Optimal. Leaf size=86 \[ -\frac {2 b^{3/2} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5} \]
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Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {325, 313, 335, 275, 228} \[ -\frac {2 b^{3/2} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 228
Rule 275
Rule 313
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [4]{a-b x^4}} \, dx &=-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}+\frac {(2 b) \int \frac {1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{5 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}+\frac {\left (2 b \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3} \, dx}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac {\left (2 b \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 )}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac {\left (b \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 )}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac {2 b^{3/2} \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 )}{5 a^{3/2} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.60 \[ -\frac {\sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {5}{4},\frac {1}{4};-\frac {1}{4};\frac {b x^4}{a}\right )}{5 x^5 \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{10} - a x^{6}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
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 \[ \int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{6}}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
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 \[ \int \frac {1}{x^6\,{\left (a-b\,x^4\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.55, size = 34, normalized size = 0.40 \[ - \frac {i e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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