Optimal. Leaf size=55 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}+\frac{b^2 x}{a^3}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
[Out]
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Rubi [A] time = 0.0755646, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}+\frac{b^2 x}{a^3}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b/x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ b^{2} \int \frac{1}{a^{3}}\, dx + \frac{x^{5}}{5 a} - \frac{b x^{3}}{3 a^{2}} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b/x**2),x)
[Out]
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Mathematica [A] time = 0.0430345, size = 55, normalized size = 1. \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{7/2}}+\frac{b^2 x}{a^3}-\frac{b x^3}{3 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b/x^2),x]
[Out]
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Maple [A] time = 0.003, size = 49, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,a}}-{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{b}^{2}x}{{a}^{3}}}-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(a+b/x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235821, size = 1, normalized size = 0.02 \[ \left [\frac{6 \, a^{2} x^{5} - 10 \, a b x^{3} + 15 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, a^{3}}, \frac{3 \, a^{2} x^{5} - 5 \, a b x^{3} - 15 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) + 15 \, b^{2} x}{15 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.33417, size = 95, normalized size = 1.73 \[ \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} + \frac{x^{5}}{5 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} x}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b/x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.220951, size = 74, normalized size = 1.35 \[ -\frac{b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{3 \, a^{4} x^{5} - 5 \, a^{3} b x^{3} + 15 \, a^{2} b^{2} x}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2),x, algorithm="giac")
[Out]