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

Optimal. Leaf size=33 \[ -\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left (\sqrt{a+b x}+1\right )}{b}+x \]

[Out]

x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b

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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]

x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b

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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]

-4*sqrt(a + b*x)/b + 4*log(sqrt(a + b*x) + 1)/b + 2*Integral(x, (x, sqrt(a + b*x
)))/b

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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]

(-5 + a + b*x - 4*Sqrt[a + b*x] + 4*Log[1 + Sqrt[a + b*x]])/b

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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]

x+a/b-4*(b*x+a)^(1/2)/b+4*ln(1+(b*x+a)^(1/2))/b

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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]

(b*x + a - 4*sqrt(b*x + a) + 4*log(sqrt(b*x + a) + 1))/b

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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]

(b*x - 4*sqrt(b*x + a) + 4*log(sqrt(b*x + a) + 1))/b

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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]

Piecewise((x - 4*sqrt(a + b*x)/b + 4*log(sqrt(a + b*x) + 1)/b, Ne(b, 0)), (x*(sq
rt(a) - 1)/(sqrt(a) + 1), True))

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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]

4*ln(sqrt(b*x + a) + 1)/b + ((b*x + a)*b - 4*sqrt(b*x + a)*b)/b^2