Optimal. Leaf size=231 \[ -\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {x \sqrt {a+\frac {b}{x^4}}}{a}-\frac {\sqrt {b} \sqrt {a+\frac {b}{x^4}}}{a x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {242, 325, 305, 220, 1196} \[ -\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {x \sqrt {a+\frac {b}{x^4}}}{a}-\frac {\sqrt {b} \sqrt {a+\frac {b}{x^4}}}{a x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 242
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^4}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x^4}} x}{a}-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {\sqrt {a+\frac {b}{x^4}} x}{a}-\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{\sqrt {a}}+\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{\sqrt {a}}\\ &=-\frac {\sqrt {b} \sqrt {a+\frac {b}{x^4}}}{a \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}+\frac {\sqrt {a+\frac {b}{x^4}} x}{a}+\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt [4]{b} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 49, normalized size = 0.21 \[ \frac {x \sqrt {\frac {a x^4}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^4}{b}\right )}{3 \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 113, normalized size = 0.49 \[ \frac {i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )\right ) \sqrt {b}}{\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \sqrt {a}\, x^{2}} \]
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 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 43, normalized size = 0.19 \[ \frac {x\,\sqrt {\frac {a\,x^4}{b}+1}\,\sqrt {x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ -\frac {a\,x^4}{b}\right )}{3\,\sqrt {a\,x^4+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.11, size = 41, normalized size = 0.18 \[ - \frac {x \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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