3.1835 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^3} \, dx\)

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[55])

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Rubi [A]  time = 0.0535309, 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 \[ \frac{\sqrt{1-2 x}}{110 (5 x+3)}-\frac{\sqrt{1-2 x}}{10 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

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[55])

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Rubi in Sympy [A]  time = 6.6489, size = 53, normalized size = 0.78 \[ \frac{\sqrt{- 2 x + 1}}{110 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{10 \left (5 x + 3\right )^{2}} + \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3025} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)/(3+5*x)**3,x)

[Out]

sqrt(-2*x + 1)/(110*(5*x + 3)) - sqrt(-2*x + 1)/(10*(5*x + 3)**2) + sqrt(55)*ata
nh(sqrt(55)*sqrt(-2*x + 1)/11)/3025

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Mathematica [A]  time = 0.0789328, size = 53, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} (5 x-8)}{110 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-8 + 5*x))/(110*(3 + 5*x)^2) + ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
/(55*Sqrt[55])

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Maple [A]  time = 0.014, size = 48, normalized size = 0.7 \[ 200\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{2200}}-{\frac{\sqrt{1-2\,x}}{1000}} \right ) }+{\frac{\sqrt{55}}{3025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

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

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Maxima [A]  time = 1.49177, size = 100, normalized size = 1.47 \[ -\frac{1}{6050} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{55 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/(5*x + 3)^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 = 0.215783, size = 99, normalized size = 1.46 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (5 \, x - 8\right )} \sqrt{-2 \, x + 1} +{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{6050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.32716, size = 233, normalized size = 3.43 \[ \begin{cases} \frac{\sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} - \frac{\sqrt{2}}{550 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{3 \sqrt{2}}{500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{11 \sqrt{2}}{2500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{\sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} + \frac{\sqrt{2} i}{550 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{3 \sqrt{2} i}{500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{11 \sqrt{2} i}{2500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

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*Abs(1/(x + 3/5))/10 > 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*sqr
t(2)*I/(500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + 11*sqrt(2)*I/(2500*s
qrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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GIAC/XCAS [A]  time = 0.212118, size = 92, normalized size = 1.35 \[ -\frac{1}{6050} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{220 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6050*sqrt(55)*ln(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