3.1845 \(\int \frac{x^2}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]

[Out]

-((b*x)/a^2) + x^3/(3*a) + (b^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(5/2)

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Rubi [A]  time = 0.0652061, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^2),x]

[Out]

-((b*x)/a^2) + x^3/(3*a) + (b^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(5/2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3 a} - \frac{\int b\, dx}{a^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(3*a) - Integral(b, x)/a**2 + b**(3/2)*atan(sqrt(a)*x/sqrt(b))/a**(5/2)

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Mathematica [A]  time = 0.0361789, size = 42, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{5/2}}-\frac{b x}{a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x^2),x]

[Out]

-((b*x)/a^2) + x^3/(3*a) + (b^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(5/2)

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Maple [A]  time = 0.001, size = 38, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,a}}-{\frac{bx}{{a}^{2}}}+{\frac{{b}^{2}}{{a}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229478, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{3} + 3 \, b \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) - 6 \, b x}{6 \, a^{2}}, \frac{a x^{3} + 3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) - 3 \, b x}{3 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.28698, size = 80, normalized size = 1.9 \[ - \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac{x^{3}}{3 a} - \frac{b x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x**2),x)

[Out]

-sqrt(-b**3/a**5)*log(-a**2*sqrt(-b**3/a**5)/b + x)/2 + sqrt(-b**3/a**5)*log(a**
2*sqrt(-b**3/a**5)/b + x)/2 + x**3/(3*a) - b*x/a**2

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GIAC/XCAS [A]  time = 0.220723, size = 54, normalized size = 1.29 \[ \frac{b^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{a^{2} x^{3} - 3 \, a b x}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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