Integrand size = 15, antiderivative size = 129 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=-\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac {8 b^{5/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}} \]
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Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {289, 287, 342, 281, 202} \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\frac {8 b^{5/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}} \]
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Rule 202
Rule 281
Rule 287
Rule 289
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}-\frac {(10 b) \int \frac {1}{x^6 \left (a+b x^4\right )^{5/4}} \, dx}{9 a} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}+\frac {\left (4 b^2\right ) \int \frac {1}{x^2 \left (a+b x^4\right )^{5/4}} \, dx}{3 a^2} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (8 b^3\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{3 a^3} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (8 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}{3 a^3 \sqrt [4]{a+b x^4}} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac {\left (8 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 )}{3 a^3 \sqrt [4]{a+b x^4}} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac {\left (4 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 )}{3 a^3 \sqrt [4]{a+b x^4}} \\ & = -\frac {1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac {2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac {4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac {8 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 )}{3 a^{7/2} \sqrt [4]{a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {5}{4},-\frac {5}{4},-\frac {b x^4}{a}\right )}{9 a x^9 \sqrt [4]{a+b x^4}} \]
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\[\int \frac {1}{x^{10} \left (b \,x^{4}+a \right )^{\frac {5}{4}}}d x\]
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\[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{10}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\frac {\Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, \frac {5}{4} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{4}} x^{9} \Gamma \left (- \frac {5}{4}\right )} \]
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\[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{10}} \,d x } \]
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\[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{10}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx=\int \frac {1}{x^{10}\,{\left (b\,x^4+a\right )}^{5/4}} \,d x \]
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