3.1942 \(\int (1-2 x)^{5/2} (3+5 x)^3 \, dx\)

Optimal. Leaf size=53 \[ \frac{125}{104} (1-2 x)^{13/2}-\frac{75}{8} (1-2 x)^{11/2}+\frac{605}{24} (1-2 x)^{9/2}-\frac{1331}{56} (1-2 x)^{7/2} \]

[Out]

(-1331*(1 - 2*x)^(7/2))/56 + (605*(1 - 2*x)^(9/2))/24 - (75*(1 - 2*x)^(11/2))/8
+ (125*(1 - 2*x)^(13/2))/104

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Rubi [A]  time = 0.0318697, antiderivative size = 53, 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 \[ \frac{125}{104} (1-2 x)^{13/2}-\frac{75}{8} (1-2 x)^{11/2}+\frac{605}{24} (1-2 x)^{9/2}-\frac{1331}{56} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(-1331*(1 - 2*x)^(7/2))/56 + (605*(1 - 2*x)^(9/2))/24 - (75*(1 - 2*x)^(11/2))/8
+ (125*(1 - 2*x)^(13/2))/104

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Rubi in Sympy [A]  time = 6.01234, size = 46, normalized size = 0.87 \[ \frac{125 \left (- 2 x + 1\right )^{\frac{13}{2}}}{104} - \frac{75 \left (- 2 x + 1\right )^{\frac{11}{2}}}{8} + \frac{605 \left (- 2 x + 1\right )^{\frac{9}{2}}}{24} - \frac{1331 \left (- 2 x + 1\right )^{\frac{7}{2}}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*(-2*x + 1)**(13/2)/104 - 75*(-2*x + 1)**(11/2)/8 + 605*(-2*x + 1)**(9/2)/24
- 1331*(-2*x + 1)**(7/2)/56

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Mathematica [A]  time = 0.0367001, size = 28, normalized size = 0.53 \[ -\frac{1}{273} (1-2 x)^{7/2} \left (2625 x^3+6300 x^2+5495 x+1838\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(7/2)*(1838 + 5495*x + 6300*x^2 + 2625*x^3))/273

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{2625\,{x}^{3}+6300\,{x}^{2}+5495\,x+1838}{273} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^3,x)

[Out]

-1/273*(2625*x^3+6300*x^2+5495*x+1838)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.33905, size = 50, normalized size = 0.94 \[ \frac{125}{104} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{75}{8} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{605}{24} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{1331}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/104*(-2*x + 1)^(13/2) - 75/8*(-2*x + 1)^(11/2) + 605/24*(-2*x + 1)^(9/2) - 1
331/56*(-2*x + 1)^(7/2)

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Fricas [A]  time = 0.222156, size = 53, normalized size = 1. \[ \frac{1}{273} \,{\left (21000 \, x^{6} + 18900 \, x^{5} - 15890 \, x^{4} - 16061 \, x^{3} + 4614 \, x^{2} + 5533 \, x - 1838\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/273*(21000*x^6 + 18900*x^5 - 15890*x^4 - 16061*x^3 + 4614*x^2 + 5533*x - 1838)
*sqrt(-2*x + 1)

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Sympy [A]  time = 2.91186, size = 100, normalized size = 1.89 \[ \frac{1000 x^{6} \sqrt{- 2 x + 1}}{13} + \frac{900 x^{5} \sqrt{- 2 x + 1}}{13} - \frac{2270 x^{4} \sqrt{- 2 x + 1}}{39} - \frac{16061 x^{3} \sqrt{- 2 x + 1}}{273} + \frac{1538 x^{2} \sqrt{- 2 x + 1}}{91} + \frac{5533 x \sqrt{- 2 x + 1}}{273} - \frac{1838 \sqrt{- 2 x + 1}}{273} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1000*x**6*sqrt(-2*x + 1)/13 + 900*x**5*sqrt(-2*x + 1)/13 - 2270*x**4*sqrt(-2*x +
1)/39 - 16061*x**3*sqrt(-2*x + 1)/273 + 1538*x**2*sqrt(-2*x + 1)/91 + 5533*x*sq
rt(-2*x + 1)/273 - 1838*sqrt(-2*x + 1)/273

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GIAC/XCAS [A]  time = 0.212275, size = 88, normalized size = 1.66 \[ \frac{125}{104} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{75}{8} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{605}{24} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{1331}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/104*(2*x - 1)^6*sqrt(-2*x + 1) + 75/8*(2*x - 1)^5*sqrt(-2*x + 1) + 605/24*(2
*x - 1)^4*sqrt(-2*x + 1) + 1331/56*(2*x - 1)^3*sqrt(-2*x + 1)