3.2133 \(\int \frac{(2+3 x) (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]

[Out]

847/(24*(1 - 2*x)^(3/2)) - 1133/(8*Sqrt[1 - 2*x]) - (505*Sqrt[1 - 2*x])/8 + (25*
(1 - 2*x)^(3/2))/8

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Rubi [A]  time = 0.0535463, 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 \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

847/(24*(1 - 2*x)^(3/2)) - 1133/(8*Sqrt[1 - 2*x]) - (505*Sqrt[1 - 2*x])/8 + (25*
(1 - 2*x)^(3/2))/8

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Rubi in Sympy [A]  time = 7.07097, size = 46, normalized size = 0.87 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{8} - \frac{505 \sqrt{- 2 x + 1}}{8} - \frac{1133}{8 \sqrt{- 2 x + 1}} + \frac{847}{24 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(3/2)/8 - 505*sqrt(-2*x + 1)/8 - 1133/(8*sqrt(-2*x + 1)) + 847/(2
4*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0433295, size = 28, normalized size = 0.53 \[ -\frac{75 x^3+645 x^2-1551 x+499}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(499 - 1551*x + 645*x^2 + 75*x^3)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{75\,{x}^{3}+645\,{x}^{2}-1551\,x+499}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(75*x^3+645*x^2-1551*x+499)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.3326, size = 45, normalized size = 0.85 \[ \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) + 11/12*(309*x - 116)/(-2*x + 1)^(3
/2)

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Fricas [A]  time = 0.205161, size = 42, normalized size = 0.79 \[ \frac{75 \, x^{3} + 645 \, x^{2} - 1551 \, x + 499}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(75*x^3 + 645*x^2 - 1551*x + 499)/((2*x - 1)*sqrt(-2*x + 1))

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Sympy [A]  time = 1.20071, size = 102, normalized size = 1.92 \[ \frac{75 x^{3}}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{645 x^{2}}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} - \frac{1551 x}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{499}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75*x**3/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) + 645*x**2/(6*x*sqrt(-2*x + 1) -
3*sqrt(-2*x + 1)) - 1551*x/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) + 499/(6*x*s
qrt(-2*x + 1) - 3*sqrt(-2*x + 1))

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GIAC/XCAS [A]  time = 0.209978, size = 54, normalized size = 1.02 \[ \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) - 11/12*(309*x - 116)/((2*x - 1)*sq
rt(-2*x + 1))