3.2047 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{3 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{\sqrt{1-2 x}}{22 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

[Out]

-Sqrt[1 - 2*x]/(22*(3 + 5*x)^2) - (3*Sqrt[1 - 2*x])/(242*(3 + 5*x)) - (3*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/(121*Sqrt[55])

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Rubi [A]  time = 0.0598688, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{3 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{\sqrt{1-2 x}}{22 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(22*(3 + 5*x)^2) - (3*Sqrt[1 - 2*x])/(242*(3 + 5*x)) - (3*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/(121*Sqrt[55])

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Rubi in Sympy [A]  time = 6.09774, size = 58, normalized size = 0.85 \[ - \frac{3 \sqrt{- 2 x + 1}}{242 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{22 \left (5 x + 3\right )^{2}} - \frac{3 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6655} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)/(242*(5*x + 3)) - sqrt(-2*x + 1)/(22*(5*x + 3)**2) - 3*sqrt(55
)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/6655

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Mathematica [A]  time = 0.0775396, size = 53, normalized size = 0.78 \[ -\frac{5 \sqrt{1-2 x} (3 x+4)}{242 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(4 + 3*x))/(242*(3 + 5*x)^2) - (3*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/(121*Sqrt[55])

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Maple [A]  time = 0.009, size = 52, normalized size = 0.8 \[ -{\frac{2}{11\, \left ( -6-10\,x \right ) ^{2}}\sqrt{1-2\,x}}+{\frac{3}{-726-1210\,x}\sqrt{1-2\,x}}-{\frac{3\,\sqrt{55}}{6655}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*(1-2*x)^(1/2)/(-6-10*x)^2+3/121*(1-2*x)^(1/2)/(-6-10*x)-3/6655*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51663, size = 100, normalized size = 1.47 \[ \frac{3}{13310} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{5 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}}{121 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/13310*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 5/121*(3*(-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 = 0.21916, size = 101, normalized size = 1.49 \[ -\frac{\sqrt{55}{\left (5 \, \sqrt{55}{\left (3 \, x + 4\right )} \sqrt{-2 \, x + 1} - 3 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{13310 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13310*sqrt(55)*(5*sqrt(55)*(3*x + 4)*sqrt(-2*x + 1) - 3*(25*x^2 + 30*x + 9)*l
og((sqrt(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 = 5.10855, size = 233, normalized size = 3.43 \[ \begin{cases} - \frac{3 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{6655} + \frac{3 \sqrt{2}}{1210 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{1100 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{\sqrt{2}}{500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{3 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{6655} - \frac{3 \sqrt{2} i}{1210 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{1100 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} i}{500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/6655 + 3*sqrt(2)/(121
0*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(1100*sqrt(-1 + 11/(10*(
x + 3/5)))*(x + 3/5)**(3/2)) - sqrt(2)/(500*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/
5)**(5/2)), 11*Abs(1/(x + 3/5))/10 > 1), (3*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x
+ 3/5)))/6655 - 3*sqrt(2)*I/(1210*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) +
sqrt(2)*I/(1100*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + sqrt(2)*I/(500*s
qrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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GIAC/XCAS [A]  time = 0.218386, size = 92, normalized size = 1.35 \[ \frac{3}{13310} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{5 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/13310*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 5/484*(3*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))/(5*x + 3)^2