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

Optimal. Leaf size=71 \[ \frac{\frac{2 a}{x}+b}{\left (b^2-4 a c\right ) \left (\frac{a}{x^2}+\frac{b}{x}+c\right )}-\frac{4 a \tanh ^{-1}\left (\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0925557, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\frac{2 a}{x}+b}{\left (b^2-4 a c\right ) \left (\frac{a}{x^2}+\frac{b}{x}+c\right )}-\frac{4 a \tanh ^{-1}\left (\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7961, size = 60, normalized size = 0.85 \[ - \frac{4 a \operatorname{atanh}{\left (\frac{\frac{2 a}{x} + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\frac{2 a}{x} + b}{\left (- 4 a c + b^{2}\right ) \left (\frac{a}{x^{2}} + \frac{b}{x} + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.150376, size = 81, normalized size = 1.14 \[ \frac{a (b-2 c x)+b^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac{4 a \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 97, normalized size = 1.4 \[{\frac{1}{c{x}^{2}+bx+a} \left ( -{\frac{ \left ( 2\,ac-{b}^{2} \right ) x}{ \left ( 4\,ac-{b}^{2} \right ) c}}+{\frac{ab}{ \left ( 4\,ac-{b}^{2} \right ) c}} \right ) }+4\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^2/x^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274394, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (a b +{\left (b^{2} - 2 \, a c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{4 \,{\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (a b +{\left (b^{2} - 2 \, a c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.83179, size = 280, normalized size = 3.94 \[ - 2 a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a^{2} b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 2 a b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} + 2 a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a^{2} b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} - \frac{- a b + x \left (2 a c - b^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-32*a**3*c**2*sqrt(-1/(4*a*c - b**2)**3
) + 16*a**2*b**2*c*sqrt(-1/(4*a*c - b**2)**3) - 2*a*b**4*sqrt(-1/(4*a*c - b**2)*
*3) + 2*a*b)/(4*a*c)) + 2*a*sqrt(-1/(4*a*c - b**2)**3)*log(x + (32*a**3*c**2*sqr
t(-1/(4*a*c - b**2)**3) - 16*a**2*b**2*c*sqrt(-1/(4*a*c - b**2)**3) + 2*a*b**4*s
qrt(-1/(4*a*c - b**2)**3) + 2*a*b)/(4*a*c)) - (-a*b + x*(2*a*c - b**2))/(4*a**2*
c**2 - a*b**2*c + x**2*(4*a*c**3 - b**2*c**2) + x*(4*a*b*c**2 - b**3*c))

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GIAC/XCAS [A]  time = 0.283716, size = 119, normalized size = 1.68 \[ -\frac{4 \, a \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{b^{2} x - 2 \, a c x + a b}{{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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