Optimal. Leaf size=89 \[ \frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{x \sqrt{b^2 x^4+1}}{b x^2+1} \]
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Rubi [A] time = 0.0135721, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1196} \[ \frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{x \sqrt{b^2 x^4+1}}{b x^2+1} \]
Antiderivative was successfully verified.
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Rule 1196
Rubi steps
\begin{align*} \int \frac{1-b x^2}{\sqrt{1+b^2 x^4}} \, dx &=-\frac{x \sqrt{1+b^2 x^4}}{1+b x^2}+\frac{\left (1+b x^2\right ) \sqrt{\frac{1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{1+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0116996, size = 47, normalized size = 0.53 \[ x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right )-\frac{1}{3} b x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.153, size = 120, normalized size = 1.4 \begin{align*}{-i\sqrt{1-ib{x}^{2}}\sqrt{1+ib{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{ib},i \right ) -{\it EllipticE} \left ( x\sqrt{ib},i \right ) \right ){\frac{1}{\sqrt{ib}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+1}}}}+{\sqrt{1-ib{x}^{2}}\sqrt{1+ib{x}^{2}}{\it EllipticF} \left ( x\sqrt{ib},i \right ){\frac{1}{\sqrt{ib}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+1}}}} \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{b x^{2} - 1}{\sqrt{b^{2} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.83919, size = 66, normalized size = 0.74 \begin{align*} - \frac{b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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