3.2485 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0164657, 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 \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.74643, size = 20, normalized size = 0.91 \[ - \frac{2 \sqrt{- 2 x + 1}}{11 \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)/(11*sqrt(5*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0196828, size = 22, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 17, normalized size = 0.8 \[ -{\frac{2}{11}\sqrt{1-2\,x}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*(1-2*x)^(1/2)/(3+5*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49566, size = 28, normalized size = 1.27 \[ -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{11 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*sqrt(-10*x^2 - x + 3)/(5*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.220966, size = 22, normalized size = 1. \[ -\frac{2 \, \sqrt{-2 \, x + 1}}{11 \, \sqrt{5 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*sqrt(-2*x + 1)/sqrt(5*x + 3)

_______________________________________________________________________________________

Sympy [A]  time = 1.84773, size = 54, normalized size = 2.45 \[ \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/55, 11*Abs(1/(x + 3/5))/10 >
1), (-2*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/55, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229835, size = 82, normalized size = 3.73 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{110 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{55 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/110*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/55*sqrt(1
0)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))