Optimal. Leaf size=33 \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0575947, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]
Antiderivative was successfully verified.
[In] Int[(-1 + Sqrt[a + b*x])/(1 + Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 \sqrt{a + b x}}{b} + \frac{4 \log{\left (\sqrt{a + b x} + 1 \right )}}{b} + \frac{2 \int ^{\sqrt{a + b x}} x\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+(b*x+a)**(1/2))/(1+(b*x+a)**(1/2)),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0221819, size = 35, normalized size = 1.06 \[ \frac{-4 \sqrt{a+b x}+4 \log \left (\sqrt{a+b x}+1\right )+a+b x-5}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + Sqrt[a + b*x])/(1 + Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 35, normalized size = 1.1 \[ x+{\frac{a}{b}}-4\,{\frac{\sqrt{bx+a}}{b}}+4\,{\frac{\ln \left ( 1+\sqrt{bx+a} \right ) }{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+(b*x+a)^(1/2))/(1+(b*x+a)^(1/2)),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.694523, size = 41, normalized size = 1.24 \[ \frac{b x + a - 4 \, \sqrt{b x + a} + 4 \, \log \left (\sqrt{b x + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(b*x + a) - 1)/(sqrt(b*x + a) + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.265543, size = 39, normalized size = 1.18 \[ \frac{b x - 4 \, \sqrt{b x + a} + 4 \, \log \left (\sqrt{b x + a} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(b*x + a) - 1)/(sqrt(b*x + a) + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.67776, size = 42, normalized size = 1.27 \[ \begin{cases} x - \frac{4 \sqrt{a + b x}}{b} + \frac{4 \log{\left (\sqrt{a + b x} + 1 \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (\sqrt{a} - 1\right )}{\sqrt{a} + 1} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+(b*x+a)**(1/2))/(1+(b*x+a)**(1/2)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.275703, size = 51, normalized size = 1.55 \[ \frac{4 \,{\rm ln}\left (\sqrt{b x + a} + 1\right )}{b} + \frac{{\left (b x + a\right )} b - 4 \, \sqrt{b x + a} b}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(b*x + a) - 1)/(sqrt(b*x + a) + 1),x, algorithm="giac")
[Out]