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 = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 228
Rule 229
Rule 325
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*}
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Mathematica [C] time = 0.01, 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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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{4} - a x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.08, size = 41, normalized size = 0.52 \[ -\frac {2\,{\left (1-\frac {a}{b\,x^2}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^2}\right )}{3\,x\,{\left (a-b\,x^2\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.90, size = 29, normalized size = 0.37 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt [4]{a} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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