3.5 \(\int \frac{1}{\left (2 \sqrt{x}+\sqrt{1+x}\right )^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{4 \sqrt{x} \sqrt{x+1}}{3 (1-3 x)}+\frac{8}{9 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{8}{9} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{10}{9} \tanh ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{x+1}}\right ) \]

[Out]

8/(9*(1 - 3*x)) - (4*Sqrt[x]*Sqrt[1 + x])/(3*(1 - 3*x)) - (8*ArcSinh[Sqrt[x]])/9
+ (10*ArcTanh[(2*Sqrt[x])/Sqrt[1 + x]])/9 + (5*Log[1 - 3*x])/9

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Rubi [A]  time = 0.145885, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{4 \sqrt{x} \sqrt{x+1}}{3 (1-3 x)}+\frac{8}{9 (1-3 x)}+\frac{5}{9} \log (1-3 x)-\frac{8}{9} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{10}{9} \tanh ^{-1}\left (\frac{2 \sqrt{x}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*Sqrt[x] + Sqrt[1 + x])^(-2),x]

[Out]

8/(9*(1 - 3*x)) - (4*Sqrt[x]*Sqrt[1 + x])/(3*(1 - 3*x)) - (8*ArcSinh[Sqrt[x]])/9
+ (10*ArcTanh[(2*Sqrt[x])/Sqrt[1 + x]])/9 + (5*Log[1 - 3*x])/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \int ^{\sqrt{x + 1}} \frac{x}{\left (x + 2 \sqrt{x^{2} - 1}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**(1/2)+(1+x)**(1/2))**2,x)

[Out]

2*Integral(x/(x + 2*sqrt(x**2 - 1))**2, (x, sqrt(x + 1)))

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Mathematica [A]  time = 0.145594, size = 65, normalized size = 0.88 \[ \frac{1}{9} \left (\frac{-12 \sqrt{x} \sqrt{x+1}+(5-15 x) \log (1-3 x)+8}{1-3 x}-8 \sinh ^{-1}\left (\sqrt{x}\right )+10 \tanh ^{-1}\left (2 \sqrt{\frac{x}{x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*Sqrt[x] + Sqrt[1 + x])^(-2),x]

[Out]

(-8*ArcSinh[Sqrt[x]] + 10*ArcTanh[2*Sqrt[x/(1 + x)]] + (8 - 12*Sqrt[x]*Sqrt[1 +
x] + (5 - 15*x)*Log[1 - 3*x])/(1 - 3*x))/9

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Maple [B]  time = 0.023, size = 115, normalized size = 1.6 \[ -{\frac{8}{27\,x-9}}+{\frac{5\,\ln \left ( 3\,x-1 \right ) }{9}}-{\frac{1}{27\,x-9}\sqrt{x}\sqrt{1+x} \left ( 12\,\ln \left ( 1/2+x+\sqrt{ \left ( 1+x \right ) x} \right ) x-15\,{\it Artanh} \left ( 1/4\,{\frac{1+5\,x}{\sqrt{ \left ( 1+x \right ) x}}} \right ) x-4\,\ln \left ( 1/2+x+\sqrt{ \left ( 1+x \right ) x} \right ) +5\,{\it Artanh} \left ( 1/4\,{\frac{1+5\,x}{\sqrt{ \left ( 1+x \right ) x}}} \right ) -12\,\sqrt{ \left ( 1+x \right ) x} \right ){\frac{1}{\sqrt{ \left ( 1+x \right ) x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^(1/2)+(1+x)^(1/2))^2,x)

[Out]

-8/9/(3*x-1)+5/9*ln(3*x-1)-1/9*x^(1/2)*(1+x)^(1/2)*(12*ln(1/2+x+((1+x)*x)^(1/2))
*x-15*arctanh(1/4*(1+5*x)/((1+x)*x)^(1/2))*x-4*ln(1/2+x+((1+x)*x)^(1/2))+5*arcta
nh(1/4*(1+5*x)/((1+x)*x)^(1/2))-12*((1+x)*x)^(1/2))/((1+x)*x)^(1/2)/(3*x-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\sqrt{x + 1} + 2 \, \sqrt{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x + 1) + 2*sqrt(x))^(-2),x, algorithm="maxima")

[Out]

integrate((sqrt(x + 1) + 2*sqrt(x))^(-2), x)

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Fricas [A]  time = 0.224661, size = 274, normalized size = 3.7 \[ \frac{2 \,{\left (5 \,{\left (3 \, x - 1\right )} \log \left (3 \, x - 1\right ) - 18\right )} \sqrt{x + 1} \sqrt{x} - 5 \,{\left (2 \,{\left (3 \, x - 1\right )} \sqrt{x + 1} \sqrt{x} - 6 \, x^{2} - x + 1\right )} \log \left (3 \, \sqrt{x + 1} \sqrt{x} - 3 \, x - 1\right ) + 4 \,{\left (2 \,{\left (3 \, x - 1\right )} \sqrt{x + 1} \sqrt{x} - 6 \, x^{2} - x + 1\right )} \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) + 5 \,{\left (2 \,{\left (3 \, x - 1\right )} \sqrt{x + 1} \sqrt{x} - 6 \, x^{2} - x + 1\right )} \log \left (\sqrt{x + 1} \sqrt{x} - x + 1\right ) - 5 \,{\left (6 \, x^{2} + x - 1\right )} \log \left (3 \, x - 1\right ) + 36 \, x + 12}{9 \,{\left (2 \,{\left (3 \, x - 1\right )} \sqrt{x + 1} \sqrt{x} - 6 \, x^{2} - x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x + 1) + 2*sqrt(x))^(-2),x, algorithm="fricas")

[Out]

1/9*(2*(5*(3*x - 1)*log(3*x - 1) - 18)*sqrt(x + 1)*sqrt(x) - 5*(2*(3*x - 1)*sqrt
(x + 1)*sqrt(x) - 6*x^2 - x + 1)*log(3*sqrt(x + 1)*sqrt(x) - 3*x - 1) + 4*(2*(3*
x - 1)*sqrt(x + 1)*sqrt(x) - 6*x^2 - x + 1)*log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)
+ 5*(2*(3*x - 1)*sqrt(x + 1)*sqrt(x) - 6*x^2 - x + 1)*log(sqrt(x + 1)*sqrt(x) -
x + 1) - 5*(6*x^2 + x - 1)*log(3*x - 1) + 36*x + 12)/(2*(3*x - 1)*sqrt(x + 1)*s
qrt(x) - 6*x^2 - x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (2 \sqrt{x} + \sqrt{x + 1}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*x**(1/2)+(1+x)**(1/2))**2,x)

[Out]

Integral((2*sqrt(x) + sqrt(x + 1))**(-2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x + 1) + 2*sqrt(x))^(-2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError