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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {63, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 28, normalized size = 1.47 \begin {gather*} -\frac {2 \log \left (\sqrt {a+b x+4}-\sqrt {a+b x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.05, size = 31, normalized size = 1.63 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + a + 4} \sqrt {b x + a} - a - 2\right )}{b} \end {gather*}

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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giac [A]  time = 1.02, size = 24, normalized size = 1.26 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + a + 4} - \sqrt {b x + a}\right )}{b} \end {gather*}

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(sqrt(b*x + a + 4) - sqrt(b*x + a))/b

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maple [B]  time = 0.01, size = 86, normalized size = 4.53 \begin {gather*} \frac {\sqrt {\left (b x +a \right ) \left (b x +a +4\right )}\, \ln \left (\frac {b^{2} x +\frac {a b}{2}+\frac {\left (a +4\right ) b}{2}}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (a +4\right ) a +\left (a b +\left (a +4\right ) b \right ) x}\right )}{\sqrt {b x +a}\, \sqrt {b x +a +4}\, \sqrt {b^{2}}} \end {gather*}

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 [B]  time = 1.36, size = 48, normalized size = 2.53 \begin {gather*} \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} \end {gather*}

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]

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

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mupad [B]  time = 0.31, size = 50, normalized size = 2.63 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x} \sqrt {a + b x + 4}}\, dx \end {gather*}

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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