3.493 \(\int \frac{\sqrt{a+b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0355879, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.06434, size = 41, normalized size = 0.91 \[ 2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} - \frac{2 \sqrt{a + b x}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0218209, size = 48, normalized size = 1.07 \[ 2 \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/x^(3/2),x]

[Out]

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

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Maple [A]  time = 0.102, size = 61, normalized size = 1.4 \[ -2\,{\frac{\sqrt{bx+a}}{\sqrt{x}}}+{1\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253428, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \, \sqrt{b x + a} \sqrt{x}}{x}, \frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) - \sqrt{b x + a} \sqrt{x}\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.86608, size = 68, normalized size = 1.51 \[ - \frac{2 \sqrt{a}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{2 b \sqrt{x}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 12.5019, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/x^(3/2),x, algorithm="giac")

[Out]

sage0*x