3.15.18 \(\int \frac {1}{\sqrt {1+b x} \sqrt {5+b x}} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {63, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {b x+1}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + b*x]*Sqrt[5 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[1 + 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 {1+b x} \sqrt {5+b x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,\sqrt {1+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {1+b x}\right )}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 39, normalized size = 2.05 \begin {gather*} \frac {2 \sqrt {b x+1} \sin ^{-1}\left (\frac {1}{2} \sqrt {-b x-1}\right )}{b \sqrt {-b x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + b*x]*Sqrt[5 + b*x]),x]

[Out]

(2*Sqrt[1 + b*x]*ArcSin[Sqrt[-1 - b*x]/2])/(b*Sqrt[-1 - b*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+5}}{\sqrt {b x+1}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[1 + b*x]*Sqrt[5 + b*x]),x]

[Out]

(2*ArcTanh[Sqrt[5 + b*x]/Sqrt[1 + b*x]])/b

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 27, normalized size = 1.42 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + 5} \sqrt {b x + 1} - 3\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+1)^(1/2)/(b*x+5)^(1/2),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + 5)*sqrt(b*x + 1) - 3)/b

________________________________________________________________________________________

giac [A]  time = 0.96, size = 23, normalized size = 1.21 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 5} - \sqrt {b x + 1}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+1)^(1/2)/(b*x+5)^(1/2),x, algorithm="giac")

[Out]

-2*log(sqrt(b*x + 5) - sqrt(b*x + 1))/b

________________________________________________________________________________________

maple [B]  time = 0.01, size = 66, normalized size = 3.47 \begin {gather*} \frac {\sqrt {\left (b x +1\right ) \left (b x +5\right )}\, \ln \left (\frac {b^{2} x +3 b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+6 b x +5}\right )}{\sqrt {b x +1}\, \sqrt {b x +5}\, \sqrt {b^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+1)^(1/2)/(b*x+5)^(1/2),x)

[Out]

((b*x+1)*(b*x+5))^(1/2)/(b*x+1)^(1/2)/(b*x+5)^(1/2)*ln((b^2*x+3*b)/(b^2)^(1/2)+(b^2*x^2+6*b*x+5)^(1/2))/(b^2)^
(1/2)

________________________________________________________________________________________

maxima [B]  time = 1.39, size = 33, normalized size = 1.74 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 6 \, b x + 5} b + 6 \, b\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+1)^(1/2)/(b*x+5)^(1/2),x, algorithm="maxima")

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 + 6*b*x + 5)*b + 6*b)/b

________________________________________________________________________________________

mupad [B]  time = 0.33, size = 43, normalized size = 2.26 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {5}-\sqrt {b\,x+5}\right )}{\left (\sqrt {b\,x+1}-1\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + 1)^(1/2)*(b*x + 5)^(1/2)),x)

[Out]

(4*atan((b*(5^(1/2) - (b*x + 5)^(1/2)))/(((b*x + 1)^(1/2) - 1)*(-b^2)^(1/2))))/(-b^2)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x + 1} \sqrt {b x + 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+1)**(1/2)/(b*x+5)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x + 1)*sqrt(b*x + 5)), x)

________________________________________________________________________________________