3.1777 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x) \, dx\)

Optimal. Leaf size=53 \[ \frac{5}{8} (1-2 x)^{9/2}-\frac{309}{56} (1-2 x)^{7/2}+\frac{707}{40} (1-2 x)^{5/2}-\frac{539}{24} (1-2 x)^{3/2} \]

[Out]

(-539*(1 - 2*x)^(3/2))/24 + (707*(1 - 2*x)^(5/2))/40 - (309*(1 - 2*x)^(7/2))/56
+ (5*(1 - 2*x)^(9/2))/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0455995, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{5}{8} (1-2 x)^{9/2}-\frac{309}{56} (1-2 x)^{7/2}+\frac{707}{40} (1-2 x)^{5/2}-\frac{539}{24} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-539*(1 - 2*x)^(3/2))/24 + (707*(1 - 2*x)^(5/2))/40 - (309*(1 - 2*x)^(7/2))/56
+ (5*(1 - 2*x)^(9/2))/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.8185, size = 46, normalized size = 0.87 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{309 \left (- 2 x + 1\right )^{\frac{7}{2}}}{56} + \frac{707 \left (- 2 x + 1\right )^{\frac{5}{2}}}{40} - \frac{539 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(9/2)/8 - 309*(-2*x + 1)**(7/2)/56 + 707*(-2*x + 1)**(5/2)/40 - 53
9*(-2*x + 1)**(3/2)/24

_______________________________________________________________________________________

Mathematica [A]  time = 0.0280065, size = 33, normalized size = 0.62 \[ \frac{1}{105} \sqrt{1-2 x} \left (1050 x^4+2535 x^3+2046 x^2+244 x-1016\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-1016 + 244*x + 2046*x^2 + 2535*x^3 + 1050*x^4))/105

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{525\,{x}^{3}+1530\,{x}^{2}+1788\,x+1016}{105} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(525*x^3+1530*x^2+1788*x+1016)*(1-2*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.35691, size = 50, normalized size = 0.94 \[ \frac{5}{8} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{309}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{707}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{539}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*(-2*x + 1)^(9/2) - 309/56*(-2*x + 1)^(7/2) + 707/40*(-2*x + 1)^(5/2) - 539/2
4*(-2*x + 1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.204373, size = 39, normalized size = 0.74 \[ \frac{1}{105} \,{\left (1050 \, x^{4} + 2535 \, x^{3} + 2046 \, x^{2} + 244 \, x - 1016\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(1050*x^4 + 2535*x^3 + 2046*x^2 + 244*x - 1016)*sqrt(-2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 3.03171, size = 46, normalized size = 0.87 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{309 \left (- 2 x + 1\right )^{\frac{7}{2}}}{56} + \frac{707 \left (- 2 x + 1\right )^{\frac{5}{2}}}{40} - \frac{539 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(9/2)/8 - 309*(-2*x + 1)**(7/2)/56 + 707*(-2*x + 1)**(5/2)/40 - 53
9*(-2*x + 1)**(3/2)/24

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213263, size = 78, normalized size = 1.47 \[ \frac{5}{8} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{309}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{707}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{539}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*(2*x - 1)^4*sqrt(-2*x + 1) + 309/56*(2*x - 1)^3*sqrt(-2*x + 1) + 707/40*(2*x
 - 1)^2*sqrt(-2*x + 1) - 539/24*(-2*x + 1)^(3/2)