3.848 \(\int \frac{1}{\sqrt{x (a+b x)}} \, dx\)

Optimal. Leaf size=28 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/Sqrt[b]

_______________________________________________________________________________________

Rubi [A]  time = 0.0252943, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[x*(a + b*x)],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/Sqrt[b]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.25238, size = 26, normalized size = 0.93 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/sqrt(b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00978284, size = 56, normalized size = 2. \[ \frac{2 \sqrt{x} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{x (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[x*(a + b*x)],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 29, normalized size = 1. \[{1\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x*(b*x+a))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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((b*x + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.272221, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right )}{\sqrt{b}}, \frac{2 \, \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log((2*b*x + a)*sqrt(b) + 2*sqrt(b*x^2 + a*x)*b)/sqrt(b), 2*arctan(sqrt(b*x^2 +
a*x)*sqrt(-b)/(b*x))/sqrt(-b)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(x*(a + b*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273528, size = 47, normalized size = 1.68 \[ -\frac{{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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