Optimal. Leaf size=110 \[ -\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}-\frac {1}{2 a x \sqrt {a+\frac {b}{x^4}}} \]
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Rubi [A] time = 0.04, 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, 199, 220} \[ -\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}-\frac {1}{2 a x \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 220
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{3/2} x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (a+b x^4\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}} x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}} x}-\frac {\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 )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.52 \[ \frac {\sqrt {\frac {a x^4}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^4}{b}\right )-1}{2 a x \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a^{2} x^{8} + 2 \, a b x^{4} + b^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 113, normalized size = 1.03 \[ -\frac {\left (a \,x^{4}+b \right ) \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x -\sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )\right )}{2 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,x^{6}} \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 39, normalized size = 0.35 \[ -\frac {{\left (\frac {b}{a}+x^4\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -\frac {b}{a\,x^4}\right )}{x\,{\left (a\,x^4+b\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.50, size = 37, normalized size = 0.34 \[ - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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