3.1916 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx\)

Optimal. Leaf size=40 \[ -\frac{15}{44} (1-2 x)^{11/2}+\frac{17}{9} (1-2 x)^{9/2}-\frac{11}{4} (1-2 x)^{7/2} \]

[Out]

(-11*(1 - 2*x)^(7/2))/4 + (17*(1 - 2*x)^(9/2))/9 - (15*(1 - 2*x)^(11/2))/44

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Rubi [A]  time = 0.0375503, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{15}{44} (1-2 x)^{11/2}+\frac{17}{9} (1-2 x)^{9/2}-\frac{11}{4} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(1 - 2*x)^(7/2))/4 + (17*(1 - 2*x)^(9/2))/9 - (15*(1 - 2*x)^(11/2))/44

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Rubi in Sympy [A]  time = 5.85238, size = 34, normalized size = 0.85 \[ - \frac{15 \left (- 2 x + 1\right )^{\frac{11}{2}}}{44} + \frac{17 \left (- 2 x + 1\right )^{\frac{9}{2}}}{9} - \frac{11 \left (- 2 x + 1\right )^{\frac{7}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*(-2*x + 1)**(11/2)/44 + 17*(-2*x + 1)**(9/2)/9 - 11*(-2*x + 1)**(7/2)/4

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Mathematica [A]  time = 0.0152654, size = 23, normalized size = 0.57 \[ -\frac{1}{99} (1-2 x)^{7/2} \left (135 x^2+239 x+119\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(7/2)*(119 + 239*x + 135*x^2))/99

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Maple [A]  time = 0.005, size = 20, normalized size = 0.5 \[ -{\frac{135\,{x}^{2}+239\,x+119}{99} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/99*(135*x^2+239*x+119)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.46945, size = 38, normalized size = 0.95 \[ -\frac{15}{44} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{17}{9} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{11}{4} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/44*(-2*x + 1)^(11/2) + 17/9*(-2*x + 1)^(9/2) - 11/4*(-2*x + 1)^(7/2)

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Fricas [A]  time = 0.202956, size = 46, normalized size = 1.15 \[ \frac{1}{99} \,{\left (1080 \, x^{5} + 292 \, x^{4} - 1106 \, x^{3} - 129 \, x^{2} + 475 \, x - 119\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/99*(1080*x^5 + 292*x^4 - 1106*x^3 - 129*x^2 + 475*x - 119)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.25485, size = 85, normalized size = 2.12 \[ \frac{120 x^{5} \sqrt{- 2 x + 1}}{11} + \frac{292 x^{4} \sqrt{- 2 x + 1}}{99} - \frac{1106 x^{3} \sqrt{- 2 x + 1}}{99} - \frac{43 x^{2} \sqrt{- 2 x + 1}}{33} + \frac{475 x \sqrt{- 2 x + 1}}{99} - \frac{119 \sqrt{- 2 x + 1}}{99} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

120*x**5*sqrt(-2*x + 1)/11 + 292*x**4*sqrt(-2*x + 1)/99 - 1106*x**3*sqrt(-2*x +
1)/99 - 43*x**2*sqrt(-2*x + 1)/33 + 475*x*sqrt(-2*x + 1)/99 - 119*sqrt(-2*x + 1)
/99

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GIAC/XCAS [A]  time = 0.209733, size = 66, normalized size = 1.65 \[ \frac{15}{44} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{17}{9} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{11}{4} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/44*(2*x - 1)^5*sqrt(-2*x + 1) + 17/9*(2*x - 1)^4*sqrt(-2*x + 1) + 11/4*(2*x -
1)^3*sqrt(-2*x + 1)