3.1915 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x) \, dx\)

Optimal. Leaf size=53 \[ \frac{45}{104} (1-2 x)^{13/2}-\frac{309}{88} (1-2 x)^{11/2}+\frac{707}{72} (1-2 x)^{9/2}-\frac{77}{8} (1-2 x)^{7/2} \]

[Out]

(-77*(1 - 2*x)^(7/2))/8 + (707*(1 - 2*x)^(9/2))/72 - (309*(1 - 2*x)^(11/2))/88 +
(45*(1 - 2*x)^(13/2))/104

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Rubi [A]  time = 0.0510248, 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 \[ \frac{45}{104} (1-2 x)^{13/2}-\frac{309}{88} (1-2 x)^{11/2}+\frac{707}{72} (1-2 x)^{9/2}-\frac{77}{8} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(-77*(1 - 2*x)^(7/2))/8 + (707*(1 - 2*x)^(9/2))/72 - (309*(1 - 2*x)^(11/2))/88 +
(45*(1 - 2*x)^(13/2))/104

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Rubi in Sympy [A]  time = 7.02017, size = 46, normalized size = 0.87 \[ \frac{45 \left (- 2 x + 1\right )^{\frac{13}{2}}}{104} - \frac{309 \left (- 2 x + 1\right )^{\frac{11}{2}}}{88} + \frac{707 \left (- 2 x + 1\right )^{\frac{9}{2}}}{72} - \frac{77 \left (- 2 x + 1\right )^{\frac{7}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45*(-2*x + 1)**(13/2)/104 - 309*(-2*x + 1)**(11/2)/88 + 707*(-2*x + 1)**(9/2)/72
- 77*(-2*x + 1)**(7/2)/8

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Mathematica [A]  time = 0.0428262, size = 28, normalized size = 0.53 \[ -\frac{(1-2 x)^{7/2} \left (4455 x^3+11394 x^2+10540 x+3712\right )}{1287} \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(7/2)*(3712 + 10540*x + 11394*x^2 + 4455*x^3))/1287

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{4455\,{x}^{3}+11394\,{x}^{2}+10540\,x+3712}{1287} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1287*(4455*x^3+11394*x^2+10540*x+3712)*(1-2*x)^(7/2)

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Maxima [A]  time = 1.39561, size = 50, normalized size = 0.94 \[ \frac{45}{104} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{309}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{707}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{77}{8} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45/104*(-2*x + 1)^(13/2) - 309/88*(-2*x + 1)^(11/2) + 707/72*(-2*x + 1)^(9/2) -
77/8*(-2*x + 1)^(7/2)

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Fricas [A]  time = 0.223651, size = 53, normalized size = 1. \[ \frac{1}{1287} \,{\left (35640 \, x^{6} + 37692 \, x^{5} - 25678 \, x^{4} - 32875 \, x^{3} + 7302 \, x^{2} + 11732 \, x - 3712\right )} \sqrt{-2 \, x + 1} \]

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]

1/1287*(35640*x^6 + 37692*x^5 - 25678*x^4 - 32875*x^3 + 7302*x^2 + 11732*x - 371
2)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.90691, size = 100, normalized size = 1.89 \[ \frac{360 x^{6} \sqrt{- 2 x + 1}}{13} + \frac{4188 x^{5} \sqrt{- 2 x + 1}}{143} - \frac{25678 x^{4} \sqrt{- 2 x + 1}}{1287} - \frac{32875 x^{3} \sqrt{- 2 x + 1}}{1287} + \frac{2434 x^{2} \sqrt{- 2 x + 1}}{429} + \frac{11732 x \sqrt{- 2 x + 1}}{1287} - \frac{3712 \sqrt{- 2 x + 1}}{1287} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

360*x**6*sqrt(-2*x + 1)/13 + 4188*x**5*sqrt(-2*x + 1)/143 - 25678*x**4*sqrt(-2*x
+ 1)/1287 - 32875*x**3*sqrt(-2*x + 1)/1287 + 2434*x**2*sqrt(-2*x + 1)/429 + 117
32*x*sqrt(-2*x + 1)/1287 - 3712*sqrt(-2*x + 1)/1287

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GIAC/XCAS [A]  time = 0.211124, size = 88, normalized size = 1.66 \[ \frac{45}{104} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{309}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{707}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{77}{8} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45/104*(2*x - 1)^6*sqrt(-2*x + 1) + 309/88*(2*x - 1)^5*sqrt(-2*x + 1) + 707/72*(
2*x - 1)^4*sqrt(-2*x + 1) + 77/8*(2*x - 1)^3*sqrt(-2*x + 1)