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

Optimal. Leaf size=238 \[ \frac{3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

(3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*
x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^2 - 4*a*c)^2*(a + b*
x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2
*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[a + b*x + c*x^2])
/(2*c^4)

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Rubi [A]  time = 0.637574, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571 \[ \frac{3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 b \log \left (a+b x+c x^2\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]  Int[(c + a/x^2 + b/x)^(-3),x]

[Out]

(3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*
x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^2 - 4*a*c)^2*(a + b*
x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2
*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[a + b*x + c*x^2])
/(2*c^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{30 a^{2} x}{c \left (- 4 a c + b^{2}\right )^{2}} - \frac{21 a b^{2} x}{c^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{3 b^{4} x}{c^{3} \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b \left (- 6 a c + b^{2}\right ) \int x\, dx}{c^{2} \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b \log{\left (a + b x + c x^{2} \right )}}{2 c^{4}} + \frac{x^{5} \left (2 a + b x\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{x^{3} \left (2 a \left (- 10 a c + b^{2}\right ) + 2 b x \left (- 7 a c + b^{2}\right )\right )}{2 c \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{6 \left (- 10 a^{3} c^{3} + 15 a^{2} b^{2} c^{2} - 5 a b^{4} c + \frac{b^{6}}{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{4} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30*a**2*x/(c*(-4*a*c + b**2)**2) - 21*a*b**2*x/(c**2*(-4*a*c + b**2)**2) + 3*b**
4*x/(c**3*(-4*a*c + b**2)**2) - 3*b*(-6*a*c + b**2)*Integral(x, x)/(c**2*(-4*a*c
+ b**2)**2) - 3*b*log(a + b*x + c*x**2)/(2*c**4) + x**5*(2*a + b*x)/(2*(-4*a*c
+ b**2)*(a + b*x + c*x**2)**2) + x**3*(2*a*(-10*a*c + b**2) + 2*b*x*(-7*a*c + b*
*2))/(2*c*(-4*a*c + b**2)**2*(a + b*x + c*x**2)) - 6*(-10*a**3*c**3 + 15*a**2*b*
*2*c**2 - 5*a*b**4*c + b**6/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(c**4*(-4*
a*c + b**2)**(5/2))

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Mathematica [A]  time = 0.649345, size = 260, normalized size = 1.09 \[ \frac{\frac{a^3 c^2 (2 c x-5 b)+a^2 b^2 c (5 b-9 c x)-a b^4 (b-6 c x)+b^6 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{6 c \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{-78 a^3 b c^3+36 a^3 c^4 x+61 a^2 b^3 c^2-102 a^2 b^2 c^3 x-14 a b^5 c+48 a b^4 c^2 x+b^7-6 b^6 c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-3 b c \log (a+x (b+c x))+2 c^2 x}{2 c^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*x + (b^7 - 14*a*b^5*c + 61*a^2*b^3*c^2 - 78*a^3*b*c^3 - 6*b^6*c*x + 48*a*
b^4*c^2*x - 102*a^2*b^2*c^3*x + 36*a^3*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))
) + (-(b^6*x) + a^2*b^2*c*(5*b - 9*c*x) - a*b^4*(b - 6*c*x) + a^3*c^2*(-5*b + 2*
c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*c*(b^6 - 10*a*b^4*c + 30*a^2*b^2*
c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) -
3*b*c*Log[a + x*(b + c*x)])/(2*c^5)

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Maple [B]  time = 0.027, size = 1524, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^3,x)

[Out]

x/c^3+18/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^3-51/c/(c*x^2+b*x+a)^2
/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^2*b^2+24/c^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b
^2*c+b^4)*x^3*a*b^4-3/c^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^6-21/
c/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^3-41/2/c^2/(c*x^2+b*x+a)^2*
b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^2+17/c^3/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*
a*b^2*c+b^4)*x^2*a-5/2/c^4/(c*x^2+b*x+a)^2*b^7/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+14
/c/(c*x^2+b*x+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x-71/c^2/(c*x^2+b*x+a)^2*a^3/(
16*a^2*c^2-8*a*b^2*c+b^4)*x*b^2+38/c^3/(c*x^2+b*x+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c
+b^4)*x*b^4-5/c^4/(c*x^2+b*x+a)^2*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6-29/c^2/(c*x
^2+b*x+a)^2*b*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)+18/c^3/(c*x^2+b*x+a)^2*b^3*a^3/(16*
a^2*c^2-8*a*b^2*c+b^4)-5/2/c^4/(c*x^2+b*x+a)^2*b^5*a^2/(16*a^2*c^2-8*a*b^2*c+b^4
)-24/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))
*a^2*b+12/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))*a*b^3-3/2/c^4/(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^5-60/c/(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*x*(16*a^2*c^2-8*a*b^2*c+b^4)+(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))*a^3+90/c^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*x*(16*a^2*c^2-8*a*b^2*c+b^
4)+(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^2*a^2-30/c^3/(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*x*(16*a
^2*c^2-8*a*b^2*c+b^4)+(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))*a*b^4+3/c^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)*arc
tan((2*c*x*(16*a^2*c^2-8*a*b^2*c+b^4)+(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^6

