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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]