3.1522 \(\int \frac{1}{\sqrt{a+b x} \sqrt{4+a+b x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

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Rubi [A]  time = 0.0061986, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {63, 215} \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

Rule 63

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 215

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*} \int \frac{1}{\sqrt{a+b x} \sqrt{4+a+b x}} \, dx &=\frac{2 \operatorname{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]  time = 0.0089983, size = 19, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

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Maple [B]  time = 0.008, size = 86, normalized size = 4.5 \begin{align*}{\sqrt{ \left ( bx+a \right ) \left ( bx+a+4 \right ) }\ln \left ({ \left ({\frac{ab}{2}}+{\frac{b \left ( a+4 \right ) }{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+b \left ( a+4 \right ) \right ) x+a \left ( a+4 \right ) } \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bx+a+4}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.10401, size = 76, normalized size = 4. \begin{align*} -\frac{\log \left (-b x + \sqrt{b x + a + 4} \sqrt{b x + a} - a - 2\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \sqrt{a + b x + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.24562, size = 34, normalized size = 1.79 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b x + a + 4} + \sqrt{b x + a} \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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