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

Optimal. Leaf size=33 \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

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Rubi [A]  time = 0.0464142, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0255826, size = 33, normalized size = 1. \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

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Maple [B]  time = 0.016, size = 58, normalized size = 1.8 \[ -{\frac{1}{2}\sqrt{2+x}\sqrt{3+x} \left ( -2\,\sqrt{{x}^{2}+5\,x+6}+5\,\ln \left ( 5/2+x+\sqrt{{x}^{2}+5\,x+6} \right ) \right ){\frac{1}{\sqrt{{x}^{2}+5\,x+6}}}}+{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2+x)^(1/2)*(3+x)^(1/2)*(-2*(x^2+5*x+6)^(1/2)+5*ln(5/2+x+(x^2+5*x+6)^(1/2))
)/(x^2+5*x+6)^(1/2)+1/2*x^2

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Maxima [A]  time = 0.756415, size = 49, normalized size = 1.48 \[ \frac{1}{2} \, x^{2} + \sqrt{x^{2} + 5 \, x + 6} - \frac{5}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} + 5 \, x + 6} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + sqrt(x^2 + 5*x + 6) - 5/2*log(2*x + 2*sqrt(x^2 + 5*x + 6) + 5)

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Fricas [A]  time = 0.270809, size = 128, normalized size = 3.88 \[ -\frac{4 \, x^{3} - 2 \,{\left (2 \, x^{2} - 4 \, x - 5\right )} \sqrt{x + 3} \sqrt{x + 2} + 2 \, x^{2} - 10 \,{\left (2 \, \sqrt{x + 3} \sqrt{x + 2} - 2 \, x - 5\right )} \log \left (2 \, \sqrt{x + 3} \sqrt{x + 2} - 2 \, x - 5\right ) - 30 \, x - 23}{4 \,{\left (2 \, \sqrt{x + 3} \sqrt{x + 2} - 2 \, x - 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*x^3 - 2*(2*x^2 - 4*x - 5)*sqrt(x + 3)*sqrt(x + 2) + 2*x^2 - 10*(2*sqrt(x
+ 3)*sqrt(x + 2) - 2*x - 5)*log(2*sqrt(x + 3)*sqrt(x + 2) - 2*x - 5) - 30*x - 2
3)/(2*sqrt(x + 3)*sqrt(x + 2) - 2*x - 5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(sqrt(x + 2)*sqrt(x + 3) + 1)/(sqrt(x + 2)*sqrt(x + 3)), x)

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GIAC/XCAS [A]  time = 0.293439, size = 54, normalized size = 1.64 \[ \frac{1}{2} \,{\left (x + 3\right )}^{2} + \sqrt{x + 3} \sqrt{x + 2} - 3 \, x + 5 \,{\rm ln}\left ({\left | -\sqrt{x + 3} + \sqrt{x + 2} \right |}\right ) - 9 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x + 3)^2 + sqrt(x + 3)*sqrt(x + 2) - 3*x + 5*ln(abs(-sqrt(x + 3) + sqrt(x +
2))) - 9