Optimal. Leaf size=32 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b} \]
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Rubi [A] time = 0.0191468, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.14984, size = 31, normalized size = 0.97 \[ \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0153896, size = 30, normalized size = 0.94 \[ \frac{x \sqrt{(a+b x)^2} (2 a+b x)}{2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0., size = 27, normalized size = 0.8 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2),x)
[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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201458, size = 14, normalized size = 0.44 \[ \frac{1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.120876, size = 8, normalized size = 0.25 \[ a x + \frac{b x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210134, size = 45, normalized size = 1.41 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )}{\rm sign}\left (b x + a\right ) + \frac{a^{2}{\rm sign}\left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2),x, algorithm="giac")
[Out]