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

Optimal. Leaf size=29 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.025614, antiderivative size = 29, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(-a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.86987, size = 29, normalized size = 1. \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00796534, size = 29, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 19, normalized size = 0.7 \[ -2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x-a)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218475, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{a b}{\left (b x + a\right )}}{b x - a}\right )}{\sqrt{a b}}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{-a b} \sqrt{x}}\right )}{\sqrt{-a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.96851, size = 65, normalized size = 2.24 \[ \begin{cases} - \frac{2 \operatorname{acoth}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} \sqrt{b}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} \sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x-a)/x**(1/2),x)

[Out]

Piecewise((-2*acoth(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt(a)*sqrt(b)), Abs(a/(b*x)) >
1), (-2*atanh(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt(a)*sqrt(b)), True))

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GIAC/XCAS [A]  time = 0.215732, size = 27, normalized size = 0.93 \[ \frac{2 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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