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

Optimal. Leaf size=62 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.106115, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.8998, size = 54, normalized size = 0.87 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (a + b x + c x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112243, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a}}-{\frac{b}{a}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)/x^3,x)

[Out]

ln(x)/a-1/2*ln(c*x^2+b*x+a)/a-1/a*b/(4*a*c-b^2)^(1/2)*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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282766, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \, b \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.62289, size = 564, normalized size = 9.1 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac{\log{\left (x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))*log(x + (24*a**4*c**2*(-
b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 14*a**3*b**2*c*(-b*sq
rt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a**3*c**2*(-b*sqrt(-4*
a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) + 2*a**2*b**4*(-b*sqrt(-4*a*c + b**2
)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2*c*(-b*sqrt(-4*a*c + b**2)/(2*
a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c**2 + 11*a*b**2*c - 2*b**4)/(9*a*b*c**2
- 2*b**3*c)) + (b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))*log(x + (2
4*a**4*c**2*(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 14*a**3*
b**2*c*(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a**3*c**2*
(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) + 2*a**2*b**4*(b*sqrt(-4*
a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2*c*(b*sqrt(-4*a*c +
b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c**2 + 11*a*b**2*c - 2*b**4)/(9*
a*b*c**2 - 2*b**3*c)) + log(x)/a

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GIAC/XCAS [A]  time = 0.305839, size = 84, normalized size = 1.35 \[ -\frac{b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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