3.1852 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{\sqrt{1-2 x}}{110 (5 x+3)}-\frac{\sqrt{1-2 x}}{10 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

[Out]

-Sqrt[1 - 2*x]/(10*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(110*(3 + 5*x)) + ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]/(55*Sqrt[5
5])

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Rubi [A]  time = 0.0138872, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 51, 63, 206} \[ \frac{\sqrt{1-2 x}}{110 (5 x+3)}-\frac{\sqrt{1-2 x}}{10 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/(3 + 5*x)^3,x]

[Out]

-Sqrt[1 - 2*x]/(10*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(110*(3 + 5*x)) + ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]/(55*Sqrt[5
5])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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{\sqrt{1-2 x}}{(3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x}}{10 (3+5 x)^2}-\frac{1}{10} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{110 (3+5 x)}-\frac{1}{110} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{\sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{110 (3+5 x)}+\frac{1}{110} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{\sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{110 (3+5 x)}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0048647, size = 30, normalized size = 0.44 \[ -\frac{8 (1-2 x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{5}{11} (1-2 x)\right )}{3993} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(1 - 2*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (5*(1 - 2*x))/11])/3993

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Maple [A]  time = 0.007, size = 48, normalized size = 0.7 \begin{align*} 200\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{2200}}-{\frac{\sqrt{1-2\,x}}{1000}} \right ) }+{\frac{\sqrt{55}}{3025}{\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((1-2*x)^(1/2)/(3+5*x)^3,x)

[Out]

200*(-1/2200*(1-2*x)^(3/2)-1/1000*(1-2*x)^(1/2))/(-10*x-6)^2+1/3025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1
/2)

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Maxima [A]  time = 1.50052, size = 100, normalized size = 1.47 \begin{align*} -\frac{1}{6050} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{55 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/55*(5*(-2*x + 1)^(3/2)
+ 11*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 1.74754, size = 189, normalized size = 2.78 \begin{align*} \frac{\sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (5 \, x - 8\right )} \sqrt{-2 \, x + 1}}{6050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6050*(sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(5*x - 8)*sqrt(-2
*x + 1))/(25*x^2 + 30*x + 9)

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Sympy [B]  time = 2.69362, size = 231, normalized size = 3.4 \begin{align*} \begin{cases} \frac{\sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} - \frac{\sqrt{2}}{550 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{3 \sqrt{2}}{500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{11 \sqrt{2}}{2500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{\sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} + \frac{\sqrt{2} i}{550 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{3 \sqrt{2} i}{500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{11 \sqrt{2} i}{2500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/3025 - sqrt(2)/(550*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(
x + 3/5)) + 3*sqrt(2)/(500*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) - 11*sqrt(2)/(2500*sqrt(-1 + 11/(10*
(x + 3/5)))*(x + 3/5)**(5/2)), 11/(10*Abs(x + 3/5)) > 1), (-sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/3025
+ sqrt(2)*I/(550*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - 3*sqrt(2)*I/(500*sqrt(1 - 11/(10*(x + 3/5)))*(x
+ 3/5)**(3/2)) + 11*sqrt(2)*I/(2500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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Giac [A]  time = 2.39951, size = 92, normalized size = 1.35 \begin{align*} -\frac{1}{6050} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{220 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/220*(5*(-2*x
+ 1)^(3/2) + 11*sqrt(-2*x + 1))/(5*x + 3)^2