Optimal. Leaf size=108 \[ \frac {2 b^{5/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {277, 325, 237, 335, 275, 232} \[ \frac {2 b^{5/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a-b x^4\right )^{3/4}}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}-\frac {\sqrt [4]{a-b x^4}}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 232
Rule 237
Rule 275
Rule 277
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^8} \, dx &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}-\frac {1}{7} b \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}-\frac {\left (2 b^2\right ) \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{21 a}\\ &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}-\frac {\left (2 b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{21 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}+\frac {\left (2 b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{21 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}+\frac {\left (b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{21 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{7 x^7}+\frac {b \sqrt [4]{a-b x^4}}{21 a x^3}+\frac {2 b^{5/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 a^{3/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.48 \[ -\frac {\sqrt [4]{a-b x^4} \, _2F_1\left (-\frac {7}{4},-\frac {1}{4};-\frac {3}{4};\frac {b x^4}{a}\right )}{7 x^7 \sqrt [4]{1-\frac {b x^4}{a}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{8}}\, dx \]
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 \[ \int \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^4\right )}^{1/4}}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.39, size = 34, normalized size = 0.31 \[ \frac {i \sqrt [4]{b} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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