3.1989 \(\int \frac{3+5 x}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{6} (1-2 x)^{3/2}-\frac{11}{2} \sqrt{1-2 x} \]

[Out]

(-11*Sqrt[1 - 2*x])/2 + (5*(1 - 2*x)^(3/2))/6

_______________________________________________________________________________________

Rubi [A]  time = 0.0218363, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{5}{6} (1-2 x)^{3/2}-\frac{11}{2} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/Sqrt[1 - 2*x],x]

[Out]

(-11*Sqrt[1 - 2*x])/2 + (5*(1 - 2*x)^(3/2))/6

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.75337, size = 22, normalized size = 0.81 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6} - \frac{11 \sqrt{- 2 x + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(3/2)/6 - 11*sqrt(-2*x + 1)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00588417, size = 18, normalized size = 0.67 \[ -\frac{1}{3} \sqrt{1-2 x} (5 x+14) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(14 + 5*x))/3

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[ -{\frac{14+5\,x}{3}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^(1/2),x)

[Out]

-1/3*(14+5*x)*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34606, size = 26, normalized size = 0.96 \[ \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{2} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*(-2*x + 1)^(3/2) - 11/2*sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.206348, size = 19, normalized size = 0.7 \[ -\frac{1}{3} \,{\left (5 \, x + 14\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(5*x + 14)*sqrt(-2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 1.54853, size = 88, normalized size = 3.26 \[ \begin{cases} - \frac{\sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{3} - \frac{11 \sqrt{5} i \sqrt{10 x - 5}}{15} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{\sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{3} - \frac{11 \sqrt{5} \sqrt{- 10 x + 5}}{15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/3 - 11*sqrt(5)*I*sqrt(10*x - 5)/1
5, 10*Abs(x + 3/5)/11 > 1), (-sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/3 - 11*sqrt(5)*s
qrt(-10*x + 5)/15, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207046, size = 26, normalized size = 0.96 \[ \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{2} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*(-2*x + 1)^(3/2) - 11/2*sqrt(-2*x + 1)