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

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

[Out]

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

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Rubi [A]  time = 0.0221095, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.41655, size = 26, normalized size = 0.93 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0103403, size = 31, normalized size = 1.11 \[ \frac{2 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 48, normalized size = 1.7 \[{1\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(b*x+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221275, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{\sqrt{b}}, \frac{2 \, \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.01145, size = 22, normalized size = 0.79 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError