3.488 \(\int \frac{x^2}{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0480662, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0002, size = 36, normalized size = 0.8 \[ - \frac{x}{2 b \left (a + b x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

-x/(2*b*(a + b*x**2)) + atan(sqrt(b)*x/sqrt(a))/(2*sqrt(a)*b**(3/2))

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Mathematica [A]  time = 0.0362842, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

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

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Maple [A]  time = 0.01, size = 36, normalized size = 0.8 \[ -{\frac{x}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{1}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262935, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x^{2} + a\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \, \sqrt{-a b} x}{4 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{-a b}}, \frac{{\left (b x^{2} + a\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - \sqrt{a b} x}{2 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.28289, size = 78, normalized size = 1.73 \[ - \frac{x}{2 a b + 2 b^{2} x^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

-x/(2*a*b + 2*b**2*x**2) - sqrt(-1/(a*b**3))*log(-a*b*sqrt(-1/(a*b**3)) + x)/4 +
sqrt(-1/(a*b**3))*log(a*b*sqrt(-1/(a*b**3)) + x)/4

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GIAC/XCAS [A]  time = 0.269045, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} - \frac{x}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="giac")

[Out]

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