Optimal. Leaf size=92 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{11/2}}-\frac{9 b^3 x}{2 a^5}+\frac{3 b^2 x^3}{2 a^4}-\frac{9 b x^5}{10 a^3}+\frac{9 x^7}{14 a^2}-\frac{x^9}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.110552, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{11/2}}-\frac{9 b^3 x}{2 a^5}+\frac{3 b^2 x^3}{2 a^4}-\frac{9 b x^5}{10 a^3}+\frac{9 x^7}{14 a^2}-\frac{x^9}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
[In] Int[x^6/(a + b/x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{9 b^{3} \int \frac{1}{a^{4}}\, dx}{2 a} - \frac{x^{9}}{2 a \left (a x^{2} + b\right )} + \frac{9 x^{7}}{14 a^{2}} - \frac{9 b x^{5}}{10 a^{3}} + \frac{3 b^{2} x^{3}}{2 a^{4}} + \frac{9 b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(a+b/x**2)**2,x)
[Out]
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Mathematica [A] time = 0.102102, size = 82, normalized size = 0.89 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{11/2}}+\frac{x \left (10 a^3 x^6-28 a^2 b x^4-\frac{35 b^4}{a x^2+b}+70 a b^2 x^2-280 b^3\right )}{70 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(a + b/x^2)^2,x]
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Maple [A] time = 0.007, size = 78, normalized size = 0.9 \[{\frac{{x}^{7}}{7\,{a}^{2}}}-{\frac{2\,b{x}^{5}}{5\,{a}^{3}}}+{\frac{{b}^{2}{x}^{3}}{{a}^{4}}}-4\,{\frac{{b}^{3}x}{{a}^{5}}}-{\frac{{b}^{4}x}{2\,{a}^{5} \left ( a{x}^{2}+b \right ) }}+{\frac{9\,{b}^{4}}{2\,{a}^{5}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(a+b/x^2)^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^6/(a + b/x^2)^2,x, algorithm="maxima")
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Fricas [A] time = 0.239665, size = 1, normalized size = 0.01 \[ \left [\frac{20 \, a^{4} x^{9} - 36 \, a^{3} b x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a b^{3} x^{3} - 630 \, b^{4} x + 315 \,{\left (a b^{3} x^{2} + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{140 \,{\left (a^{6} x^{2} + a^{5} b\right )}}, \frac{10 \, a^{4} x^{9} - 18 \, a^{3} b x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a b^{3} x^{3} - 315 \, b^{4} x + 315 \,{\left (a b^{3} x^{2} + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right )}{70 \,{\left (a^{6} x^{2} + a^{5} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(a + b/x^2)^2,x, algorithm="fricas")
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Sympy [A] time = 1.83235, size = 134, normalized size = 1.46 \[ - \frac{b^{4} x}{2 a^{6} x^{2} + 2 a^{5} b} - \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac{x^{7}}{7 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{b^{2} x^{3}}{a^{4}} - \frac{4 b^{3} x}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(a+b/x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.226671, size = 113, normalized size = 1.23 \[ \frac{9 \, b^{4} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} - \frac{b^{4} x}{2 \,{\left (a x^{2} + b\right )} a^{5}} + \frac{5 \, a^{12} x^{7} - 14 \, a^{11} b x^{5} + 35 \, a^{10} b^{2} x^{3} - 140 \, a^{9} b^{3} x}{35 \, a^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(a + b/x^2)^2,x, algorithm="giac")
[Out]