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

Optimal. Leaf size=118 \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.220944, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(c + a/x^2 + b/x),x]

[Out]

((b^2 - a*c)*x)/c^3 - (b*x^2)/(2*c^2) + x^3/(3*c) - ((b^4 - 4*a*b^2*c + 2*a^2*c^
2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 2
*a*c)*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. \[ - \frac{b \int x\, dx}{c^{2}} - \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{4}} + \left (- a c + b^{2}\right ) \int \frac{1}{c^{3}}\, dx + \frac{x^{3}}{3 c} - \frac{\left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{4} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.143475, size = 112, normalized size = 0.95 \[ \frac{\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))+c x \left (-6 a c+6 b^2-3 b c x+2 c^2 x^2\right )}{6 c^4} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(6*b^2 - 6*a*c - 3*b*c*x + 2*c^2*x^2) + (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*Ar
cTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*(b^3 - 2*a*b*c)*Log
[a + x*(b + c*x)])/(6*c^4)

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Maple [A]  time = 0.005, size = 190, normalized size = 1.6 \[{\frac{{x}^{3}}{3\,c}}-{\frac{b{x}^{2}}{2\,{c}^{2}}}-{\frac{ax}{{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ab}{{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{c}^{4}}}+2\,{\frac{{a}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c+a/x^2+b/x),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280483, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, c^{3} x^{3} - 3 \, b c^{2} x^{2} + 6 \,{\left (b^{2} c - a c^{2}\right )} x - 3 \,{\left (b^{3} - 2 \, a b c\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} c^{4}}, \frac{6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c^{3} x^{3} - 3 \, b c^{2} x^{2} + 6 \,{\left (b^{2} c - a c^{2}\right )} x - 3 \,{\left (b^{3} - 2 \, a b c\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*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)) +
(2*c^3*x^3 - 3*b*c^2*x^2 + 6*(b^2*c - a*c^2)*x - 3*(b^3 - 2*a*b*c)*log(c*x^2 + b
*x + a))*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*c^4), 1/6*(6*(b^4 - 4*a*b^2*c + 2
*a^2*c^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*c^3*x^3 - 3
*b*c^2*x^2 + 6*(b^2*c - a*c^2)*x - 3*(b^3 - 2*a*b*c)*log(c*x^2 + b*x + a))*sqrt(
-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)]

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Sympy [A]  time = 3.48875, size = 496, normalized size = 4.2 \[ - \frac{b x^{2}}{2 c^{2}} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \frac{x^{3}}{3 c} - \frac{x \left (a c - b^{2}\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x**2/(2*c**2) + (b*(2*a*c - b**2)/(2*c**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2
- 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2)))*log(x + (-3*a**2*b*c + a*b**3 + 4
*a*c**4*(b*(2*a*c - b**2)/(2*c**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2
*c + b**4)/(2*c**4*(4*a*c - b**2))) - b**2*c**3*(b*(2*a*c - b**2)/(2*c**4) - sqr
t(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2))))/(2*
a**2*c**2 - 4*a*b**2*c + b**4)) + (b*(2*a*c - b**2)/(2*c**4) + sqrt(-4*a*c + b**
2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2)))*log(x + (-3*a**2*b
*c + a*b**3 + 4*a*c**4*(b*(2*a*c - b**2)/(2*c**4) + sqrt(-4*a*c + b**2)*(2*a**2*
c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2))) - b**2*c**3*(b*(2*a*c - b**2)
/(2*c**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c
- b**2))))/(2*a**2*c**2 - 4*a*b**2*c + b**4)) + x**3/(3*c) - x*(a*c - b**2)/c**
3

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GIAC/XCAS [A]  time = 0.296501, size = 153, normalized size = 1.3 \[ \frac{2 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x - 6 \, a c x}{6 \, c^{3}} - \frac{{\left (b^{3} - 2 \, a b c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*c^2*x^3 - 3*b*c*x^2 + 6*b^2*x - 6*a*c*x)/c^3 - 1/2*(b^3 - 2*a*b*c)*ln(c*x
^2 + b*x + a)/c^4 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)