Optimal. Leaf size=83 \[ \frac{1}{2} x \sqrt [4]{a-b x^4}-\frac{\sqrt{a} \sqrt{b} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{2 \left (a-b x^4\right )^{3/4}} \]
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Rubi [A] time = 0.0326889, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {195, 237, 335, 275, 232} \[ \frac{1}{2} x \sqrt [4]{a-b x^4}-\frac{\sqrt{a} \sqrt{b} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \sqrt [4]{a-b x^4} \, dx &=\frac{1}{2} x \sqrt [4]{a-b x^4}+\frac{1}{2} a \int \frac{1}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=\frac{1}{2} x \sqrt [4]{a-b x^4}+\frac{\left (a \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1-\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{2 \left (a-b x^4\right )^{3/4}}\\ &=\frac{1}{2} x \sqrt [4]{a-b x^4}-\frac{\left (a \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{2 \left (a-b x^4\right )^{3/4}}\\ &=\frac{1}{2} x \sqrt [4]{a-b x^4}-\frac{\left (a \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{4 \left (a-b x^4\right )^{3/4}}\\ &=\frac{1}{2} x \sqrt [4]{a-b x^4}-\frac{\sqrt{a} \sqrt{b} \left (1-\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0048972, size = 47, normalized size = 0.57 \[ \frac{x \sqrt [4]{a-b x^4} \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b x^4}{a}\right )}{\sqrt [4]{1-\frac{b x^4}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{-b{x}^{4}+a}\, dx \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{\left (-b x^{4} + a\right )}^{\frac{1}{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 ({\left (-b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.811925, size = 39, normalized size = 0.47 \begin{align*} \frac{\sqrt [4]{a} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \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{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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