Optimal. Leaf size=44 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
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Rubi [A] time = 0.0786467, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 3.98391, size = 29, normalized size = 0.66 \[ \frac{\sqrt{\left (a + b x^{2}\right )^{2}} \log{\left (a + b x^{2} \right )}}{2 b \left (a + b x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/((b*x**2+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0118819, size = 35, normalized size = 0.8 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Maple [A] time = 0.007, size = 32, normalized size = 0.7 \[{\frac{ \left ( b{x}^{2}+a \right ) \ln \left ( b{x}^{2}+a \right ) }{2\,b}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/((b*x^2+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.698751, size = 23, normalized size = 0.52 \[ \frac{1}{2} \, \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((b*x^2 + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258657, size = 18, normalized size = 0.41 \[ \frac{\log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((b*x^2 + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.292081, size = 10, normalized size = 0.23 \[ \frac{\log{\left (a + b x^{2} \right )}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x**2+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270073, size = 30, normalized size = 0.68 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right ){\rm sign}\left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((b*x^2 + a)^2),x, algorithm="giac")
[Out]