Optimal. Leaf size=168 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}}} \]
[Out]
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Rubi [F] time = 0.675554, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3},x\right ) \]
Verification is Not applicable to the result.
[In] Int[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)
[Out]
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Mathematica [C] time = 0.0963126, size = 95, normalized size = 0.57 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b+b\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 b+\text{$\#$1} b}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]
[Out]
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Maple [C] time = 0.235, size = 67, normalized size = 0.4 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{6}+3\,b{{\it \_Z}}^{4}+3\,b{{\it \_Z}}^{2}+b \right ) }{\frac{ \left ({{\it \_R}}^{4}+2\,{{\it \_R}}^{2}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}a+{{\it \_R}}^{5}b+2\,{{\it \_R}}^{3}b+{\it \_R}\,b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{2}}{a x^{6} +{\left (x^{2} + 1\right )}^{3} b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.13289, size = 42, normalized size = 0.25 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a b^{5} + 46656 b^{6}\right ) + 3888 t^{4} b^{4} + 108 t^{2} b^{2} + 1, \left ( t \mapsto t \log{\left (6 t b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{2}}{a x^{6} +{\left (x^{2} + 1\right )}^{3} b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="giac")
[Out]