Optimal. Leaf size=18 \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]
[Out]
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Rubi [A] time = 0.115315, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*x^2 + Sqrt[a + b*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 4.47737, size = 14, normalized size = 0.78 \[ \frac{\log{\left (\sqrt{a + b x^{2}} + 1 \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*x**2+(b*x**2+a)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0170817, size = 18, normalized size = 1. \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*x^2 + Sqrt[a + b*x^2]),x]
[Out]
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Maple [B] time = 0.057, size = 1059, normalized size = 58.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*x^2+(b*x^2+a)^(1/2)),x)
[Out]
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Maxima [A] time = 0.728139, size = 22, normalized size = 1.22 \[ \frac{\log \left (\sqrt{b x^{2} + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a + sqrt(b*x^2 + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276914, size = 90, normalized size = 5. \[ \frac{2 \, \log \left (b x^{2} + a - 1\right ) + \log \left (\frac{b x^{2} + a + 2 \, \sqrt{b x^{2} + a} + 1}{x^{2}}\right ) - \log \left (\frac{b x^{2} + a - 2 \, \sqrt{b x^{2} + a} + 1}{x^{2}}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a + sqrt(b*x^2 + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{a + b x^{2} + \sqrt{a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*x**2+(b*x**2+a)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.261379, size = 22, normalized size = 1.22 \[ \frac{{\rm ln}\left (\sqrt{b x^{2} + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a + sqrt(b*x^2 + a)),x, algorithm="giac")
[Out]