Optimal. Leaf size=82 \[ \frac{a^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{3 \sqrt{b} \left (a-b x^4\right )^{3/4}}+\frac{1}{3} x^2 \sqrt [4]{a-b x^4} \]
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Rubi [A] time = 0.0427203, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {275, 195, 233, 232} \[ \frac{a^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a-b x^4\right )^{3/4}}+\frac{1}{3} x^2 \sqrt [4]{a-b x^4} \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 233
Rule 232
Rubi steps
\begin{align*} \int x \sqrt [4]{a-b x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt [4]{a-b x^2} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^2 \sqrt [4]{a-b x^4}+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^2 \sqrt [4]{a-b x^4}+\frac{\left (a \left (1-\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{6 \left (a-b x^4\right )^{3/4}}\\ &=\frac{1}{3} x^2 \sqrt [4]{a-b x^4}+\frac{a^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a-b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0103342, size = 52, normalized size = 0.63 \[ \frac{x^2 \sqrt [4]{a-b x^4} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^4}{a}\right )}{2 \sqrt [4]{1-\frac{b x^4}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int x\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}} x\,{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, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.961479, size = 31, normalized size = 0.38 \begin{align*} \frac{\sqrt [4]{a} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2} \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}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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