3.2531 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]

[Out]

-65/(24*(3 + 2*x)^(3/2)) + 109/(8*Sqrt[3 + 2*x]) + (47*Sqrt[3 + 2*x])/8 - (3 + 2
*x)^(3/2)/8

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Rubi [A]  time = 0.0550816, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-65/(24*(3 + 2*x)^(3/2)) + 109/(8*Sqrt[3 + 2*x]) + (47*Sqrt[3 + 2*x])/8 - (3 + 2
*x)^(3/2)/8

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Rubi in Sympy [A]  time = 9.33706, size = 44, normalized size = 0.83 \[ - \frac{\left (2 x + 3\right )^{\frac{3}{2}}}{8} + \frac{47 \sqrt{2 x + 3}}{8} + \frac{109}{8 \sqrt{2 x + 3}} - \frac{65}{24 \left (2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(3/2)/8 + 47*sqrt(2*x + 3)/8 + 109/(8*sqrt(2*x + 3)) - 65/(24*(2*x +
3)**(3/2))

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Mathematica [A]  time = 0.0213701, size = 28, normalized size = 0.53 \[ -\frac{3 x^3-57 x^2-273 x-263}{3 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(-263 - 273*x - 57*x^2 + 3*x^3)/(3*(3 + 2*x)^(3/2))

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{3\,{x}^{3}-57\,{x}^{2}-273\,x-263}{3} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*x^3-57*x^2-273*x-263)/(3+2*x)^(3/2)

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Maxima [A]  time = 0.707586, size = 45, normalized size = 0.85 \[ -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)

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Fricas [A]  time = 0.275379, size = 32, normalized size = 0.6 \[ -\frac{3 \, x^{3} - 57 \, x^{2} - 273 \, x - 263}{3 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*x^3 - 57*x^2 - 273*x - 263)/(2*x + 3)^(3/2)

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Sympy [A]  time = 2.31233, size = 102, normalized size = 1.92 \[ - \frac{3 x^{3}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{57 x^{2}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{273 x}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{263}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**3/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 57*x**2/(6*x*sqrt(2*x + 3) + 9*s
qrt(2*x + 3)) + 273*x/(6*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)) + 263/(6*x*sqrt(2*x
+ 3) + 9*sqrt(2*x + 3))

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GIAC/XCAS [A]  time = 0.269465, size = 45, normalized size = 0.85 \[ -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(2*x + 3)^(3/2) + 47/8*sqrt(2*x + 3) + 1/12*(327*x + 458)/(2*x + 3)^(3/2)