Optimal. Leaf size=110 \[ \frac {b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+\frac {b}{x^4}}}+\frac {x^3 \sqrt {a+\frac {b}{x^4}}}{3 a} \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {335, 325, 220} \[ \frac {b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+\frac {b}{x^4}}}+\frac {x^3 \sqrt {a+\frac {b}{x^4}}}{3 a} \]
Antiderivative was successfully verified.
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Rule 220
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+\frac {b}{x^4}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x^4}} x^3}{3 a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{3 a}\\ &=\frac {\sqrt {a+\frac {b}{x^4}} x^3}{3 a}+\frac {b^{3/4} \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 )}{6 a^{5/4} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.58 \[ \frac {-b \sqrt {\frac {a x^4}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^4}{b}\right )+a x^4+b}{3 a x \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {a + \frac {b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 124, normalized size = 1.13 \[ \frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,x^{5}+\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b x -\sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, b \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )}{3 \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,x^{2}} \]
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 \[ \int \frac {x^{2}}{\sqrt {a + \frac {b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x^4}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.30, size = 42, normalized size = 0.38 \[ - \frac {x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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