Optimal. Leaf size=35 \[ -\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.0497282, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 7.74039, size = 36, normalized size = 1.03 \[ - \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0107399, size = 31, normalized size = 0.89 \[ -\frac{\sqrt{(a+b x)^2} (a+2 b x)}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]
[Out]
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Maple [A] time = 0.004, size = 28, normalized size = 0.8 \[ -{\frac{2\,bx+a}{2\,{x}^{2} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+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 + a)^2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227406, size = 15, normalized size = 0.43 \[ -\frac{2 \, b x + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.08921, size = 12, normalized size = 0.34 \[ - \frac{a + 2 b x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.208175, size = 53, normalized size = 1.51 \[ -\frac{b^{2}{\rm sign}\left (b x + a\right )}{2 \, a} - \frac{2 \, b x{\rm sign}\left (b x + a\right ) + a{\rm sign}\left (b x + a\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/x^3,x, algorithm="giac")
[Out]