Optimal. Leaf size=75 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.0682473, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{2}\right )^{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x**2+a)**2)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0161505, size = 39, normalized size = 0.52 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a-2 b x^2 \log (x)\right )}{2 x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^3,x]
[Out]
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Maple [A] time = 0.012, size = 38, normalized size = 0.5 \[{\frac{2\,b\ln \left ( x \right ){x}^{2}-a}{ \left ( 2\,b{x}^{2}+2\,a \right ){x}^{2}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x^2+a)^2)^(1/2)/x^3,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(sqrt((b*x^2 + a)^2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.259938, size = 23, normalized size = 0.31 \[ \frac{2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.0711, size = 10, normalized size = 0.13 \[ - \frac{a}{2 x^{2}} + b \log{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x**2+a)**2)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.269956, size = 61, normalized size = 0.81 \[ \frac{1}{2} \, b{\rm ln}\left (x^{2}\right ){\rm sign}\left (b x^{2} + a\right ) - \frac{b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a{\rm sign}\left (b x^{2} + a\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)/x^3,x, algorithm="giac")
[Out]