Optimal. Leaf size=100 \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]
[Out]
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Rubi [A] time = 0.100349, 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 \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]
Antiderivative was successfully verified.
[In] Int[x^8/Sqrt[a - b*x^4],x]
[Out]
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Rubi in Sympy [A] time = 12.8651, size = 88, normalized size = 0.88 \[ \frac{5 a^{\frac{9}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{21 b^{\frac{9}{4}} \sqrt{a - b x^{4}}} - \frac{5 a x \sqrt{a - b x^{4}}}{21 b^{2}} - \frac{x^{5} \sqrt{a - b x^{4}}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(-b*x**4+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.160883, size = 122, normalized size = 1.22 \[ \frac{x \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \left (-5 a^2+2 a b x^4+3 b^2 x^8\right )-5 i a^2 \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{21 b^2 \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^8/Sqrt[a - b*x^4],x]
[Out]
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Maple [A] time = 0.027, size = 107, normalized size = 1.1 \[ -{\frac{{x}^{5}}{7\,b}\sqrt{-b{x}^{4}+a}}-{\frac{5\,ax}{21\,{b}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{5\,{a}^{2}}{21\,{b}^{2}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(-b*x^4+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{\sqrt{-b x^{4} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.27009, size = 39, normalized size = 0.39 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(-b*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="giac")
[Out]