Optimal. Leaf size=109 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]
[Out]
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Rubi [A] time = 0.128659, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
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Rubi in Sympy [A] time = 19.6964, size = 94, normalized size = 0.86 \[ - \frac{\sqrt{\sqrt{a} - \sqrt{b}} \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} - \sqrt{b}}} \right )}}{2 a^{\frac{3}{4}} \sqrt{b}} + \frac{\sqrt{\sqrt{a} + \sqrt{b}} \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} + \sqrt{b}}} \right )}}{2 a^{\frac{3}{4}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a*x**4+2*a*x**2+a-b),x)
[Out]
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Mathematica [A] time = 0.189585, size = 128, normalized size = 1.17 \[ \frac{\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right )}{\sqrt{a-\sqrt{a} \sqrt{b}}}}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
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Maple [A] time = 0.017, size = 134, normalized size = 1.2 \[{\frac{1}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}+{\frac{a}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{1}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}+{\frac{a}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a*x^4+2*a*x^2+a-b),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{a x^{4} + 2 \, a x^{2} + a - b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277647, size = 360, normalized size = 3.3 \[ \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) + \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.07186, size = 44, normalized size = 0.4 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} + 32 t^{2} a^{2} b + a - b, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a*x**4+2*a*x**2+a-b),x)
[Out]
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GIAC/XCAS [A] time = 2.25456, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="giac")
[Out]