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

Optimal. Leaf size=66 \[ \frac{2 a+b x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0764737, antiderivative size = 66, 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 \[ \frac{2 a+b x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7373, size = 60, normalized size = 0.91 \[ - \frac{2 b \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{2 a + b x}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.104742, size = 69, normalized size = 1.05 \[ \frac{2 a+b x}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 70, normalized size = 1.1 \[{\frac{-bx-2\,a}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}-2\,{\frac{b}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^2/x^3,x)

[Out]

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

_______________________________________________________________________________________

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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.265909, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (b c x^{2} + b^{2} x + a b\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (b x + 2 \, a\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (b c x^{2} + b^{2} x + a b\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (b x + 2 \, a\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.65729, size = 252, normalized size = 3.82 \[ b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )} - b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )} - \frac{2 a + b x}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-16*a**2*b*c**2*sqrt(-1/(4*a*c - b**2)**3)
+ 8*a*b**3*c*sqrt(-1/(4*a*c - b**2)**3) - b**5*sqrt(-1/(4*a*c - b**2)**3) + b**
2)/(2*b*c)) - b*sqrt(-1/(4*a*c - b**2)**3)*log(x + (16*a**2*b*c**2*sqrt(-1/(4*a*
c - b**2)**3) - 8*a*b**3*c*sqrt(-1/(4*a*c - b**2)**3) + b**5*sqrt(-1/(4*a*c - b*
*2)**3) + b**2)/(2*b*c)) - (2*a + b*x)/(4*a**2*c - a*b**2 + x**2*(4*a*c**2 - b**
2*c) + x*(4*a*b*c - b**3))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271158, size = 103, normalized size = 1.56 \[ \frac{2 \, b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{b x + 2 \, a}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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