Optimal. Leaf size=68 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac{x^2}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.0705767, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac{x^2}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 8.34, size = 60, normalized size = 0.88 \[ \frac{x^{2}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{3 x^{2}}{16 a^{2} \left (a + c x^{4}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(c*x**4+a)**3,x)
[Out]
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Mathematica [A] time = 0.0785993, size = 58, normalized size = 0.85 \[ \frac{1}{16} \left (\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{a^{5/2} \sqrt{c}}+\frac{5 a x^2+3 c x^6}{a^2 \left (a+c x^4\right )^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + c*x^4)^3,x]
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Maple [A] time = 0.009, size = 57, normalized size = 0.8 \[{\frac{{x}^{2}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{3\,{x}^{2}}{16\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3}{16\,{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(c*x^4+a)^3,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/(c*x^4 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23838, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) + 2 \,{\left (3 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{-a c}}{32 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \sqrt{-a c}}, -\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) -{\left (3 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{a c}}{16 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \sqrt{a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.94295, size = 110, normalized size = 1.62 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x^{2} \right )}}{32} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x^{2} \right )}}{32} + \frac{5 a x^{2} + 3 c x^{6}}{16 a^{4} + 32 a^{3} c x^{4} + 16 a^{2} c^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x**4+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.221665, size = 66, normalized size = 0.97 \[ \frac{3 \, \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{2}} + \frac{3 \, c x^{6} + 5 \, a x^{2}}{16 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a)^3,x, algorithm="giac")
[Out]