3.9.49 \(\int \frac {1-x}{1+\sqrt {x}} \, dx\) [849]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1412, 26, 45} \begin {gather*} x-\frac {2 x^{3/2}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)/(1 + Sqrt[x]),x]

[Out]

x - (2*x^(3/2))/3

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 45

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])

Rule 1412

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} x-\frac {2 x^{3/2}}{3} \end {gather*}

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 = 0.52, size = 8, normalized size = 0.73

method	result	size
derivativedivides	\(x -\frac {2 x^{\frac {3}{2}}}{3}\)	\(8\)
default	\(x -\frac {2 x^{\frac {3}{2}}}{3}\)	\(8\)
trager	\(-1+x -\frac {2 x^{\frac {3}{2}}}{3}\)	\(9\)
meijerg	\(2 \sqrt {x}-\frac {\sqrt {x}\, \left (4 x -6 \sqrt {x}+12\right )}{6}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)/(1+x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 7, normalized size = 0.64 \begin {gather*} -\frac {2}{3} \, x^{\frac {3}{2}} + x \end {gather*}

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 = 0.37, size = 7, normalized size = 0.64 \begin {gather*} -\frac {2}{3} \, x^{\frac {3}{2}} + x \end {gather*}

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 = 0.05, size = 8, normalized size = 0.73 \begin {gather*} - \frac {2 x^{\frac {3}{2}}}{3} + x \end {gather*}

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 = 2.50, size = 7, normalized size = 0.64 \begin {gather*} -\frac {2}{3} \, x^{\frac {3}{2}} + x \end {gather*}

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

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Mupad [B]
time = 0.03, size = 7, normalized size = 0.64 \begin {gather*} x-\frac {2\,x^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 1)/(x^(1/2) + 1),x)

[Out]

x - (2*x^(3/2))/3

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