3.1790 \(\int \sqrt{1-2 x} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{28} (1-2 x)^{7/2}+\frac{11}{2} (1-2 x)^{5/2}-\frac{121}{12} (1-2 x)^{3/2} \]

[Out]

(-121*(1 - 2*x)^(3/2))/12 + (11*(1 - 2*x)^(5/2))/2 - (25*(1 - 2*x)^(7/2))/28

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Rubi [A]  time = 0.0248892, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{25}{28} (1-2 x)^{7/2}+\frac{11}{2} (1-2 x)^{5/2}-\frac{121}{12} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-121*(1 - 2*x)^(3/2))/12 + (11*(1 - 2*x)^(5/2))/2 - (25*(1 - 2*x)^(7/2))/28

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Rubi in Sympy [A]  time = 4.97057, size = 34, normalized size = 0.85 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{7}{2}}}{28} + \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2} - \frac{121 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(7/2)/28 + 11*(-2*x + 1)**(5/2)/2 - 121*(-2*x + 1)**(3/2)/12

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Mathematica [A]  time = 0.0192825, size = 28, normalized size = 0.7 \[ \frac{1}{21} \sqrt{1-2 x} \left (150 x^3+237 x^2+74 x-115\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-115 + 74*x + 237*x^2 + 150*x^3))/21

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Maple [A]  time = 0.006, size = 20, normalized size = 0.5 \[ -{\frac{75\,{x}^{2}+156\,x+115}{21} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2*(1-2*x)^(1/2),x)

[Out]

-1/21*(75*x^2+156*x+115)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.34405, size = 38, normalized size = 0.95 \[ -\frac{25}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{11}{2} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{121}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/28*(-2*x + 1)^(7/2) + 11/2*(-2*x + 1)^(5/2) - 121/12*(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.213896, size = 32, normalized size = 0.8 \[ \frac{1}{21} \,{\left (150 \, x^{3} + 237 \, x^{2} + 74 \, x - 115\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(150*x^3 + 237*x^2 + 74*x - 115)*sqrt(-2*x + 1)

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Sympy [A]  time = 4.67384, size = 187, normalized size = 4.68 \[ \begin{cases} \frac{10 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{7} - \frac{11 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{35} - \frac{242 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{525} - \frac{2662 \sqrt{5} i \sqrt{10 x - 5}}{2625} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{10 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{3}}{7} - \frac{11 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{35} - \frac{242 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{525} - \frac{2662 \sqrt{5} \sqrt{- 10 x + 5}}{2625} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((10*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/7 - 11*sqrt(5)*I*(x + 3/5)**
2*sqrt(10*x - 5)/35 - 242*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/525 - 2662*sqrt(5)*
I*sqrt(10*x - 5)/2625, 10*Abs(x + 3/5)/11 > 1), (10*sqrt(5)*sqrt(-10*x + 5)*(x +
3/5)**3/7 - 11*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)**2/35 - 242*sqrt(5)*sqrt(-10*x
+ 5)*(x + 3/5)/525 - 2662*sqrt(5)*sqrt(-10*x + 5)/2625, True))

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GIAC/XCAS [A]  time = 0.209909, size = 57, normalized size = 1.42 \[ \frac{25}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{11}{2} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{121}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/28*(2*x - 1)^3*sqrt(-2*x + 1) + 11/2*(2*x - 1)^2*sqrt(-2*x + 1) - 121/12*(-2*
x + 1)^(3/2)