3.2305 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

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Rubi [A]  time = 0.0158401, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

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Rubi in Sympy [A]  time = 2.84319, size = 20, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{33 \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)/(33*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.0297559, size = 22, normalized size = 1. \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.004, size = 17, normalized size = 0.8 \[ -{\frac{2}{33} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/33*(1-2*x)^(3/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.50095, size = 65, normalized size = 2.95 \[ -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{4 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) + 4/165*sqrt(-10*x^2 - x + 3)/(5
*x + 3)

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Fricas [A]  time = 0.213066, size = 45, normalized size = 2.05 \[ \frac{2 \, \sqrt{5 \, x + 3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*sqrt(5*x + 3)*(2*x - 1)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 5.09661, size = 100, normalized size = 4.55 \[ \begin{cases} \frac{4 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{4 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825} - \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375 \left (x + \frac{3}{5}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((4*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/825 - 2*sqrt(10)*sqrt(-1 + 11
/(10*(x + 3/5)))/(375*(x + 3/5)), 11*Abs(1/(x + 3/5))/10 > 1), (4*sqrt(10)*I*sqr
t(1 - 11/(10*(x + 3/5)))/825 - 2*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(375*(x
+ 3/5)), True))

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GIAC/XCAS [A]  time = 0.234299, size = 176, normalized size = 8. \[ -\frac{1}{13200} \, \sqrt{5}{\left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{12 \, \sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{16 \,{\left (\frac{3 \, \sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4 \, \sqrt{2}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13200*sqrt(5)*(sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2)
- 12*sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 16*(3*sqrt(2)
*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4*sqrt(2))*(5*x + 3)^(3/2)/(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3)