Optimal. Leaf size=53 \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {224, 221} \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-b x^4}} \, dx &=\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{\sqrt {a-b x^4}}\\ &=\frac {\sqrt [4]{a} \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 )}{\sqrt [4]{b} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 72, normalized size = 1.36 \[ -\frac {i \sqrt {1-\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{4} - a}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 1.21 \[ \frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}} \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 38, normalized size = 0.72 \[ \frac {x\,\sqrt {1-\frac {b\,x^4}{a}}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {b\,x^4}{a}\right )}{\sqrt {a-b\,x^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 37, normalized size = 0.70 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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