3.4.9 \(\int \frac {\sqrt {2 x+x^2}}{1+x} \, dx\) [309]

Optimal. Leaf size=26 \[ \sqrt {2 x+x^2}-\tan ^{-1}\left (\sqrt {2 x+x^2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {699, 702, 209} \begin {gather*} \sqrt {x^2+2 x}-\text {ArcTan}\left (\sqrt {x^2+2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[2*x + x^2]/(1 + x),x]

[Out]

Sqrt[2*x + x^2] - ArcTan[Sqrt[2*x + x^2]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 699

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 702

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {2 x+x^2}}{1+x} \, dx &=\sqrt {2 x+x^2}-\int \frac {1}{(1+x) \sqrt {2 x+x^2}} \, dx\\ &=\sqrt {2 x+x^2}-4 \text {Subst}\left (\int \frac {1}{4+4 x^2} \, dx,x,\sqrt {2 x+x^2}\right )\\ &=\sqrt {2 x+x^2}-\tan ^{-1}\left (\sqrt {2 x+x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 44, normalized size = 1.69 \begin {gather*} \sqrt {x (2+x)} \left (1+\frac {2 \tan ^{-1}\left (1+x-\sqrt {x} \sqrt {2+x}\right )}{\sqrt {x} \sqrt {2+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2*x + x^2]/(1 + x),x]

[Out]

Sqrt[x*(2 + x)]*(1 + (2*ArcTan[1 + x - Sqrt[x]*Sqrt[2 + x]])/(Sqrt[x]*Sqrt[2 + x]))

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Maple [A]
time = 0.43, size = 21, normalized size = 0.81

method	result	size
default	\(\sqrt {\left (x +1\right )^{2}-1}+\arctan \left (\frac {1}{\sqrt {\left (x +1\right )^{2}-1}}\right )\)	\(21\)
risch	\(\frac {x \left (2+x \right )}{\sqrt {x \left (2+x \right )}}+\arctan \left (\frac {1}{\sqrt {\left (x +1\right )^{2}-1}}\right )\)	\(24\)
trager	\(\sqrt {x^{2}+2 x}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{2}+2 x}}{x +1}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x+1)^2-1)^(1/2)+arctan(1/((x+1)^2-1)^(1/2))

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Maxima [A]
time = 0.50, size = 17, normalized size = 0.65 \begin {gather*} \sqrt {x^{2} + 2 \, x} + \arcsin \left (\frac {1}{{\left | x + 1 \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 2*x) + arcsin(1/abs(x + 1))

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Fricas [A]
time = 1.93, size = 27, normalized size = 1.04 \begin {gather*} \sqrt {x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 2*x) - 2*arctan(-x + sqrt(x^2 + 2*x) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x + 2\right )}}{x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(x + 2))/(x + 1), x)

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Giac [A]
time = 1.65, size = 27, normalized size = 1.04 \begin {gather*} \sqrt {x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 2*x) - 2*arctan(-x + sqrt(x^2 + 2*x) - 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x^2+2\,x}}{x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2)^(1/2)/(x + 1),x)

[Out]

int((2*x + x^2)^(1/2)/(x + 1), x)

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