3.1.36 \(\int \sqrt {\frac {1+x}{3+2 x}} \, dx\) [36]

Optimal. Leaf size=44 \[ \frac {1}{2} \sqrt {1+x} \sqrt {3+2 x}-\frac {\sinh ^{-1}\left (\sqrt {2} \sqrt {1+x}\right )}{2 \sqrt {2}} \]

[Out]

-1/4*arcsinh(2^(1/2)*(1+x)^(1/2))*2^(1/2)+1/2*(1+x)^(1/2)*(3+2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1978, 52, 56, 221} \begin {gather*} \frac {1}{2} \sqrt {x+1} \sqrt {2 x+3}-\frac {\sinh ^{-1}\left (\sqrt {2} \sqrt {x+1}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x)/(3 + 2*x)],x]

[Out]

(Sqrt[1 + x]*Sqrt[3 + 2*x])/2 - ArcSinh[Sqrt[2]*Sqrt[1 + x]]/(2*Sqrt[2])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

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

Rule 221

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

Rule 1978

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

Rubi steps

\begin {align*} \int \sqrt {\frac {1+x}{3+2 x}} \, dx &=\int \frac {\sqrt {1+x}}{\sqrt {3+2 x}} \, dx\\ &=\frac {1}{2} \sqrt {1+x} \sqrt {3+2 x}-\frac {1}{4} \int \frac {1}{\sqrt {1+x} \sqrt {3+2 x}} \, dx\\ &=\frac {1}{2} \sqrt {1+x} \sqrt {3+2 x}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+2 x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} \sqrt {1+x} \sqrt {3+2 x}-\frac {\sinh ^{-1}\left (\sqrt {2} \sqrt {1+x}\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 75, normalized size = 1.70 \begin {gather*} \frac {\sqrt {\frac {1+x}{3+2 x}} \left (\sqrt {1+x} (3+2 x)-\sqrt {6+4 x} \tanh ^{-1}\left (\frac {\sqrt {2+2 x}}{-1+\sqrt {3+2 x}}\right )\right )}{2 \sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x)/(3 + 2*x)],x]

[Out]

(Sqrt[(1 + x)/(3 + 2*x)]*(Sqrt[1 + x]*(3 + 2*x) - Sqrt[6 + 4*x]*ArcTanh[Sqrt[2 + 2*x]/(-1 + Sqrt[3 + 2*x])]))/
(2*Sqrt[1 + x])

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[(x + 1)/(2*x + 3)],x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(30)=60\).
time = 0.08, size = 75, normalized size = 1.70

method	result	size
default	\(-\frac {\sqrt {\frac {1+x}{3+2 x}}\, \left (3+2 x \right ) \left (\ln \left (\frac {5 \sqrt {2}}{4}+x \sqrt {2}+\sqrt {2 x^{2}+5 x +3}\right ) \sqrt {2}-4 \sqrt {2 x^{2}+5 x +3}\right )}{8 \sqrt {\left (3+2 x \right ) \left (1+x \right )}}\)	\(75\)
risch	\(\frac {\left (3+2 x \right ) \sqrt {\frac {1+x}{3+2 x}}}{2}-\frac {\ln \left (\frac {\left (\frac {5}{2}+2 x \right ) \sqrt {2}}{2}+\sqrt {2 x^{2}+5 x +3}\right ) \sqrt {2}\, \sqrt {\frac {1+x}{3+2 x}}\, \sqrt {\left (3+2 x \right ) \left (1+x \right )}}{8 \left (1+x \right )}\)	\(80\)
trager	\(3 \left (\frac {1}{2}+\frac {x}{3}\right ) \sqrt {-\frac {-1-x}{3+2 x}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (8 \sqrt {-\frac {-1-x}{3+2 x}}\, x +4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +12 \sqrt {-\frac {-1-x}{3+2 x}}+5 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{8}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1+x)/(3+2*x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/8*((1+x)/(3+2*x))^(1/2)*(3+2*x)*(ln(5/4*2^(1/2)+x*2^(1/2)+(2*x^2+5*x+3)^(1/2))*2^(1/2)-4*(2*x^2+5*x+3)^(1/2
))/((3+2*x)*(1+x))^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).
time = 0.33, size = 80, normalized size = 1.82 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \sqrt {\frac {x + 1}{2 \, x + 3}}}{\sqrt {2} + 2 \, \sqrt {\frac {x + 1}{2 \, x + 3}}}\right ) - \frac {\sqrt {\frac {x + 1}{2 \, x + 3}}}{2 \, {\left (\frac {2 \, {\left (x + 1\right )}}{2 \, x + 3} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*log(-(sqrt(2) - 2*sqrt((x + 1)/(2*x + 3)))/(sqrt(2) + 2*sqrt((x + 1)/(2*x + 3)))) - 1/2*sqrt((x +
1)/(2*x + 3))/(2*(x + 1)/(2*x + 3) - 1)

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Fricas [A]
time = 0.30, size = 55, normalized size = 1.25 \begin {gather*} \frac {1}{2} \, {\left (2 \, x + 3\right )} \sqrt {\frac {x + 1}{2 \, x + 3}} + \frac {1}{8} \, \sqrt {2} \log \left (2 \, \sqrt {2} {\left (2 \, x + 3\right )} \sqrt {\frac {x + 1}{2 \, x + 3}} - 4 \, x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*x + 3)*sqrt((x + 1)/(2*x + 3)) + 1/8*sqrt(2)*log(2*sqrt(2)*(2*x + 3)*sqrt((x + 1)/(2*x + 3)) - 4*x - 5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {x + 1}{2 x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))**(1/2),x)

[Out]

Integral(sqrt((x + 1)/(2*x + 3)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
time = 0.01, size = 74, normalized size = 1.68 \begin {gather*} \frac {1}{2} \sqrt {2 x^{2}+5 x+3} \mathrm {sign}\left (2 x+3\right )+\frac {\mathrm {sign}\left (2 x+3\right ) \ln \left |2 \sqrt {2} \left (-\sqrt {2} x+\sqrt {2 x^{2}+5 x+3}\right )-5\right |}{4 \sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(3+2*x))^(1/2),x)

[Out]

1/8*sqrt(2)*log(abs(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 5*x + 3)) - 5))*sgn(2*x + 3) + 1/2*sqrt(2*x^2 + 5*x +
3)*sgn(2*x + 3)

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Mupad [B]
time = 0.21, size = 57, normalized size = 1.30 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sqrt {\frac {x+1}{2\,x+3}}\right )}{4}-\frac {\sqrt {\frac {x+1}{2\,x+3}}}{2\,\left (\frac {2\,x+2}{2\,x+3}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)/(2*x + 3))^(1/2),x)

[Out]

- (2^(1/2)*atanh(2^(1/2)*((x + 1)/(2*x + 3))^(1/2)))/4 - ((x + 1)/(2*x + 3))^(1/2)/(2*((2*x + 2)/(2*x + 3) - 1
))

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