Optimal. Leaf size=85 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x}{16 a^2 b \left (a+b x^2\right )}+\frac{x}{24 a b \left (a+b x^2\right )^2}-\frac{x}{6 b \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.100726, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x}{16 a^2 b \left (a+b x^2\right )}+\frac{x}{24 a b \left (a+b x^2\right )^2}-\frac{x}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 20.3255, size = 68, normalized size = 0.8 \[ - \frac{x}{6 b \left (a + b x^{2}\right )^{3}} + \frac{x}{24 a b \left (a + b x^{2}\right )^{2}} + \frac{x}{16 a^{2} b \left (a + b x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.0755909, size = 69, normalized size = 0.81 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{-3 a^2 x+8 a b x^3+3 b^2 x^5}{48 a^2 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Maple [A] time = 0.012, size = 58, normalized size = 0.7 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{3}} \left ({\frac{b{x}^{5}}{16\,{a}^{2}}}+{\frac{{x}^{3}}{6\,a}}-{\frac{x}{16\,b}} \right ) }+{\frac{1}{16\,{a}^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b^2*x^4+2*a*b*x^2+a^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^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275281, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x\right )} \sqrt{-a b}}{96 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x\right )} \sqrt{a b}}{48 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.15994, size = 139, normalized size = 1.64 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{32} + \frac{- 3 a^{2} x + 8 a b x^{3} + 3 b^{2} x^{5}}{48 a^{5} b + 144 a^{4} b^{2} x^{2} + 144 a^{3} b^{3} x^{4} + 48 a^{2} b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271589, size = 84, normalized size = 0.99 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} a^{2} b} + \frac{3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x}{48 \,{\left (b x^{2} + a\right )}^{3} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")
[Out]