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

Optimal. Leaf size=38 \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0312646, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x^2]/(a + b*x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} x} + \frac{\sqrt{c x^{2}} \int \frac{1}{b}\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c*x**2)*log(a + b*x)/(b**2*x) + sqrt(c*x**2)*Integral(1/b, x)/x

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Mathematica [A]  time = 0.0109076, size = 28, normalized size = 0.74 \[ \frac{c x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 29, normalized size = 0.8 \[ -{\frac{a\ln \left ( bx+a \right ) -bx}{{b}^{2}x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(1/2)/(b*x+a),x)

[Out]

-(c*x^2)^(1/2)*(a*ln(b*x+a)-b*x)/b^2/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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204184, size = 36, normalized size = 0.95 \[ \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x - a*log(b*x + a))/(b^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*x**2)/(a + b*x), x)

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GIAC/XCAS [A]  time = 0.207965, size = 50, normalized size = 1.32 \[ \sqrt{c}{\left (\frac{x{\rm sign}\left (x\right )}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{2}} + \frac{a{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*(x*sign(x)/b - a*ln(abs(b*x + a))*sign(x)/b^2 + a*ln(abs(a))*sign(x)/b^2
)