Optimal. Leaf size=51 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{c}{2 a^2 x^2}-\frac{1}{6 a x^6} \]
[Out]
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Rubi [A] time = 0.0680764, 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 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{c}{2 a^2 x^2}-\frac{1}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a + c*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 12.6732, size = 44, normalized size = 0.86 \[ - \frac{1}{6 a x^{6}} + \frac{c}{2 a^{2} x^{2}} + \frac{c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0774301, size = 88, normalized size = 1.73 \[ -\frac{3 c^{3/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 c^{3/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{a} \left (a-3 c x^4\right )}{6 a^{5/2} x^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(a + c*x^4)),x]
[Out]
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Maple [A] time = 0.008, size = 43, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{6}a}}+{\frac{c}{2\,{a}^{2}{x}^{2}}}+{\frac{{c}^{2}}{2\,{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(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(1/((c*x^4 + a)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228595, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, c x^{6} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} + 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) + 6 \, c x^{4} - 2 \, a}{12 \, a^{2} x^{6}}, -\frac{3 \, c x^{6} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) - 3 \, c x^{4} + a}{6 \, a^{2} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.7149, size = 90, normalized size = 1.76 \[ - \frac{\sqrt{- \frac{c^{3}}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac{\sqrt{- \frac{c^{3}}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac{- a + 3 c x^{4}}{6 a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(c*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.224902, size = 58, normalized size = 1.14 \[ \frac{c^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a^{2}} + \frac{3 \, c x^{4} - a}{6 \, a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*x^7),x, algorithm="giac")
[Out]