Integrand size = 15, antiderivative size = 129 \[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=-\frac {4 b^2}{15 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}+\frac {2 b \left (a+b x^4\right )^{3/4}}{15 a^2 x^5}+\frac {4 b^{5/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a+b x^4}} \]
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Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {331, 318, 287, 342, 281, 202} \[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\frac {4 b^{5/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a+b x^4}}-\frac {4 b^2}{15 a^2 x \sqrt [4]{a+b x^4}}+\frac {2 b \left (a+b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9} \]
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Rule 202
Rule 281
Rule 287
Rule 318
Rule 331
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\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 {4 b^2}{15 a^2 x \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^3\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{15 a^2} \\ & = -\frac {4 b^2}{15 a^2 x \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 ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{15 a^2 \sqrt [4]{a+b x^4}} \\ & = -\frac {4 b^2}{15 a^2 x \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}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{15 a^2 \sqrt [4]{a+b x^4}} \\ & = -\frac {4 b^2}{15 a^2 x \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}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{15 a^2 \sqrt [4]{a+b x^4}} \\ & = -\frac {4 b^2}{15 a^2 x \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} \cot ^{-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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},-\frac {5}{4},-\frac {b x^4}{a}\right )}{9 x^9 \sqrt [4]{a+b x^4}} \]
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\[\int \frac {1}{x^{10} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x\]
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\[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\frac {\Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, \frac {1}{4} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} x^{9} \Gamma \left (- \frac {5}{4}\right )} \]
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\[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \]
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\[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx=\int \frac {1}{x^{10}\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]
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