Integrand size = 15, antiderivative size = 44 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=-\frac {\left (a+b x^4\right )^{5/4}}{9 a x^9}+\frac {4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=\frac {4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac {\left (a+b x^4\right )^{5/4}}{9 a x^9} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-5 a^2-a b x^4+4 b^2 x^8\right )}{45 a^2 x^9} \]
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Time = 4.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-4 b \,x^{4}+5 a \right )}{45 a^{2} x^{9}}\) | \(28\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-4 b \,x^{4}+5 a \right )}{45 a^{2} x^{9}}\) | \(28\) |
trager | \(-\frac {\left (-4 b^{2} x^{8}+a b \,x^{4}+5 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 a^{2} x^{9}}\) | \(38\) |
risch | \(-\frac {\left (-4 b^{2} x^{8}+a b \,x^{4}+5 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 a^{2} x^{9}}\) | \(38\) |
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none
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=\frac {{\left (4 \, b^{2} x^{8} - a b x^{4} - 5 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, a^{2} x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (37) = 74\).
Time = 0.66 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=- \frac {5 \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{16 x^{8} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{16 a x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{4 a^{2} \Gamma \left (- \frac {1}{4}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=\frac {\frac {9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} - \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}}{45 \, a^{2}} \]
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\[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{10}} \,d x } \]
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Time = 5.89 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx=-\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (5\,a^2+a\,b\,x^4-4\,b^2\,x^8\right )}{45\,a^2\,x^9} \]
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