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

Optimal. Leaf size=41 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]

[Out]

-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.0367683, antiderivative size = 41, 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 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[a + b*x]),x]

[Out]

-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2)

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Rubi in Sympy [A]  time = 2.2741, size = 32, normalized size = 0.78 \[ - \frac{\sqrt{a + b x}}{a x} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(a*x) + b*atanh(sqrt(a + b*x)/sqrt(a))/a**(3/2)

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Mathematica [A]  time = 0.0230001, size = 41, normalized size = 1. \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2)

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Maple [A]  time = 0.005, size = 40, normalized size = 1. \[ 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{axb}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(1/2),x)

[Out]

2*b*(-1/2*(b*x+a)^(1/2)/a/x/b+1/2/a^(3/2)*arctanh((b*x+a)^(1/2)/a^(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(b*x + a)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218636, size = 1, normalized size = 0.02 \[ \left [\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \, \sqrt{b x + a} \sqrt{a}}{2 \, a^{\frac{3}{2}} x}, -\frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) + \sqrt{b x + a} \sqrt{-a}}{\sqrt{-a} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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))/(a^(3/2)*x), -(b*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + sqrt(b*x + a)*sqrt
(-a))/(sqrt(-a)*a*x)]

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Sympy [A]  time = 3.45636, size = 44, normalized size = 1.07 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*sqrt(a/(b*x) + 1)/(a*sqrt(x)) + b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(
3/2)

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GIAC/XCAS [A]  time = 0.200883, size = 63, normalized size = 1.54 \[ -\frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x + a} b}{a x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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