Optimal. Leaf size=65 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a+b x^2\right )}-\frac{x}{4 b \left (a+b x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.0577224, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a+b x^2\right )}-\frac{x}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 8.40751, size = 51, normalized size = 0.78 \[ - \frac{x}{4 b \left (a + b x^{2}\right )^{2}} + \frac{x}{8 a b \left (a + b x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0522065, size = 58, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (b x^2-a\right )}{\left (a+b x^2\right )^2}+\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*x^2)^3,x]
[Out]
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Maple [A] time = 0.011, size = 49, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{{x}^{3}}{8\,a}}-{\frac{x}{8\,b}} \right ) }+{\frac{1}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a)^3,x)
[Out]
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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*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22503, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (b x^{3} - a x\right )} \sqrt{-a b}}{16 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{-a b}}, \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (b x^{3} - a x\right )} \sqrt{a b}}{8 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.78902, size = 110, normalized size = 1.69 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (- a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{- a x + b x^{3}}{8 a^{3} b + 16 a^{2} b^{2} x^{2} + 8 a b^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214209, size = 68, normalized size = 1.05 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b} + \frac{b x^{3} - a x}{8 \,{\left (b x^{2} + a\right )}^{2} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^2 + a)^3,x, algorithm="giac")
[Out]