3.4.30 \(\int \frac {1}{\sqrt {x}+\sqrt {1+x}} \, dx\) [330]

Optimal. Leaf size=21 \[ -\frac {2 x^{3/2}}{3}+\frac {2}{3} (1+x)^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2131, 30, 32} \begin {gather*} \frac {2}{3} (x+1)^{3/2}-\frac {2 x^{3/2}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x] + Sqrt[1 + x])^(-1),x]

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2131

Int[(u_.)/((d_.)*(x_)^(n_.) + (c_.)*Sqrt[(a_.) + (b_.)*(x_)^(p_.)]), x_Symbol] :> Dist[-b/(a*d), Int[u*x^n, x]
, x] + Dist[1/(a*c), Int[u*Sqrt[a + b*x^(2*n)], x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^
2 - d^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 x^{3/2}}{3}+\frac {2}{3} (1+x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x] + Sqrt[1 + x])^(-1),x]

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

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Mathics [A]
time = 2.35, size = 27, normalized size = 1.29 \begin {gather*} \frac {2 \left (1+\sqrt {x} \sqrt {1+x}+2 x\right )}{3 \sqrt {x}+3 \sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(Sqrt[x] + Sqrt[1 + x]),x]')

[Out]

2 (1 + Sqrt[x] Sqrt[1 + x] + 2 x) / (3 (Sqrt[x] + Sqrt[1 + x]))

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Maple [A]
time = 0.01, size = 14, normalized size = 0.67

method	result	size
default	\(-\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(14\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }\, x^{\frac {3}{2}}}{3}-\frac {2 \sqrt {\pi }\, x^{\frac {3}{2}} \left (2+\frac {2}{x}\right ) \sqrt {1+\frac {1}{x}}}{3}}{2 \sqrt {\pi }}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + 1) + sqrt(x)), x)

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Fricas [A]
time = 0.32, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\)
time = 0.17, size = 63, normalized size = 3.00 \begin {gather*} \frac {2 \sqrt {x} \sqrt {x + 1}}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {4 x}{3 \sqrt {x} + 3 \sqrt {x + 1}} + \frac {2}{3 \sqrt {x} + 3 \sqrt {x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 38, normalized size = 1.81 \begin {gather*} 2 \left (-\frac {1}{3} \sqrt {x} x+2 \left (\frac {1}{6} \sqrt {x} \sqrt {x}+\frac 1{6}\right ) \sqrt {x+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)

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Mupad [B]
time = 0.20, size = 21, normalized size = 1.00 \begin {gather*} \frac {2\,x\,\sqrt {x+1}}{3}+\frac {2\,\sqrt {x+1}}{3}-\frac {2\,x^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)^(1/2) + x^(1/2)),x)

[Out]

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

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