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26988, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(3*(a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a
*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b
^3*c^3 - 20*a^3*b*c^4)*x^3 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3
- 40*a^4*c^4)*x^2 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*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)) + (5*a^2*b^5 - 36*a^3*b^3*c + 58*a^4*b*c^
2 - 2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^5 - 4*(b^5*c^2 - 8*a*b^3*c^3 + 16*a
^2*b*c^4)*x^4 + 2*(2*b^6*c - 18*a*b^4*c^2 + 51*a^2*b^2*c^3 - 50*a^3*c^4)*x^3 + (
5*b^7 - 38*a*b^5*c + 73*a^2*b^3*c^2 - 22*a^3*b*c^3)*x^2 + 2*(5*a*b^6 - 39*a^2*b^
4*c + 79*a^3*b^2*c^2 - 30*a^4*c^3)*x + 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 +
(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^4 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^
2*c^3)*x^3 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^2 + 2*(a*b^6 - 8*a^2*b^4*c + 16*
a^3*b^2*c^2)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^4 - 8*a^3*b
^2*c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*
a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*
b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x)*sqrt(b^2 - 4*a*c)), 1/2*(6*(a^2*b^6 -
10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b
^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c
^4)*x^3 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^2 +
2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x)*arctan(-sqrt(-b^2 +
4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (5*a^2*b^5 - 36*a^3*b^3*c + 58*a^4*b*c^2 -
2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^5 - 4*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b
*c^4)*x^4 + 2*(2*b^6*c - 18*a*b^4*c^2 + 51*a^2*b^2*c^3 - 50*a^3*c^4)*x^3 + (5*b^
7 - 38*a*b^5*c + 73*a^2*b^3*c^2 - 22*a^3*b*c^3)*x^2 + 2*(5*a*b^6 - 39*a^2*b^4*c
+ 79*a^3*b^2*c^2 - 30*a^4*c^3)*x + 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + (b^
5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^4 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^
3)*x^3 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^2 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*
b^2*c^2)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^4 - 8*a^3*b^2*
c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*a*b
^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5
*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 16.2446, size = 1714, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 +
10*a*b**4*c - b**6)/(2*c**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)))*log(x + (-66*a**3*b*c**2 -
64*a**3*c**6*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2
*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**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))) + 27*a**2*b**3
*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3
- 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**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))) - 3*
a*b**5 - 12*a*b**4*c**4*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**
3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**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))) + b
**6*c**3*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**
2*c**2 + 10*a*b**4*c - b**6)/(2*c**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))))/(60*a**3*c**3 - 9
0*a**2*b**2*c**2 + 30*a*b**4*c - 3*b**6)) + (-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b*
*2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**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)))*log(x + (-66*a**3*b*c**2 - 64*a**3*c**6*(-3*b/(2*c**4) + 3*sqrt
(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*
c**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))) + 27*a**2*b**3*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4
) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c -
b**6)/(2*c**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))) - 3*a*b**5 - 12*a*b**4*c**4*(-3*b/(2*c**
4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c
- b**6)/(2*c**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))) + b**6*c**3*(-3*b/(2*c**4) + 3*sqrt(-(4
*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**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))))/(60*a**3*c**3 - 90*a**2*b**2*c**2 + 30*a*b**4*c - 3*b
**6)) + (-58*a**4*b*c**2 + 36*a**3*b**3*c - 5*a**2*b**5 + x**3*(36*a**3*c**4 - 1
02*a**2*b**2*c**3 + 48*a*b**4*c**2 - 6*b**6*c) + x**2*(-42*a**3*b*c**3 - 41*a**2
*b**3*c**2 + 34*a*b**5*c - 5*b**7) + x*(28*a**4*c**3 - 142*a**3*b**2*c**2 + 76*a
**2*b**4*c - 10*a*b**6))/(32*a**4*c**6 - 16*a**3*b**2*c**5 + 2*a**2*b**4*c**4 +
x**4*(32*a**2*c**8 - 16*a*b**2*c**7 + 2*b**4*c**6) + x**3*(64*a**2*b*c**7 - 32*a
*b**3*c**6 + 4*b**5*c**5) + x**2*(64*a**3*c**7 - 12*a*b**4*c**5 + 2*b**6*c**4) +
x*(64*a**3*b*c**6 - 32*a**2*b**3*c**5 + 4*a*b**5*c**4)) + x/c**3

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GIAC/XCAS [A]  time = 0.268229, size = 381, normalized size = 1.6 \[ \frac{3 \,{\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x}{c^{3}} - \frac{3 \, b{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac{5 \, a^{2} b^{5} - 36 \, a^{3} b^{3} c + 58 \, a^{4} b c^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 17 \, a^{2} b^{2} c^{3} - 6 \, a^{3} c^{4}\right )} x^{3} +{\left (5 \, b^{7} - 34 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} + 42 \, a^{3} b c^{3}\right )} x^{2} + 2 \,{\left (5 \, a b^{6} - 38 \, a^{2} b^{4} c + 71 \, a^{3} b^{2} c^{2} - 14 \, a^{4} c^{3}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + b)/sqrt(-b^2
+ 4*a*c))/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + x/c^3 - 3/
2*b*ln(c*x^2 + b*x + a)/c^4 - 1/2*(5*a^2*b^5 - 36*a^3*b^3*c + 58*a^4*b*c^2 + 6*(
b^6*c - 8*a*b^4*c^2 + 17*a^2*b^2*c^3 - 6*a^3*c^4)*x^3 + (5*b^7 - 34*a*b^5*c + 41
*a^2*b^3*c^2 + 42*a^3*b*c^3)*x^2 + 2*(5*a*b^6 - 38*a^2*b^4*c + 71*a^3*b^2*c^2 -
14*a^4*c^3)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^4)