3.473 \(\int \frac{1}{x^{3/2} (-a+b x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.035425, antiderivative size = 40, 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 \[ \frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(-a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.7452, size = 36, normalized size = 0.9 \[ \frac{2}{a \sqrt{x}} - \frac{2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0207675, size = 40, normalized size = 1. \[ \frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 32, normalized size = 0.8 \[ -2\,{\frac{b}{a\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+2\,{\frac{1}{a\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x-a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218691, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{x} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 2}{a \sqrt{x}}, \frac{2 \,{\left (\sqrt{x} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) + 1\right )}}{a \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x - a)*x^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.14282, size = 78, normalized size = 1.95 \[ \begin{cases} \frac{2}{a \sqrt{x}} - \frac{2 \sqrt{b} \operatorname{acoth}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{2}{a \sqrt{x}} - \frac{2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(b*x-a),x)

[Out]

Piecewise((2/(a*sqrt(x)) - 2*sqrt(b)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/a**(3/2), Ab
s(b*x/a) > 1), (2/(a*sqrt(x)) - 2*sqrt(b)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/a**(3/2
), True))

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GIAC/XCAS [A]  time = 0.231607, size = 45, normalized size = 1.12 \[ \frac{2 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} + \frac{2}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x - a)*x^(3/2)),x, algorithm="giac")

[Out]

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