3.649 \(\int \frac{\left (2 \sqrt{3-x}+\frac{3}{\sqrt{1+x}}\right )^2}{x} \, dx\)

Optimal. Leaf size=56 \[ -4 x+21 \log (x)-9 \log (x+1)+12 \sin ^{-1}\left (\frac{1-x}{2}\right )-24 \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{x+1}}{\sqrt{3-x}}\right ) \]

[Out]

-4*x + 12*ArcSin[(1 - x)/2] - 24*Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[1 + x])/Sqrt[3 -
x]] + 21*Log[x] - 9*Log[1 + x]

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Rubi [A]  time = 0.408798, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37 \[ -4 x+21 \log (x)-9 \log (x+1)+12 \sin ^{-1}\left (\frac{1-x}{2}\right )-24 \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{x+1}}{\sqrt{3-x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*Sqrt[3 - x] + 3/Sqrt[1 + x])^2/x,x]

[Out]

-4*x + 12*ArcSin[(1 - x)/2] - 24*Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[1 + x])/Sqrt[3 -
x]] + 21*Log[x] - 9*Log[1 + x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.104082, size = 69, normalized size = 1.23 \[ -12 \sqrt{3} \log \left (\sqrt{-3 x^2+6 x+9}+x+3\right )-4 x+3 \left (7+4 \sqrt{3}\right ) \log (x)-9 \log (x+1)+12 \tan ^{-1}\left (\frac{1-x}{\sqrt{-(x-3) (x+1)}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-4*x + 12*ArcTan[(1 - x)/Sqrt[-((-3 + x)*(1 + x))]] + 3*(7 + 4*Sqrt[3])*Log[x] -
9*Log[1 + x] - 12*Sqrt[3]*Log[3 + x + Sqrt[9 + 6*x - 3*x^2]]

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Maple [A]  time = 0.025, size = 76, normalized size = 1.4 \[ -9\,\ln \left ( 1+x \right ) +21\,\ln \left ( x \right ) +12\,{\frac{\sqrt{3-x}\sqrt{1+x}}{\sqrt{-{x}^{2}+2\,x+3}} \left ( -\arcsin \left ( -1/2+x/2 \right ) -\sqrt{3}{\it Artanh} \left ( 1/3\,{\frac{ \left ( 3+x \right ) \sqrt{3}}{\sqrt{-{x}^{2}+2\,x+3}}} \right ) \right ) }-4\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x)

[Out]

-9*ln(1+x)+21*ln(x)+12*(3-x)^(1/2)*(1+x)^(1/2)/(-x^2+2*x+3)^(1/2)*(-arcsin(-1/2+
1/2*x)-3^(1/2)*arctanh(1/3*(3+x)*3^(1/2)/(-x^2+2*x+3)^(1/2)))-4*x

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Maxima [A]  time = 0.843976, size = 77, normalized size = 1.38 \[ -12 \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{-x^{2} + 2 \, x + 3}}{{\left | x \right |}} + \frac{6}{{\left | x \right |}} + 2\right ) - 4 \, x + 12 \, \arcsin \left (-\frac{1}{2} \, x + \frac{1}{2}\right ) - 9 \, \log \left (x + 1\right ) + 21 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*sqrt(-x + 3) + 3/sqrt(x + 1))^2/x,x, algorithm="maxima")

[Out]

-12*sqrt(3)*log(2*sqrt(3)*sqrt(-x^2 + 2*x + 3)/abs(x) + 6/abs(x) + 2) - 4*x + 12
*arcsin(-1/2*x + 1/2) - 9*log(x + 1) + 21*log(x)

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Fricas [A]  time = 0.283814, size = 96, normalized size = 1.71 \[ 6 \, \sqrt{3} \log \left (-\frac{\sqrt{3}{\left (x + 3\right )} \sqrt{x + 1} \sqrt{-x + 3} + x^{2} - 6 \, x - 9}{x^{2}}\right ) - 4 \, x - 12 \, \arctan \left (\frac{x - 1}{\sqrt{x + 1} \sqrt{-x + 3}}\right ) - 9 \, \log \left (x + 1\right ) + 21 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(3)*log(-(sqrt(3)*(x + 3)*sqrt(x + 1)*sqrt(-x + 3) + x^2 - 6*x - 9)/x^2) -
4*x - 12*arctan((x - 1)/(sqrt(x + 1)*sqrt(-x + 3))) - 9*log(x + 1) + 21*log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (2 \sqrt{- x + 3} \sqrt{x + 1} + 3\right )^{2}}{x \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*(3-x)**(1/2)+3/(1+x)**(1/2))**2/x,x)

[Out]

Integral((2*sqrt(-x + 3)*sqrt(x + 1) + 3)**2/(x*(x + 1)), 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((2*sqrt(-x + 3) + 3/sqrt(x + 1))^2/x,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError