\(\int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx\) [1523]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {1}{2} \sqrt {2+b x}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, 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 = {65, 221} \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {1}{2} \sqrt {b x+2}\right )}{b} \]

[In]

Int[1/(Sqrt[2 + b*x]*Sqrt[6 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]/2])/b

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {6+b x}}{\sqrt {2+b x}}\right )}{b} \]

[In]

Integrate[1/(Sqrt[2 + b*x]*Sqrt[6 + b*x]),x]

[Out]

(2*ArcTanh[Sqrt[6 + b*x]/Sqrt[2 + b*x]])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(15)=30\).

Time = 0.52 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.47

method	result	size
default	\(\frac {\sqrt {\left (b x +2\right ) \left (b x +6\right )}\, \ln \left (\frac {b^{2} x +4 b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+8 b x +12}\right )}{\sqrt {b x +2}\, \sqrt {b x +6}\, \sqrt {b^{2}}}\)	\(66\)

[In]

int(1/(b*x+2)^(1/2)/(b*x+6)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*x+2)*(b*x+6))^(1/2)/(b*x+2)^(1/2)/(b*x+6)^(1/2)*ln((b^2*x+4*b)/(b^2)^(1/2)+(b^2*x^2+8*b*x+12)^(1/2))/(b^2)
^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=-\frac {\log \left (-b x + \sqrt {b x + 6} \sqrt {b x + 2} - 4\right )}{b} \]

[In]

integrate(1/(b*x+2)^(1/2)/(b*x+6)^(1/2),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + 6)*sqrt(b*x + 2) - 4)/b

Sympy [F]

\[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=\int \frac {1}{\sqrt {b x + 2} \sqrt {b x + 6}}\, dx \]

[In]

integrate(1/(b*x+2)**(1/2)/(b*x+6)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x + 2)*sqrt(b*x + 6)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 8 \, b x + 12} b + 8 \, b\right )}{b} \]

[In]

integrate(1/(b*x+2)^(1/2)/(b*x+6)^(1/2),x, algorithm="maxima")

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 + 8*b*x + 12)*b + 8*b)/b

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=-\frac {2 \, \log \left (\sqrt {b x + 6} - \sqrt {b x + 2}\right )}{b} \]

[In]

integrate(1/(b*x+2)^(1/2)/(b*x+6)^(1/2),x, algorithm="giac")

[Out]

-2*log(sqrt(b*x + 6) - sqrt(b*x + 2))/b

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.47 \[ \int \frac {1}{\sqrt {2+b x} \sqrt {6+b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {6}-\sqrt {b\,x+6}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]

[In]

int(1/((b*x + 2)^(1/2)*(b*x + 6)^(1/2)),x)

[Out]

-(4*atan((b*(6^(1/2) - (b*x + 6)^(1/2)))/((2^(1/2) - (b*x + 2)^(1/2))*(-b^2)^(1/2))))/(-b^2)^(1/2)