3.2121 \(\int \frac{(2+3 x)^2 (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{15}{8} (1-2 x)^{3/2}-\frac{309}{8} \sqrt{1-2 x}-\frac{707}{8 \sqrt{1-2 x}}+\frac{539}{24 (1-2 x)^{3/2}} \]

[Out]

539/(24*(1 - 2*x)^(3/2)) - 707/(8*Sqrt[1 - 2*x]) - (309*Sqrt[1 - 2*x])/8 + (15*(
1 - 2*x)^(3/2))/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0537984, 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 \[ \frac{15}{8} (1-2 x)^{3/2}-\frac{309}{8} \sqrt{1-2 x}-\frac{707}{8 \sqrt{1-2 x}}+\frac{539}{24 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

539/(24*(1 - 2*x)^(3/2)) - 707/(8*Sqrt[1 - 2*x]) - (309*Sqrt[1 - 2*x])/8 + (15*(
1 - 2*x)^(3/2))/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.23615, size = 46, normalized size = 0.87 \[ \frac{15 \left (- 2 x + 1\right )^{\frac{3}{2}}}{8} - \frac{309 \sqrt{- 2 x + 1}}{8} - \frac{707}{8 \sqrt{- 2 x + 1}} + \frac{539}{24 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*(-2*x + 1)**(3/2)/8 - 309*sqrt(-2*x + 1)/8 - 707/(8*sqrt(-2*x + 1)) + 539/(24
*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0397672, size = 28, normalized size = 0.53 \[ -\frac{45 x^3+396 x^2-960 x+308}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(308 - 960*x + 396*x^2 + 45*x^3)/(3*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{45\,{x}^{3}+396\,{x}^{2}-960\,x+308}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(45*x^3+396*x^2-960*x+308)/(1-2*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34887, size = 45, normalized size = 0.85 \[ \frac{15}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{309}{8} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (303 \, x - 113\right )}}{12 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/8*(-2*x + 1)^(3/2) - 309/8*sqrt(-2*x + 1) + 7/12*(303*x - 113)/(-2*x + 1)^(3/
2)

_______________________________________________________________________________________

Fricas [A]  time = 0.215588, size = 42, normalized size = 0.79 \[ \frac{45 \, x^{3} + 396 \, x^{2} - 960 \, x + 308}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(45*x^3 + 396*x^2 - 960*x + 308)/((2*x - 1)*sqrt(-2*x + 1))

_______________________________________________________________________________________

Sympy [A]  time = 1.20602, size = 102, normalized size = 1.92 \[ \frac{45 x^{3}}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{396 x^{2}}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} - \frac{960 x}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{308}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45*x**3/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) + 396*x**2/(6*x*sqrt(-2*x + 1) -
3*sqrt(-2*x + 1)) - 960*x/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) + 308/(6*x*sq
rt(-2*x + 1) - 3*sqrt(-2*x + 1))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21489, size = 54, normalized size = 1.02 \[ \frac{15}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{309}{8} \, \sqrt{-2 \, x + 1} - \frac{7 \,{\left (303 \, x - 113\right )}}{12 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/8*(-2*x + 1)^(3/2) - 309/8*sqrt(-2*x + 1) - 7/12*(303*x - 113)/((2*x - 1)*sqr
t(-2*x + 1))