Integrand size = 16, antiderivative size = 131 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=-\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {a^{5/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 b^{3/2} \left (a-b x^4\right )^{3/4}} \]
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Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {285, 327, 243, 342, 281, 238} \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=-\frac {a^{5/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b} \]
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Rule 238
Rule 243
Rule 281
Rule 285
Rule 327
Rule 342
Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} x^9 \sqrt [4]{a-b x^4}+\frac {1}{10} a \int \frac {x^8}{\left (a-b x^4\right )^{3/4}} \, dx \\ & = -\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}+\frac {a^2 \int \frac {x^4}{\left (a-b x^4\right )^{3/4}} \, dx}{12 b} \\ & = -\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}+\frac {a^3 \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{24 b^2} \\ & = -\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}+\frac {\left (a^3 \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}{24 b^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {\left (a^3 \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 )}{24 b^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {\left (a^3 \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 )}{48 b^2 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {a^{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 )}{24 b^{3/2} \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 = 7.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.73 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\frac {x \sqrt [4]{a-b x^4} \left (-\sqrt [4]{1-\frac {b x^4}{a}} \left (5 a^2+a b x^4-6 b^2 x^8\right )+5 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )\right )}{60 b^2 \sqrt [4]{1-\frac {b x^4}{a}}} \]
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\[\int x^{8} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}d x\]
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\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.31 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
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\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
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Timed out. \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int x^8\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]
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