Optimal. Leaf size=100 \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]
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Rubi [A] time = 0.0341451, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {321, 224, 221} \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{a-b x^4}} \, dx &=-\frac{x^5 \sqrt{a-b x^4}}{7 b}+\frac{(5 a) \int \frac{x^4}{\sqrt{a-b x^4}} \, dx}{7 b}\\ &=-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{21 b^2}\\ &=-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b}+\frac{\left (5 a^2 \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{21 b^2 \sqrt{a-b x^4}}\\ &=-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b}+\frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.019846, size = 80, normalized size = 0.8 \[ \frac{5 a^2 x \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b x^4}{a}\right )-5 a^2 x+2 a b x^5+3 b^2 x^9}{21 b^2 \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 107, normalized size = 1.1 \begin{align*} -{\frac{{x}^{5}}{7\,b}\sqrt{-b{x}^{4}+a}}-{\frac{5\,ax}{21\,{b}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{5\,{a}^{2}}{21\,{b}^{2}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b x^{4} + a} x^{8}}{b x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33588, size = 39, normalized size = 0.39 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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