3.856 \(\int \frac{\sqrt{c x^2}}{x (a+b x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{\sqrt{c x^2} \log (a+b x)}{b x} \]

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi [A]  time = 0.0114679, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\sqrt{c x^2} \log (a+b x)}{b x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x^2]/(x*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi in Sympy [A]  time = 11.8602, size = 17, normalized size = 0.77 \[ \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x**2)*log(a + b*x)/(b*x)

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Mathematica [A]  time = 0.00718074, size = 21, normalized size = 0.95 \[ \frac{c x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*x^2]/(x*(a + b*x)),x]

[Out]

(c*x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 21, normalized size = 1. \[{\frac{\ln \left ( bx+a \right ) }{bx}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(1/2)/x/(b*x+a),x)

[Out]

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

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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(c*x^2)/((b*x + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.201458, size = 27, normalized size = 1.23 \[ \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}}{x \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*x**2)/(x*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.208499, size = 38, normalized size = 1.73 \[ \sqrt{c}{\left (\frac{{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b} - \frac{{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*(ln(abs(b*x + a))*sign(x)/b - ln(abs(a))*sign(x)/b)