^2 - 2*a*c)*Log[a + b*x +
c*x^2])/(2*a^4)

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Rubi in Sympy [A]  time = 85.6393, size = 192, normalized size = 0.95 \[ \frac{- 2 a c + b^{2} + b c x}{a x^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{- 8 a c + 3 b^{2}}{2 a^{2} x^{2} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 11 a c + 3 b^{2}\right )}{a^{3} x \left (- 4 a c + b^{2}\right )} + \frac{b \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{4} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\left (- 2 a c + 3 b^{2}\right ) \log{\left (x \right )}}{a^{4}} - \frac{\left (- 2 a c + 3 b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c+a/x**2+b/x)**2/x**7,x)

[Out]

(-2*a*c + b**2 + b*c*x)/(a*x**2*(-4*a*c + b**2)*(a + b*x + c*x**2)) - (-8*a*c +
3*b**2)/(2*a**2*x**2*(-4*a*c + b**2)) + b*(-11*a*c + 3*b**2)/(a**3*x*(-4*a*c + b
**2)) + b*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)*atanh((b + 2*c*x)/sqrt(-4*a*c +
b**2))/(a**4*(-4*a*c + b**2)**(3/2)) + (-2*a*c + 3*b**2)*log(x)/a**4 - (-2*a*c +
3*b**2)*log(a + b*x + c*x**2)/(2*a**4)

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Mathematica [A]  time = 0.772291, size = 175, normalized size = 0.87 \[ \frac{\frac{2 b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 a \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{a^2}{x^2}+2 \log (x) \left (3 b^2-2 a c\right )+\left (2 a c-3 b^2\right ) \log (a+x (b+c x))+\frac{4 a b}{x}}{2 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + a/x^2 + b/x)^2*x^7),x]

[Out]

(-(a^2/x^2) + (4*a*b)/x + (2*a*(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*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)
*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(3*b^2 - 2*a*c
)*Log[x] + (-3*b^2 + 2*a*c)*Log[a + x*(b + c*x)])/(2*a^4)

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Maple [B]  time = 0.023, size = 646, normalized size = 3.2 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{a}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}+2\,{\frac{b}{{a}^{3}x}}+3\,{\frac{{c}^{2}bx}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{3}cx}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{c}^{2}}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{b}^{2}c}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{4}}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{c}^{2}\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) }{ \left ( 4\,ac-{b}^{2} \right ){a}^{2}}}-7\,{\frac{c\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{2}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{3\,\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{4}}{2\,{a}^{4} \left ( 4\,ac-{b}^{2} \right ) }}+30\,{\frac{{c}^{2}b}{{a}^{2}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }-20\,{\frac{{b}^{3}c}{{a}^{3}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+3\,{\frac{{b}^{5}}{{a}^{4}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^2/x^7,x)

[Out]

-1/2/a^2/x^2-2/a^3*ln(x)*c+3/a^4*b^2*ln(x)+2/a^3*b/x+3/a^2/(c*x^2+b*x+a)*b*c^2/(
4*a*c-b^2)*x-1/a^3/(c*x^2+b*x+a)*b^3*c/(4*a*c-b^2)*x-2/a/(c*x^2+b*x+a)/(4*a*c-b^
2)*c^2+4/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c-1/a^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4
+4/a^2/(4*a*c-b^2)*c^2*ln((4*a*c-b^2)*(c*x^2+b*x+a))-7/a^3/(4*a*c-b^2)*c*ln((4*a
*c-b^2)*(c*x^2+b*x+a))*b^2+3/2/a^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^4
+30/a^2/(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*c^2-20/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^3*c+3/a^4/(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^5

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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^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.521776, 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 + b/x + a/x^2)^2*x^7),x, algorithm="fricas")

[Out]

