3.849 \(\int \frac{1-x}{1+\sqrt{x}} \, dx\)

Optimal. Leaf size=11 \[ x-\frac{2 x^{3/2}}{3} \]

[Out]

x - (2*x^(3/2))/3

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Rubi [A]  time = 0.008115, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1398, 26, 43} \[ x-\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)/(1 + Sqrt[x]),x]

[Out]

x - (2*x^(3/2))/3

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.000415, size = 11, normalized size = 1. \[ x-\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)/(1 + Sqrt[x]),x]

[Out]

x - (2*x^(3/2))/3

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Maple [A]  time = 0.003, size = 8, normalized size = 0.7 \begin{align*} x-{\frac{2}{3}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-2/3*x^(3/2)

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Maxima [A]  time = 1.28941, size = 9, normalized size = 0.82 \begin{align*} -\frac{2}{3} \, x^{\frac{3}{2}} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(1+x^(1/2)),x, algorithm="maxima")

[Out]

-2/3*x^(3/2) + x

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Fricas [A]  time = 1.45331, size = 24, normalized size = 2.18 \begin{align*} -\frac{2}{3} \, x^{\frac{3}{2}} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x^(3/2) + x

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Sympy [A]  time = 0.13158, size = 8, normalized size = 0.73 \begin{align*} - \frac{2 x^{\frac{3}{2}}}{3} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(1+x**(1/2)),x)

[Out]

-2*x**(3/2)/3 + x

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Giac [A]  time = 1.16135, size = 9, normalized size = 0.82 \begin{align*} -\frac{2}{3} \, x^{\frac{3}{2}} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x^(3/2) + x