Optimal. Leaf size=51 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]
[Out]
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Rubi [A] time = 0.0767684, antiderivative size = 51, 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 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]
Antiderivative was successfully verified.
[In] Int[x^9/(a + c*x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} + \frac{x^{6}}{6 c} - \frac{\int ^{x^{2}} a\, dx}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**9/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0559989, size = 48, normalized size = 0.94 \[ \frac{1}{6} \left (\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{c^{5/2}}+\frac{c x^6-3 a x^2}{c^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^9/(a + c*x^4),x]
[Out]
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Maple [A] time = 0.008, size = 43, normalized size = 0.8 \[{\frac{{x}^{6}}{6\,c}}-{\frac{a{x}^{2}}{2\,{c}^{2}}}+{\frac{{a}^{2}}{2\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^9/(c*x^4+a),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^9/(c*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237071, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, c x^{6} - 6 \, a x^{2} + 3 \, a \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{12 \, c^{2}}, \frac{c x^{6} - 3 \, a x^{2} + 3 \, a \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right )}{6 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/(c*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.39009, size = 87, normalized size = 1.71 \[ - \frac{a x^{2}}{2 c^{2}} - \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} - \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} + \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{x^{6}}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**9/(c*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.223756, size = 61, normalized size = 1.2 \[ \frac{a^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c^{2}} + \frac{c^{2} x^{6} - 3 \, a c x^{2}}{6 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/(c*x^4 + a),x, algorithm="giac")
[Out]