3.1779 \(\int \sqrt{1-2 x} (3+5 x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0186, 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 \[ \frac{1}{2} (1-2 x)^{5/2}-\frac{11}{6} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]*(3 + 5*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.09354, size = 20, normalized size = 0.74 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{2} - \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00888337, size = 18, normalized size = 0.67 \[ -\frac{1}{3} (1-2 x)^{3/2} (3 x+4) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(3/2)*(4 + 3*x))/3

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Maple [A]  time = 0.005, size = 15, normalized size = 0.6 \[ -{\frac{3\,x+4}{3} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)*(1-2*x)^(1/2),x)

[Out]

-1/3*(3*x+4)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.35328, size = 26, normalized size = 0.96 \[ \frac{1}{2} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{11}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20698, size = 26, normalized size = 0.96 \[ \frac{1}{3} \,{\left (6 \, x^{2} + 5 \, x - 4\right )} \sqrt{-2 \, x + 1} \]

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*(6*x^2 + 5*x - 4)*sqrt(-2*x + 1)

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Sympy [A]  time = 3.44916, size = 138, normalized size = 5.11 \[ \begin{cases} \frac{2 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{5} - \frac{11 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{75} - \frac{121 \sqrt{5} i \sqrt{10 x - 5}}{375} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{2 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{5} - \frac{11 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{75} - \frac{121 \sqrt{5} \sqrt{- 10 x + 5}}{375} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/5 - 11*sqrt(5)*I*(x + 3/5)*sq
rt(10*x - 5)/75 - 121*sqrt(5)*I*sqrt(10*x - 5)/375, 10*Abs(x + 3/5)/11 > 1), (2*
sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)**2/5 - 11*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/75
- 121*sqrt(5)*sqrt(-10*x + 5)/375, True))

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GIAC/XCAS [A]  time = 0.212763, size = 35, normalized size = 1.3 \[ \frac{1}{2} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{11}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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