Optimal. Leaf size=61 \[ -\frac {\sqrt {b} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^4}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {313, 335, 275, 228} \[ -\frac {\sqrt {b} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 275
Rule 313
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [4]{a-b x^4}} \, dx &=\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3} \, dx}{\sqrt [4]{a-b x^4}}\\ &=-\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,\frac {1}{x}\right )}{\sqrt [4]{a-b x^4}}\\ &=-\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^2}{b}}} \, dx,x,\frac {1}{x^2}\right )}{2 \sqrt [4]{a-b x^4}}\\ &=-\frac {\sqrt {b} \sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.82 \[ -\frac {\sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};\frac {b x^4}{a}\right )}{x \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{6} - a x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 41, normalized size = 0.67 \[ -\frac {{\left (1-\frac {a}{b\,x^4}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ \frac {a}{b\,x^4}\right )}{2\,x\,{\left (a-b\,x^4\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.66, size = 31, normalized size = 0.51 \[ \frac {i e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{2 \sqrt [4]{b} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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