3.806 \(\int \left (1+\sqrt{x}\right ) \sqrt{x} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 x^{3/2}}{3}+\frac{x^2}{2} \]

[Out]

(2*x^(3/2))/3 + x^2/2

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Rubi [A]  time = 0.00896272, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 x^{3/2}}{3}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[x])*Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + x^2/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 x^{\frac{3}{2}}}{3} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/3 + Integral(x, x)

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Mathematica [A]  time = 0.00372876, size = 17, normalized size = 1. \[ \frac{2 x^{3/2}}{3}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2))/3 + x^2/2

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Maple [A]  time = 0.002, size = 12, normalized size = 0.7 \[{\frac{2}{3}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(1+x^(1/2)),x)

[Out]

2/3*x^(3/2)+1/2*x^2

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Maxima [A]  time = 0.706601, size = 35, normalized size = 2.06 \[ \frac{1}{2} \,{\left (\sqrt{x} + 1\right )}^{4} - \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{3} +{\left (\sqrt{x} + 1\right )}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(x) + 1)^4 - 4/3*(sqrt(x) + 1)^3 + (sqrt(x) + 1)^2

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Fricas [A]  time = 0.26268, size = 15, normalized size = 0.88 \[ \frac{1}{2} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2/3*x^(3/2)

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Sympy [A]  time = 0.277931, size = 12, normalized size = 0.71 \[ \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/3 + x**2/2

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GIAC/XCAS [A]  time = 0.261648, size = 15, normalized size = 0.88 \[ \frac{1}{2} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2/3*x^(3/2)