Optimal. Leaf size=109 \[ -\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {4 b^{5/2} \sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a-b x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {331, 319, 342,
281, 234} \begin {gather*} -\frac {4 b^{5/2} 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 )}{15 a^{5/2} \sqrt [4]{a-b x^4}}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 281
Rule 319
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}+\frac {(2 b) \int \frac {1}{x^6 \sqrt [4]{a-b x^4}} \, dx}{3 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}+\frac {\left (4 b^2\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{15 a^2}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}+\frac {\left (4 b^2 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3} \, dx}{15 a^2 \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {\left (4 b^2 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,\frac {1}{x}\right )}{15 a^2 \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {\left (2 b^2 \sqrt [4]{1-\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^2}{b}}} \, dx,x,\frac {1}{x^2}\right )}{15 a^2 \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {4 b^{5/2} \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 )}{15 a^{5/2} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 52, normalized size = 0.48 \begin {gather*} -\frac {\sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {9}{4},\frac {1}{4};-\frac {5}{4};\frac {b x^4}{a}\right )}{9 x^9 \sqrt [4]{a-b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{10} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.08, size = 28, normalized size = 0.26 \begin {gather*} {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{14} - a x^{10}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 31, normalized size = 0.28 \begin {gather*} \frac {i e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{10 \sqrt [4]{b} x^{10}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{10}\,{\left (a-b\,x^4\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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