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

Optimal. Leaf size=15 \[ \frac {2}{3} (x+1)^{3/2}-x \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {371} \[ \frac {2}{3} (x+1)^{3/2}-x \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.27 \[ 2 \left (\frac {1}{3} (x+1)^{3/2}-\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + Sqrt[1 + x]),x]

[Out]

2*(-1/2*x + (1 + x)^(3/2)/3)

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IntegrateAlgebraic [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2}-x \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

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fricas [A]  time = 1.04, size = 11, normalized size = 0.73 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x

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giac [A]  time = 0.92, size = 12, normalized size = 0.80 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x - 1

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maple [A]  time = 0.02, size = 13, normalized size = 0.87

method	result	size
derivativedivides	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\)	\(13\)
default	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\)	\(13\)
trager	\(-x +\left (\frac {2}{3}+\frac {2 x}{3}\right ) \sqrt {1+x}\)	\(16\)
meijerg	\(\frac {-\frac {\sqrt {\pi }\, \left (12 x +8\right )}{6}+\frac {\sqrt {\pi }\, \left (8+8 x \right ) \sqrt {1+x}}{6}}{2 \sqrt {\pi }}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 12, normalized size = 0.80 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - x - 1

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mupad [B]  time = 0.03, size = 11, normalized size = 0.73 \[ \frac {2\,{\left (x+1\right )}^{3/2}}{3}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x + 1)^(1/2) + 1),x)

[Out]

(2*(x + 1)^(3/2))/3 - x

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sympy [B]  time = 0.97, size = 22, normalized size = 1.47 \[ \frac {2 x \sqrt {x + 1}}{3} - x + \frac {2 \sqrt {x + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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