Optimal. Leaf size=110 \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4}}{3 a x^3} \]
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Rubi [A] time = 0.0251748, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 220} \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx &=-\frac{\sqrt{a+b x^4}}{3 a x^3}-\frac{b \int \frac{1}{\sqrt{a+b x^4}} \, dx}{3 a}\\ &=-\frac{\sqrt{a+b x^4}}{3 a x^3}-\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0087508, size = 51, normalized size = 0.46 \[ -\frac{\sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right )}{3 x^3 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 93, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{b}{3\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{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{b x^{4} + a} x^{4}}\,{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{b x^{4} + a}}{b x^{8} + a x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.08434, size = 41, normalized size = 0.37 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{1}{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{b x^{4} + a} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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