3.1929 \(\int (1-2 x)^{5/2} (3+5 x)^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{44} (1-2 x)^{11/2}+\frac{55}{18} (1-2 x)^{9/2}-\frac{121}{28} (1-2 x)^{7/2} \]

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

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Rubi [A]  time = 0.0306902, 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 \[ -\frac{25}{44} (1-2 x)^{11/2}+\frac{55}{18} (1-2 x)^{9/2}-\frac{121}{28} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

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Rubi in Sympy [A]  time = 5.05594, size = 34, normalized size = 0.85 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{11}{2}}}{44} + \frac{55 \left (- 2 x + 1\right )^{\frac{9}{2}}}{18} - \frac{121 \left (- 2 x + 1\right )^{\frac{7}{2}}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(11/2)/44 + 55*(-2*x + 1)**(9/2)/18 - 121*(-2*x + 1)**(7/2)/28

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Mathematica [A]  time = 0.0340296, size = 23, normalized size = 0.57 \[ -\frac{1}{693} (1-2 x)^{7/2} \left (1575 x^2+2660 x+1271\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(7/2)*(1271 + 2660*x + 1575*x^2))/693

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Maple [A]  time = 0.006, size = 20, normalized size = 0.5 \[ -{\frac{1575\,{x}^{2}+2660\,x+1271}{693} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^2,x)

[Out]

-1/693*(1575*x^2+2660*x+1271)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.35181, size = 38, normalized size = 0.95 \[ -\frac{25}{44} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{55}{18} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{121}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/44*(-2*x + 1)^(11/2) + 55/18*(-2*x + 1)^(9/2) - 121/28*(-2*x + 1)^(7/2)

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Fricas [A]  time = 0.203623, size = 46, normalized size = 1.15 \[ \frac{1}{693} \,{\left (12600 \, x^{5} + 2380 \, x^{4} - 12302 \, x^{3} - 867 \, x^{2} + 4966 \, x - 1271\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/693*(12600*x^5 + 2380*x^4 - 12302*x^3 - 867*x^2 + 4966*x - 1271)*sqrt(-2*x + 1
)

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Sympy [A]  time = 2.2788, size = 85, normalized size = 2.12 \[ \frac{200 x^{5} \sqrt{- 2 x + 1}}{11} + \frac{340 x^{4} \sqrt{- 2 x + 1}}{99} - \frac{12302 x^{3} \sqrt{- 2 x + 1}}{693} - \frac{289 x^{2} \sqrt{- 2 x + 1}}{231} + \frac{4966 x \sqrt{- 2 x + 1}}{693} - \frac{1271 \sqrt{- 2 x + 1}}{693} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

200*x**5*sqrt(-2*x + 1)/11 + 340*x**4*sqrt(-2*x + 1)/99 - 12302*x**3*sqrt(-2*x +
1)/693 - 289*x**2*sqrt(-2*x + 1)/231 + 4966*x*sqrt(-2*x + 1)/693 - 1271*sqrt(-2
*x + 1)/693

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GIAC/XCAS [A]  time = 0.209813, size = 66, normalized size = 1.65 \[ \frac{25}{44} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{55}{18} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{121}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/44*(2*x - 1)^5*sqrt(-2*x + 1) + 55/18*(2*x - 1)^4*sqrt(-2*x + 1) + 121/28*(2*
x - 1)^3*sqrt(-2*x + 1)