[-1/2*(((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^4 + (3*b^6 - 20*a*b^4*c + 30*a
^2*b^2*c^2)*x^3 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*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^3*b^2 - 4*a^4*c - 2*(3*a*b^3*c - 11*a^2*b*c^2)*x^3 -
(6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*x^2 - 3*(a^2*b^3 - 4*a^3*b*c)*x + ((3*b^4*
c - 14*a*b^2*c^2 + 8*a^2*c^3)*x^4 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*x^3 + (3*
a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*x^2)*log(c*x^2 + b*x + a) - 2*((3*b^4*c - 14*a
*b^2*c^2 + 8*a^2*c^3)*x^4 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*x^3 + (3*a*b^4 -
14*a^2*b^2*c + 8*a^3*c^2)*x^2)*log(x))*sqrt(b^2 - 4*a*c))/(((a^4*b^2*c - 4*a^5*c
^2)*x^4 + (a^4*b^3 - 4*a^5*b*c)*x^3 + (a^5*b^2 - 4*a^6*c)*x^2)*sqrt(b^2 - 4*a*c)
), -1/2*(2*((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^4 + (3*b^6 - 20*a*b^4*c +
30*a^2*b^2*c^2)*x^3 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*x^2)*arctan(-sqrt(
-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (a^3*b^2 - 4*a^4*c - 2*(3*a*b^3*c - 1
1*a^2*b*c^2)*x^3 - (6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*x^2 - 3*(a^2*b^3 - 4*a^3
*b*c)*x + ((3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*x^4 + (3*b^5 - 14*a*b^3*c + 8*a^
2*b*c^2)*x^3 + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*x^2)*log(c*x^2 + b*x + a) -
2*((3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*x^4 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)
*x^3 + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*x^2)*log(x))*sqrt(-b^2 + 4*a*c))/(((
a^4*b^2*c - 4*a^5*c^2)*x^4 + (a^4*b^3 - 4*a^5*b*c)*x^3 + (a^5*b^2 - 4*a^6*c)*x^2
)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 59.1678, size = 4083, normalized size = 20.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+a/x**2+b/x)**2/x**7,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))
*log(x + (3072*a**14*c**6*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2
*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4))**2 - 9408*a**13*b**2*c**5*(-b*sqrt(-(4*a*c - b**2)**3
)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**
2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 9040*a**12*b**4*c**4*
(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))
**2 - 4116*a**11*b**6*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**
2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4))**2 + 3072*a**11*c**7*(-b*sqrt(-(4*a*c - b**2)**3)*(3
0*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 987*a**10*b**8*c**2*(-b*sqrt
(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 75
36*a**10*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*
b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c
- 3*b**2)/(2*a**4)) - 121*a**9*b**10*c*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**
2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*
c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 8152*a**9*b**4*c**5*(-b*sqrt(-(4*a*
c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a
**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 6*a**8*b**12
*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*
a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)
)**2 - 4343*a**8*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**
2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4)) - 6144*a**8*c**8 + 1198*a**7*b**8*c**3*(-b*sqrt(-(4*
a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48
*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 50208*a**7
*b**2*c**7 - 165*a**6*b**10*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20
*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b*
*6)) + (2*a*c - 3*b**2)/(2*a**4)) - 137792*a**6*b**4*c**6 + 9*a**5*b**12*c*(-b*s
qrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 17
6474*a**5*b**6*c**5 - 119275*a**4*b**8*c**4 + 45448*a**3*b**10*c**3 - 9846*a**2*
b**12*c**2 + 1134*a*b**14*c - 54*b**16)/(17280*a**7*b*c**8 - 69570*a**6*b**3*c**
7 + 112428*a**5*b**5*c**6 - 88605*a**4*b**7*c**5 + 37600*a**3*b**9*c**4 - 8820*a
**2*b**11*c**3 + 1080*a*b**13*c**2 - 54*b**15*c)) + (b*sqrt(-(4*a*c - b**2)**3)*
(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))*log(x + (3072*a**14*c**6*(b*
sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2
- 9408*a**13*b**2*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c +
3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a
*c - 3*b**2)/(2*a**4))**2 + 9040*a**12*b**4*c**4*(b*sqrt(-(4*a*c - b**2)**3)*(30
*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1
2*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 4116*a**11*b**6*c**3*(b*sq
rt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 +
3072*a**11*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4
)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*
b**2)/(2*a**4)) + 987*a**10*b**8*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2
- 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 7536*a**10*b**2*c**6*(b*sqrt(-(4*a*c
- b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 121*a**9*b**10
*c*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64
*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4
))**2 + 8152*a**9*b**4*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**
2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4)) + 6*a**8*b**12*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*
c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 4343*a**8*b**6*c**4*(b*sqrt(-(4*
a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48
*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 6144*a**8*
c**8 + 1198*a**7*b**8*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2
*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4)) + 50208*a**7*b**2*c**7 - 165*a**6*b**10*c**2*(b*sqrt(
-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 137792
*a**6*b**4*c**6 + 9*a**5*b**12*c*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*
a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**
6)) + (2*a*c - 3*b**2)/(2*a**4)) + 176474*a**5*b**6*c**5 - 119275*a**4*b**8*c**4
+ 45448*a**3*b**10*c**3 - 9846*a**2*b**12*c**2 + 1134*a*b**14*c - 54*b**16)/(17
280*a**7*b*c**8 - 69570*a**6*b**3*c**7 + 112428*a**5*b**5*c**6 - 88605*a**4*b**7
*c**5 + 37600*a**3*b**9*c**4 - 8820*a**2*b**11*c**3 + 1080*a*b**13*c**2 - 54*b**
15*c)) + (-4*a**3*c + a**2*b**2 + x**3*(22*a*b*c**2 - 6*b**3*c) + x**2*(-8*a**2*
c**2 + 25*a*b**2*c - 6*b**4) + x*(12*a**2*b*c - 3*a*b**3))/(x**4*(8*a**4*c**2 -
2*a**3*b**2*c) + x**3*(8*a**4*b*c - 2*a**3*b**3) + x**2*(8*a**5*c - 2*a**4*b**2)
) - (2*a*c - 3*b**2)*log(x + (-6144*a**8*c**8 + 50208*a**7*b**2*c**7 - 3072*a**7
*c**7*(2*a*c - 3*b**2) - 137792*a**6*b**4*c**6 + 7536*a**6*b**2*c**6*(2*a*c - 3*
b**2) + 3072*a**6*c**6*(2*a*c - 3*b**2)**2 + 176474*a**5*b**6*c**5 - 8152*a**5*b
**4*c**5*(2*a*c - 3*b**2) - 9408*a**5*b**2*c**5*(2*a*c - 3*b**2)**2 - 119275*a**
4*b**8*c**4 + 4343*a**4*b**6*c**4*(2*a*c - 3*b**2) + 9040*a**4*b**4*c**4*(2*a*c
- 3*b**2)**2 + 45448*a**3*b**10*c**3 - 1198*a**3*b**8*c**3*(2*a*c - 3*b**2) - 41
16*a**3*b**6*c**3*(2*a*c - 3*b**2)**2 - 9846*a**2*b**12*c**2 + 165*a**2*b**10*c*
*2*(2*a*c - 3*b**2) + 987*a**2*b**8*c**2*(2*a*c - 3*b**2)**2 + 1134*a*b**14*c -
9*a*b**12*c*(2*a*c - 3*b**2) - 121*a*b**10*c*(2*a*c - 3*b**2)**2 - 54*b**16 + 6*
b**12*(2*a*c - 3*b**2)**2)/(17280*a**7*b*c**8 - 69570*a**6*b**3*c**7 + 112428*a*
*5*b**5*c**6 - 88605*a**4*b**7*c**5 + 37600*a**3*b**9*c**4 - 8820*a**2*b**11*c**
3 + 1080*a*b**13*c**2 - 54*b**15*c))/a**4

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GIAC/XCAS [A]  time = 0.276051, size = 309, normalized size = 1.53 \[ -\frac{{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (3 \, b^{2} - 2 \, a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{4}} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{a^{3} b^{2} - 4 \, a^{4} c - 2 \,{\left (3 \, a b^{3} c - 11 \, a^{2} b c^{2}\right )} x^{3} -{\left (6 \, a b^{4} - 25 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x^{2} - 3 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^2*x^7),x, algorithm="giac")

[Out]

-(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^
4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(3*b^2 - 2*a*c)*ln(c*x^2 + b*x + a)/a
^4 + (3*b^2 - 2*a*c)*ln(abs(x))/a^4 - 1/2*(a^3*b^2 - 4*a^4*c - 2*(3*a*b^3*c - 11
*a^2*b*c^2)*x^3 - (6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*x^2 - 3*(a^2*b^3 - 4*a^3*
b*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*a^4*x^2)