3.16.31 \(\int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx\) [1531]

Optimal. Leaf size=15 \[ \frac {2 \sinh ^{-1}\left (\sqrt {1+b x}\right )}{b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, 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 = {65, 221} \begin {gather*} \frac {2 \sinh ^{-1}\left (\sqrt {b x+1}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.67 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {2+b x}}{\sqrt {1+b x}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(13)=26\).
time = 0.16, size = 66, normalized size = 4.40

method	result	size
default	\(\frac {\sqrt {\left (b x +1\right ) \left (b x +2\right )}\, \ln \left (\frac {\frac {3}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+3 b x +2}\right )}{\sqrt {b x +1}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
time = 0.31, size = 33, normalized size = 2.20 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 3 \, b x + 2} b + 3 \, 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+2)^(1/2),x, algorithm="maxima")

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 + 3*b*x + 2)*b + 3*b)/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
time = 0.89, size = 28, normalized size = 1.87 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {b x + 2} \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+2)^(1/2),x, algorithm="fricas")

[Out]

-log(-2*b*x + 2*sqrt(b*x + 2)*sqrt(b*x + 1) - 3)/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x + 1} \sqrt {b x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.53, size = 23, normalized size = 1.53 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 2} - \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+2)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.29, size = 43, normalized size = 2.87 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\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 + 2)^(1/2)),x)

[Out]

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

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