Integrand size = 16, antiderivative size = 130 \[ \int x^9 \sqrt [4]{a-b x^4} \, dx=-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {4 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{77 b^{5/2} \left (a-b x^4\right )^{3/4}} \]
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Time = 0.06 (sec) , antiderivative size = 130, 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 = {281, 285, 327, 239, 238} \[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\frac {4 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{77 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b} \]
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Rule 238
Rule 239
Rule 281
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^4 \sqrt [4]{a-b x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {1}{22} a \text {Subst}\left (\int \frac {x^4}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = -\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b} \\ & = -\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2} \\ & = -\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (2 a^3 \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {4 a^{7/2} \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 )}{77 b^{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 = 7.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\frac {x^2 \sqrt [4]{a-b x^4} \left (-\sqrt [4]{1-\frac {b x^4}{a}} \left (6 a^2+a b x^4-7 b^2 x^8\right )+6 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {b x^4}{a}\right )\right )}{77 b^2 \sqrt [4]{1-\frac {b x^4}{a}}} \]
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\[\int x^{9} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}d x\]
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\[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.24 \[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a} x^{10} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{10} \]
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\[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9} \,d x } \]
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\[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9} \,d x } \]
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Timed out. \[ \int x^9 \sqrt [4]{a-b x^4} \, dx=\int x^9\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]
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