3.1536 \(\int \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=32 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b} \]

[Out]

((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]

((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b)

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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]

(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*b)

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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]

(x*Sqrt[(a + b*x)^2]*(2*a + b*x))/(2*(a + b*x))

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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]

1/2*x*(b*x+2*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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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]

Exception raised: ValueError

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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]

1/2*b*x^2 + a*x

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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]

a*x + b*x**2/2

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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]

1/2*(b*x^2 + 2*a*x)*sign(b*x + a) + 1/2*a^2*sign(b*x + a)/b