Optimal. Leaf size=79 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{3 a x^3} \]
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Rubi [A] time = 0.0209344, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {325, 224, 221} \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 224
Rule 221
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{\left (b \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{3 a \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{3 a x^3}+\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0092002, size = 52, normalized size = 0.66 \[ -\frac{\sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{b x^4}{a}\right )}{3 x^3 \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 88, normalized size = 1.1 \begin{align*} -{\frac{1}{3\,a{x}^{3}}\sqrt{-b{x}^{4}+a}}+{\frac{b}{3\,a}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\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 [A] time = 1.12636, size = 42, normalized size = 0.53 \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^{2 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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