\(\int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx\) [846]

Optimal result

Integrand size = 19, antiderivative size = 16 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=\text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 221} \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=\text {arcsinh}\left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ) \]

[In]

Int[1/(Sqrt[1 + 2*x]*Sqrt[3 + 2*x]),x]

[Out]

ArcSinh[Sqrt[1 + 2*x]/Sqrt[2]]

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=\text {arctanh}\left (\frac {\sqrt {3+2 x}}{\sqrt {1+2 x}}\right ) \]

[In]

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

[Out]

ArcTanh[Sqrt[3 + 2*x]/Sqrt[1 + 2*x]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(13)=26\).

Time = 1.45 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.56

method	result	size
default	\(\frac {\sqrt {\left (1+2 x \right ) \left (2 x +3\right )}\, \ln \left (\frac {\left (4+4 x \right ) \sqrt {4}}{4}+\sqrt {4 x^{2}+8 x +3}\right ) \sqrt {4}}{4 \sqrt {1+2 x}\, \sqrt {2 x +3}}\)	\(57\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=-\frac {1}{2} \, \log \left (\sqrt {2 \, x + 3} \sqrt {2 \, x + 1} - 2 \, x - 2\right ) \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=\begin {cases} \operatorname {acosh}{\left (\sqrt {x + \frac {3}{2}} \right )} & \text {for}\: \left |{x + \frac {3}{2}}\right | > 1 \\- i \operatorname {asin}{\left (\sqrt {x + \frac {3}{2}} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((acosh(sqrt(x + 3/2)), Abs(x + 3/2) > 1), (-I*asin(sqrt(x + 3/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=\frac {1}{2} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 8 \, x + 3} + 8\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=-\log \left (\sqrt {2 \, x + 3} - \sqrt {2 \, x + 1}\right ) \]

[In]

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

[Out]

-log(sqrt(2*x + 3) - sqrt(2*x + 1))

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx=-2\,\mathrm {atanh}\left (\frac {\sqrt {3}-\sqrt {2\,x+3}}{\sqrt {2\,x+1}-1}\right ) \]

[In]

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

[Out]

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