Optimal. Leaf size=109 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]
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Rubi [A] time = 0.0537356, 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, Rules used = {1130, 205} \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1130
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{a-b+2 a x^2+a x^4} \, dx &=-\left (\frac{1}{2} \left (-1+\frac{\sqrt{a}}{\sqrt{b}}\right ) \int \frac{1}{a-\sqrt{a} \sqrt{b}+a x^2} \, dx\right )+\frac{1}{2} \left (1+\frac{\sqrt{a}}{\sqrt{b}}\right ) \int \frac{1}{a+\sqrt{a} \sqrt{b}+a x^2} \, dx\\ &=-\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}}\\ \end{align*}
Mathematica [A] time = 0.104889, 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.
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Maple [A] time = 0.133, size = 134, normalized size = 1.2 \begin{align*} -{\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}}}}+{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{a x^{4} + 2 \, a x^{2} + a - b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47349, size = 617, normalized size = 5.66 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.373015, size = 44, normalized size = 0.4 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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