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

Optimal. Leaf size=15 \[ \frac{\sqrt{b x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.00696891, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{b x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.91304, size = 12, normalized size = 0.8 \[ \frac{\sqrt{b x^{2}} \log{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0026693, size = 14, normalized size = 0.93 \[ \frac{b x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2]/x^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{x}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^(1/2)/x^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226209, size = 18, normalized size = 1.2 \[ \frac{\sqrt{b x^{2}} \log \left (x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(b*x**2)/x**2, x)

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GIAC/XCAS [A]  time = 0.211894, size = 12, normalized size = 0.8 \[ \sqrt{b}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*ln(abs(x))*sign(x)