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

Optimal. Leaf size=39 \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0350426, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.39471, size = 32, normalized size = 0.82 \[ - \frac{\sqrt{a + b x}}{x} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0326559, size = 39, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/x^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 37, normalized size = 1. \[ 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229515, size = 1, normalized size = 0.03 \[ \left [\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \, \sqrt{b x + a} \sqrt{a}}{2 \, \sqrt{a} x}, \frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} \sqrt{-a}}{\sqrt{-a} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(b*x*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) - 2*sqrt(b*x + a)*sqr
t(a))/(sqrt(a)*x), (b*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - sqrt(b*x + a)*sqrt(
-a))/(sqrt(-a)*x)]

_______________________________________________________________________________________

Sympy [A]  time = 2.92092, size = 44, normalized size = 1.13 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*sqrt(a/(b*x) + 1)/sqrt(x) - b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205324, size = 55, normalized size = 1.41 \[ \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x + a} b}{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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