Optimal. Leaf size=232 \[ \frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+c x^4}}+\frac{\sqrt{c} x \sqrt{a+c x^4}}{a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt{a+c x^4}}{a x} \]
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Rubi [A] time = 0.0717008, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {4, 325, 305, 220, 1196} \[ \frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+c x^4}}+\frac{\sqrt{c} x \sqrt{a+c x^4}}{a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt{a+c x^4}}{a x} \]
Antiderivative was successfully verified.
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Rule 4
Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac{1}{x^2 \sqrt{a+c x^4}} \, dx\\ &=-\frac{\sqrt{a+c x^4}}{a x}+\frac{c \int \frac{x^2}{\sqrt{a+c x^4}} \, dx}{a}\\ &=-\frac{\sqrt{a+c x^4}}{a x}+\frac{\sqrt{c} \int \frac{1}{\sqrt{a+c x^4}} \, dx}{\sqrt{a}}-\frac{\sqrt{c} \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{\sqrt{a}}\\ &=-\frac{\sqrt{a+c x^4}}{a x}+\frac{\sqrt{c} x \sqrt{a+c x^4}}{a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0089387, size = 49, normalized size = 0.21 \[ -\frac{\sqrt{\frac{c x^4}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^4}{a}\right )}{x \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.18, size = 115, normalized size = 0.5 \begin{align*} -{\frac{1}{ax}\sqrt{c{x}^{4}+a}}+{i\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+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{1}{\sqrt{c x^{4} + a} x^{2}}\,{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{\sqrt{c x^{4} + a}}{c x^{6} + a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.717127, size = 39, normalized size = 0.17 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{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{1}{\sqrt{c x^{4} + a} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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