Optimal. Leaf size=46 \[ -\frac {\left (a-b x^4\right )^{5/4}}{9 a x^9}-\frac {4 b \left (a-b x^4\right )^{5/4}}{45 a^2 x^5} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270}
\begin {gather*} -\frac {4 b \left (a-b x^4\right )^{5/4}}{45 a^2 x^5}-\frac {\left (a-b x^4\right )^{5/4}}{9 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^{10}} \, dx &=-\frac {\left (a-b x^4\right )^{5/4}}{9 a x^9}+\frac {(4 b) \int \frac {\sqrt [4]{a-b x^4}}{x^6} \, dx}{9 a}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{9 a x^9}-\frac {4 b \left (a-b x^4\right )^{5/4}}{45 a^2 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 42, normalized size = 0.91 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-5 a^2+a b x^4+4 b^2 x^8\right )}{45 a^2 x^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 29, normalized size = 0.63
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (4 b \,x^{4}+5 a \right )}{45 a^{2} x^{9}}\) | \(29\) |
trager | \(-\frac {\left (-4 b^{2} x^{8}-a b \,x^{4}+5 a^{2}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{45 a^{2} x^{9}}\) | \(40\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-4 b^{2} x^{8}-a b \,x^{4}+5 a^{2}\right )}{45 x^{9} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}} a^{2}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.80 \begin {gather*} -\frac {\frac {9 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} + \frac {5 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}}{45 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 38, normalized size = 0.83 \begin {gather*} \frac {{\left (4 \, b^{2} x^{8} + a b x^{4} - 5 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, a^{2} x^{9}} \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 = 0.70, size = 406, normalized size = 8.83 \begin {gather*} \begin {cases} - \frac {5 \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {9}{4}\right )}{16 x^{8} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {9}{4}\right )}{16 a x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {9}{4}\right )}{4 a^{2} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {5 a^{3} b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{x^{4} \left (- 16 a^{3} b x^{4} \Gamma \left (- \frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (- \frac {1}{4}\right )\right )} - \frac {6 a^{2} b^{\frac {9}{4}} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (- \frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (- \frac {1}{4}\right )} - \frac {3 a b^{\frac {13}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (- \frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (- \frac {1}{4}\right )} + \frac {4 b^{\frac {17}{4}} x^{8} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (- \frac {1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (- \frac {1}{4}\right )} & \text {otherwise} \end {cases} \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 [B]
time = 1.30, size = 38, normalized size = 0.83 \begin {gather*} \frac {{\left (a-b\,x^4\right )}^{1/4}\,\left (-5\,a^2+a\,b\,x^4+4\,b^2\,x^8\right )}{45\,a^2\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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