3.2530 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]

[Out]

-65/(8*Sqrt[3 + 2*x]) - (109*Sqrt[3 + 2*x])/8 + (47*(3 + 2*x)^(3/2))/24 - (3*(3
+ 2*x)^(5/2))/40

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Rubi [A]  time = 0.0546672, 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 \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-65/(8*Sqrt[3 + 2*x]) - (109*Sqrt[3 + 2*x])/8 + (47*(3 + 2*x)^(3/2))/24 - (3*(3
+ 2*x)^(5/2))/40

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Rubi in Sympy [A]  time = 9.28155, size = 46, normalized size = 0.87 \[ - \frac{3 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{47 \left (2 x + 3\right )^{\frac{3}{2}}}{24} - \frac{109 \sqrt{2 x + 3}}{8} - \frac{65}{8 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(2*x + 3)**(5/2)/40 + 47*(2*x + 3)**(3/2)/24 - 109*sqrt(2*x + 3)/8 - 65/(8*sq
rt(2*x + 3))

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Mathematica [A]  time = 0.0193599, size = 28, normalized size = 0.53 \[ -\frac{9 x^3-77 x^2+117 x+501}{15 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(501 + 117*x - 77*x^2 + 9*x^3)/(15*Sqrt[3 + 2*x])

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{9\,{x}^{3}-77\,{x}^{2}+117\,x+501}{15}{\frac{1}{\sqrt{3+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(9*x^3-77*x^2+117*x+501)/(3+2*x)^(1/2)

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Maxima [A]  time = 0.710076, size = 50, normalized size = 0.94 \[ -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(
2*x + 3)

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Fricas [A]  time = 0.280955, size = 32, normalized size = 0.6 \[ -\frac{9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(9*x^3 - 77*x^2 + 117*x + 501)/sqrt(2*x + 3)

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Sympy [A]  time = 16.6291, size = 320, normalized size = 6.04 \[ - \frac{3 \sqrt{3} x^{3} \sqrt{2 x + 3}}{10 \sqrt{3} x + 15 \sqrt{3}} + \frac{9 \sqrt{3} x^{2} \sqrt{2 x + 3}}{10 \sqrt{3} x + 15 \sqrt{3}} + \frac{10 \sqrt{3} x^{2} \sqrt{2 x + 3}}{6 \sqrt{3} x + 9 \sqrt{3}} - \frac{54 \sqrt{3} x \sqrt{2 x + 3}}{10 \sqrt{3} x + 15 \sqrt{3}} - \frac{60 \sqrt{3} x \sqrt{2 x + 3}}{6 \sqrt{3} x + 9 \sqrt{3}} + \frac{324 x}{10 \sqrt{3} x + 15 \sqrt{3}} + \frac{360 x}{6 \sqrt{3} x + 9 \sqrt{3}} + \frac{23 x}{\sqrt{2 x + 3}} - \frac{162 \sqrt{3} \sqrt{2 x + 3}}{10 \sqrt{3} x + 15 \sqrt{3}} - \frac{180 \sqrt{3} \sqrt{2 x + 3}}{6 \sqrt{3} x + 9 \sqrt{3}} + \frac{486}{10 \sqrt{3} x + 15 \sqrt{3}} + \frac{540}{6 \sqrt{3} x + 9 \sqrt{3}} + \frac{59}{\sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(3)*x**3*sqrt(2*x + 3)/(10*sqrt(3)*x + 15*sqrt(3)) + 9*sqrt(3)*x**2*sqrt(
2*x + 3)/(10*sqrt(3)*x + 15*sqrt(3)) + 10*sqrt(3)*x**2*sqrt(2*x + 3)/(6*sqrt(3)*
x + 9*sqrt(3)) - 54*sqrt(3)*x*sqrt(2*x + 3)/(10*sqrt(3)*x + 15*sqrt(3)) - 60*sqr
t(3)*x*sqrt(2*x + 3)/(6*sqrt(3)*x + 9*sqrt(3)) + 324*x/(10*sqrt(3)*x + 15*sqrt(3
)) + 360*x/(6*sqrt(3)*x + 9*sqrt(3)) + 23*x/sqrt(2*x + 3) - 162*sqrt(3)*sqrt(2*x
+ 3)/(10*sqrt(3)*x + 15*sqrt(3)) - 180*sqrt(3)*sqrt(2*x + 3)/(6*sqrt(3)*x + 9*s
qrt(3)) + 486/(10*sqrt(3)*x + 15*sqrt(3)) + 540/(6*sqrt(3)*x + 9*sqrt(3)) + 59/s
qrt(2*x + 3)

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GIAC/XCAS [A]  time = 0.267573, size = 50, normalized size = 0.94 \[ -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(
2*x + 3)