3.10.45 \(\int \frac {1+2 x}{\sqrt {x+x^2}} \, dx\) [945]

Optimal. Leaf size=11 \[ 2 \sqrt {x+x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {643} \begin {gather*} 2 \sqrt {x^2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)/Sqrt[x + x^2],x]

[Out]

2*Sqrt[x + x^2]

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1+2 x}{\sqrt {x+x^2}} \, dx &=2 \sqrt {x+x^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} 2 \sqrt {x (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)/Sqrt[x + x^2],x]

[Out]

2*Sqrt[x*(1 + x)]

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Maple [A]
time = 0.22, size = 10, normalized size = 0.91

method	result	size
derivativedivides	\(2 \sqrt {x^{2}+x}\)	\(10\)
default	\(2 \sqrt {x^{2}+x}\)	\(10\)
trager	\(2 \sqrt {x^{2}+x}\)	\(10\)
gosper	\(\frac {2 x \left (1+x \right )}{\sqrt {x^{2}+x}}\)	\(14\)
risch	\(\frac {2 x \left (1+x \right )}{\sqrt {x \left (1+x \right )}}\)	\(14\)
meijerg	\(\frac {2 \sqrt {\pi }\, \sqrt {x}\, \sqrt {1+x}-2 \sqrt {\pi }\, \arcsinh \left (\sqrt {x}\right )}{\sqrt {\pi }}+2 \arcsinh \left (\sqrt {x}\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 9, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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Fricas [A]
time = 0.37, size = 9, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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Sympy [A]
time = 0.06, size = 8, normalized size = 0.73 \begin {gather*} 2 \sqrt {x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x**2 + x)

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Giac [A]
time = 5.64, size = 9, normalized size = 0.82 \begin {gather*} 2 \, \sqrt {x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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Mupad [B]
time = 3.52, size = 9, normalized size = 0.82 \begin {gather*} 2\,\sqrt {x\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)/(x + x^2)^(1/2),x)

[Out]

2*(x*(x + 1))^(1/2)

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