Integrand size = 16, antiderivative size = 111 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac {4 b^{5/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}} \]
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Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {331, 243, 342, 281, 238} \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {4 b^{5/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7} \]
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Rule 238
Rule 243
Rule 281
Rule 331
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}+\frac {(6 b) \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}} \, dx}{7 a} \\ & = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}+\frac {\left (4 b^2\right ) \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{7 a^2} \\ & = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}+\frac {\left (4 b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{7 a^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac {\left (4 b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{7 a^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac {\left (2 b^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{7 a^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{7 a x^7}-\frac {2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac {4 b^{5/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},-\frac {3}{4},\frac {b x^4}{a}\right )}{7 x^7 \left (a-b x^4\right )^{3/4}} \]
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\[\int \frac {1}{x^{8} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=\frac {i e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{10 b^{\frac {3}{4}} x^{10}} \]
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\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \]
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\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx=\int \frac {1}{x^8\,{\left (a-b\,x^4\right )}^{3/4}} \,d x \]
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