3.2041 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

[Out]

(-2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Rubi [A]  time = 0.0053159, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {63, 206} \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - 2*x]*(3 + 5*x)),x]

[Out]

(-2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0035806, size = 25, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - 2*x]*(3 + 5*x)),x]

[Out]

(-2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Maple [A]  time = 0.003, size = 19, normalized size = 0.8 \begin{align*} -{\frac{2\,\sqrt{55}}{55}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*x)/(1-2*x)^(1/2),x)

[Out]

-2/55*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [B]  time = 2.3471, size = 49, normalized size = 1.96 \begin{align*} \frac{1}{55} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*x)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

1/55*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))

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Fricas [A]  time = 1.59226, size = 89, normalized size = 3.56 \begin{align*} \frac{1}{55} \, \sqrt{55} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*x)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/55*sqrt(55)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3))

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Sympy [A]  time = 1.00642, size = 61, normalized size = 2.44 \begin{align*} \begin{cases} - \frac{2 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{55} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{2 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{55} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*x)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-2*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/55, 11/(10*Abs(x + 3/5)) > 1), (2*sqrt(55)*I*asin(s
qrt(110)/(10*sqrt(x + 3/5)))/55, True))

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Giac [B]  time = 2.60141, size = 54, normalized size = 2.16 \begin{align*} -\frac{1}{55} \, \sqrt{55} \log \left (\frac{1}{5} \, \sqrt{55} + \sqrt{-2 \, x + 1}\right ) + \frac{1}{55} \, \sqrt{55} \log \left ({\left | -\frac{1}{5} \, \sqrt{55} + \sqrt{-2 \, x + 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*x)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/55*sqrt(55)*log(1/5*sqrt(55) + sqrt(-2*x + 1)) + 1/55*sqrt(55)*log(abs(-1/5*sqrt(55) + sqrt(-2*x + 1)))