Optimal. Leaf size=79 \[ -\frac{\left (a-b x^2\right )^{3/4}}{a x}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.0202877, 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, 229, 228} \[ -\frac{\left (a-b x^2\right )^{3/4}}{a x}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx &=-\frac{\left (a-b x^2\right )^{3/4}}{a x}-\frac{b \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{2 a}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{a x}-\frac{\left (b \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{2 a \sqrt [4]{a-b x^2}}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{a x}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0088997, size = 50, normalized size = 0.63 \[ -\frac{\sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{b x^2}{a}\right )}{x \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [4]{-b{x}^{2}+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 \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} 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{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{b x^{4} - a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.790655, size = 29, normalized size = 0.37 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt [4]{a} x} \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}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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