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

Optimal. Leaf size=70 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]

[Out]

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

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Rubi [A]  time = 0.105374, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(c + a/x^2 + b/x)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.1567, size = 65, normalized size = 0.93 \[ - \frac{b \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{x}{c} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103111, size = 73, normalized size = 1.04 \[ \frac{\left (b^2-2 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{c^2 \sqrt{4 a c-b^2}}-\frac{b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 101, normalized size = 1.4 \[{\frac{x}{c}}-{\frac{b\ln \left ( c{x}^{2}+bx+a \right ) }{2\,{c}^{2}}}-2\,{\frac{a}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x),x)

[Out]

x/c-1/2*b*ln(c*x^2+b*x+a)/c^2-2/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*a+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280536, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 2 \, a c\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x - b \log \left (c x^{2} + b x + a\right )\right )}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (b^{2} - 2 \, a c\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x - b \log \left (c x^{2} + b x + a\right )\right )}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((b^2 - 2*a*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)) - sqrt(b^2 - 4*a*c)*
(2*c*x - b*log(c*x^2 + b*x + a)))/(sqrt(b^2 - 4*a*c)*c^2), 1/2*(2*(b^2 - 2*a*c)*
arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + sqrt(-b^2 + 4*a*c)*(2*c*
x - b*log(c*x^2 + b*x + a)))/(sqrt(-b^2 + 4*a*c)*c^2)]

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Sympy [A]  time = 2.37063, size = 306, normalized size = 4.37 \[ \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b - 4 a c^{2} \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac{b}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b - 4 a c^{2} \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac{b}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+a/x**2+b/x),x)

[Out]

(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2)))*log(x
+ (-a*b - 4*a*c**2*(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4
*a*c - b**2))) + b**2*c*(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**
2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(2*c**2) + sqrt(-4*a*c + b**2)*(2*a*c
- b**2)/(2*c**2*(4*a*c - b**2)))*log(x + (-a*b - 4*a*c**2*(-b/(2*c**2) + sqrt(-4
*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2))) + b**2*c*(-b/(2*c**2) + sqr
t(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + x/c

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GIAC/XCAS [A]  time = 0.290691, size = 90, normalized size = 1.29 \[ \frac{x}{c} - \frac{b{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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