3.253 \(\int (1+\sqrt{x}) \sqrt{x} \, dx\)

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

[Out]

(2*x^(3/2))/3 + x^2/2

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Rubi [A]  time = 0.0030238, 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, Rules used = {14} \[ \frac{x^2}{2}+\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])*Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + x^2/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (1+\sqrt{x}\right ) \sqrt{x} \, dx &=\int \left (\sqrt{x}+x\right ) \, dx\\ &=\frac{2 x^{3/2}}{3}+\frac{x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.0024605, size = 17, normalized size = 1. \[ \frac{x^2}{2}+\frac{2 x^{3/2}}{3} \]

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 = 0.001, size = 12, normalized size = 0.7 \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(x^(1/2)+1),x)

[Out]

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

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Maxima [B]  time = 0.926366, size = 35, normalized size = 2.06 \begin{align*} \frac{1}{2} \,{\left (\sqrt{x} + 1\right )}^{4} - \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{3} +{\left (\sqrt{x} + 1\right )}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(1+x^(1/2)),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 = 2.09504, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{2} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(1+x^(1/2)),x, algorithm="fricas")

[Out]

1/2*x^2 + 2/3*x^(3/2)

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Sympy [A]  time = 0.120932, size = 12, normalized size = 0.71 \begin{align*} \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2} \end{align*}

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 [A]  time = 1.04057, size = 15, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x^{2} + \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(1+x^(1/2)),x, algorithm="giac")

[Out]

1/2*x^2 + 2/3*x^(3/2)