3.5 \(\int \sqrt{b x^2} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{2} x \sqrt{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.00703291, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2],x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{b x^{2}} \int x\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2)**(1/2),x)

[Out]

sqrt(b*x**2)*Integral(x, x)/x

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Mathematica [A]  time = 0.00142296, size = 14, normalized size = 1. \[ \frac{1}{2} x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2],x]

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \[{\frac{x}{2}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^(1/2),x)

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237769, size = 14, normalized size = 1. \[ \frac{1}{2} \, \sqrt{b x^{2}} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2),x, algorithm="fricas")

[Out]

1/2*sqrt(b*x^2)*x

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Sympy [A]  time = 0.410708, size = 14, normalized size = 1. \[ \frac{\sqrt{b} x \sqrt{x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2)**(1/2),x)

[Out]

sqrt(b)*x*sqrt(x**2)/2

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GIAC/XCAS [A]  time = 0.217012, size = 14, normalized size = 1. \[ \frac{1}{2} \, \sqrt{b} x^{2}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2),x, algorithm="giac")

[Out]

1/2*sqrt(b)*x^2*sign(x)