Optimal. Leaf size=55 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]
[Out]
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Rubi [A] time = 0.0725561, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]
Antiderivative was successfully verified.
[In] Int[x^8/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{7}{2}}} - \frac{b x^{3}}{3 c^{2}} + \frac{x^{5}}{5 c} + \frac{\int b^{2}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0432015, size = 55, normalized size = 1. \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]
Antiderivative was successfully verified.
[In] Integrate[x^8/(b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 49, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,c}}-{\frac{b{x}^{3}}{3\,{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}-{\frac{{b}^{3}}{{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(c*x^4+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^8/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.263321, size = 1, normalized size = 0.02 \[ \left [\frac{6 \, c^{2} x^{5} - 10 \, b c x^{3} + 15 \, b^{2} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, c^{3}}, \frac{3 \, c^{2} x^{5} - 5 \, b c x^{3} - 15 \, b^{2} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) + 15 \, b^{2} x}{15 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.30234, size = 95, normalized size = 1.73 \[ \frac{b^{2} x}{c^{3}} - \frac{b x^{3}}{3 c^{2}} + \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x - \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x + \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} + \frac{x^{5}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.268974, size = 74, normalized size = 1.35 \[ -\frac{b^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{3}} + \frac{3 \, c^{4} x^{5} - 5 \, b c^{3} x^{3} + 15 \, b^{2} c^{2} x}{15 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]