3.47 \(\int \frac{1}{x \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2 \sqrt{b x+c x^2}}{b x} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(b*x)

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Rubi [A]  time = 0.0293648, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \sqrt{b x+c x^2}}{b x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(b*x)

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Rubi in Sympy [A]  time = 3.83977, size = 17, normalized size = 0.81 \[ - \frac{2 \sqrt{b x + c x^{2}}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + c*x**2)/(b*x)

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Mathematica [A]  time = 0.0187811, size = 21, normalized size = 1. \[ -\frac{2 (b+c x)}{b \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*(b + c*x))/(b*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.005, size = 22, normalized size = 1.1 \[ -2\,{\frac{cx+b}{b\sqrt{c{x}^{2}+bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^2+b*x)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215935, size = 26, normalized size = 1.24 \[ -\frac{2 \, \sqrt{c x^{2} + b x}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x),x, algorithm="fricas")

[Out]

-2*sqrt(c*x^2 + b*x)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.217975, size = 31, normalized size = 1.48 \[ \frac{2}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x),x, algorithm="giac")

[Out]

2/(sqrt(c)*x - sqrt(c*x^2 + b*x))