\(\int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx\) [1522]

Optimal result

Integrand size = 20, antiderivative size = 19 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {65, 221} \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \]

[In]

Int[1/(Sqrt[a + b*x]*Sqrt[4 + a + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[a + 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 {a+b x}\right )}{b} \\ & = \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {4+a+b x}}{\sqrt {a+b x}}\right )}{b} \]

[In]

Integrate[1/(Sqrt[a + b*x]*Sqrt[4 + a + b*x]),x]

[Out]

(2*ArcTanh[Sqrt[4 + a + b*x]/Sqrt[a + b*x]])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(15)=30\).

Time = 0.65 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.53

method	result	size
default	\(\frac {\sqrt {\left (b x +a \right ) \left (b x +a +4\right )}\, \ln \left (\frac {\frac {a b}{2}+\frac {b \left (a +4\right )}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (a b +b \left (a +4\right )\right ) x +a \left (a +4\right )}\right )}{\sqrt {b x +a}\, \sqrt {b x +a +4}\, \sqrt {b^{2}}}\)	\(86\)

[In]

int(1/(b*x+a)^(1/2)/(b*x+a+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*x+a)*(b*x+a+4))^(1/2)/(b*x+a)^(1/2)/(b*x+a+4)^(1/2)*ln((1/2*a*b+1/2*b*(a+4)+b^2*x)/(b^2)^(1/2)+(b^2*x^2+(a
*b+b*(a+4))*x+a*(a+4))^(1/2))/(b^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=-\frac {\log \left (-b x + \sqrt {b x + a + 4} \sqrt {b x + a} - a - 2\right )}{b} \]

[In]

integrate(1/(b*x+a)^(1/2)/(b*x+a+4)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\int \frac {1}{\sqrt {a + b x} \sqrt {a + b x + 4}}\, dx \]

[In]

integrate(1/(b*x+a)**(1/2)/(b*x+a+4)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x)*sqrt(a + b*x + 4)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + a^{2} + 2 \, {\left (a b + 2 \, b\right )} x + 4 \, a} b + 4 \, b\right )}{b} \]

[In]

integrate(1/(b*x+a)^(1/2)/(b*x+a+4)^(1/2),x, algorithm="maxima")

[Out]

log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + a^2 + 2*(a*b + 2*b)*x + 4*a)*b + 4*b)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=-\frac {2 \, \log \left (\sqrt {b x + a + 4} - \sqrt {b x + a}\right )}{b} \]

[In]

integrate(1/(b*x+a)^(1/2)/(b*x+a+4)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx=\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a+4}-\sqrt {a+b\,x+4}\right )}{\sqrt {-b^2}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b^2}} \]

[In]

int(1/((a + b*x)^(1/2)*(a + b*x + 4)^(1/2)),x)

[Out]

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