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

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 25, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [A]  time = 1.29, size = 30, normalized size = 1.20 \begin {gather*} \frac {1}{55} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \end {gather*}

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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giac [B]  time = 1.31, size = 40, normalized size = 1.60 \begin {gather*} -\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 {gather*}

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)))

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maple [A]  time = 0.00, size = 19, normalized size = 0.76 \begin {gather*} -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{55} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 36, normalized size = 1.44 \begin {gather*} \frac {1}{55} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) \end {gather*}

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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mupad [B]  time = 0.08, size = 15, normalized size = 0.60 \begin {gather*} -\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55-110\,x}}{11}\right )}{55} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(1/2)*(5*x + 3)),x)

[Out]

-(2*55^(1/2)*atanh((55 - 110*x)^(1/2)/11))/55

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sympy [A]  time = 1.02, size = 61, normalized size = 2.44 \begin {gather*} \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 {gather*}

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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