Integrand size = 14, antiderivative size = 59 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a-b x^4\right )^{3/4}} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {281, 239, 238} \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a-b x^4\right )^{3/4}} \]
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Rule 238
Rule 239
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{2 \left (a-b x^4\right )^{3/4}} \\ & = \frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \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 = 7.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^2 \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {b x^4}{a}\right )}{2 \left (a-b x^4\right )^{3/4}} \]
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\[\int \frac {x}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.49 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^{2} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}}} \]
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\[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\int \frac {x}{{\left (a-b\,x^4\right )}^{3/4}} \,d x \]
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