3.1857 \(\int (1-2 x)^{3/2} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{36} (1-2 x)^{9/2}+\frac{55}{14} (1-2 x)^{7/2}-\frac{121}{20} (1-2 x)^{5/2} \]

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

_______________________________________________________________________________________

Rubi [A]  time = 0.0253241, 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 \[ -\frac{25}{36} (1-2 x)^{9/2}+\frac{55}{14} (1-2 x)^{7/2}-\frac{121}{20} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.32961, size = 34, normalized size = 0.85 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{9}{2}}}{36} + \frac{55 \left (- 2 x + 1\right )^{\frac{7}{2}}}{14} - \frac{121 \left (- 2 x + 1\right )^{\frac{5}{2}}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(9/2)/36 + 55*(-2*x + 1)**(7/2)/14 - 121*(-2*x + 1)**(5/2)/20

_______________________________________________________________________________________

Mathematica [A]  time = 0.0292301, size = 23, normalized size = 0.57 \[ -\frac{1}{315} (1-2 x)^{5/2} \left (875 x^2+1600 x+887\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(5/2)*(887 + 1600*x + 875*x^2))/315

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 20, normalized size = 0.5 \[ -{\frac{875\,{x}^{2}+1600\,x+887}{315} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^2,x)

[Out]

-1/315*(875*x^2+1600*x+887)*(1-2*x)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.35259, size = 38, normalized size = 0.95 \[ -\frac{25}{36} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{55}{14} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{121}{20} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/36*(-2*x + 1)^(9/2) + 55/14*(-2*x + 1)^(7/2) - 121/20*(-2*x + 1)^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.208672, size = 39, normalized size = 0.98 \[ -\frac{1}{315} \,{\left (3500 \, x^{4} + 2900 \, x^{3} - 1977 \, x^{2} - 1948 \, x + 887\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/315*(3500*x^4 + 2900*x^3 - 1977*x^2 - 1948*x + 887)*sqrt(-2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 2.74134, size = 236, normalized size = 5.9 \[ \begin{cases} - \frac{20 \sqrt{5} i \left (x + \frac{3}{5}\right )^{4} \sqrt{10 x - 5}}{9} + \frac{220 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{63} - \frac{121 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{525} - \frac{2662 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{7875} - \frac{29282 \sqrt{5} i \sqrt{10 x - 5}}{39375} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{20 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{4}}{9} + \frac{220 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{3}}{63} - \frac{121 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{525} - \frac{2662 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{7875} - \frac{29282 \sqrt{5} \sqrt{- 10 x + 5}}{39375} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-20*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/9 + 220*sqrt(5)*I*(x + 3/5)
**3*sqrt(10*x - 5)/63 - 121*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/525 - 2662*sqr
t(5)*I*(x + 3/5)*sqrt(10*x - 5)/7875 - 29282*sqrt(5)*I*sqrt(10*x - 5)/39375, 10*
Abs(x + 3/5)/11 > 1), (-20*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)**4/9 + 220*sqrt(5)*
sqrt(-10*x + 5)*(x + 3/5)**3/63 - 121*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)**2/525 -
2662*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/7875 - 29282*sqrt(5)*sqrt(-10*x + 5)/393
75, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224425, size = 66, normalized size = 1.65 \[ -\frac{25}{36} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{55}{14} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{121}{20} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/36*(2*x - 1)^4*sqrt(-2*x + 1) - 55/14*(2*x - 1)^3*sqrt(-2*x + 1) - 121/20*(2
*x - 1)^2*sqrt(-2*x + 1)