3.2526 \(\int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right ) \, dx\)

Optimal. Leaf size=53 \[ -\frac{3}{104} (2 x+3)^{13/2}+\frac{47}{88} (2 x+3)^{11/2}-\frac{109}{72} (2 x+3)^{9/2}+\frac{65}{56} (2 x+3)^{7/2} \]

[Out]

(65*(3 + 2*x)^(7/2))/56 - (109*(3 + 2*x)^(9/2))/72 + (47*(3 + 2*x)^(11/2))/88 -
(3*(3 + 2*x)^(13/2))/104

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Rubi [A]  time = 0.0544637, 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 \[ -\frac{3}{104} (2 x+3)^{13/2}+\frac{47}{88} (2 x+3)^{11/2}-\frac{109}{72} (2 x+3)^{9/2}+\frac{65}{56} (2 x+3)^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(65*(3 + 2*x)^(7/2))/56 - (109*(3 + 2*x)^(9/2))/72 + (47*(3 + 2*x)^(11/2))/88 -
(3*(3 + 2*x)^(13/2))/104

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Rubi in Sympy [A]  time = 9.60531, size = 46, normalized size = 0.87 \[ - \frac{3 \left (2 x + 3\right )^{\frac{13}{2}}}{104} + \frac{47 \left (2 x + 3\right )^{\frac{11}{2}}}{88} - \frac{109 \left (2 x + 3\right )^{\frac{9}{2}}}{72} + \frac{65 \left (2 x + 3\right )^{\frac{7}{2}}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(2*x + 3)**(13/2)/104 + 47*(2*x + 3)**(11/2)/88 - 109*(2*x + 3)**(9/2)/72 + 6
5*(2*x + 3)**(7/2)/56

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Mathematica [A]  time = 0.0221515, size = 28, normalized size = 0.53 \[ -\frac{(2 x+3)^{7/2} \left (2079 x^3-9891 x^2-16429 x-5829\right )}{9009} \]

Antiderivative was successfully verified.

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

[Out]

-((3 + 2*x)^(7/2)*(-5829 - 16429*x - 9891*x^2 + 2079*x^3))/9009

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{2079\,{x}^{3}-9891\,{x}^{2}-16429\,x-5829}{9009} \left ( 3+2\,x \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9009*(2079*x^3-9891*x^2-16429*x-5829)*(3+2*x)^(7/2)

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Maxima [A]  time = 0.711092, size = 50, normalized size = 0.94 \[ -\frac{3}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{47}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{109}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{65}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/104*(2*x + 3)^(13/2) + 47/88*(2*x + 3)^(11/2) - 109/72*(2*x + 3)^(9/2) + 65/5
6*(2*x + 3)^(7/2)

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Fricas [A]  time = 0.280618, size = 53, normalized size = 1. \[ -\frac{1}{9009} \,{\left (16632 \, x^{6} - 4284 \, x^{5} - 375242 \, x^{4} - 1116057 \, x^{3} - 1364067 \, x^{2} - 758349 \, x - 157383\right )} \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9009*(16632*x^6 - 4284*x^5 - 375242*x^4 - 1116057*x^3 - 1364067*x^2 - 758349*
x - 157383)*sqrt(2*x + 3)

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Sympy [A]  time = 5.77605, size = 100, normalized size = 1.89 \[ - \frac{24 x^{6} \sqrt{2 x + 3}}{13} + \frac{68 x^{5} \sqrt{2 x + 3}}{143} + \frac{53606 x^{4} \sqrt{2 x + 3}}{1287} + \frac{372019 x^{3} \sqrt{2 x + 3}}{3003} + \frac{151563 x^{2} \sqrt{2 x + 3}}{1001} + \frac{84261 x \sqrt{2 x + 3}}{1001} + \frac{17487 \sqrt{2 x + 3}}{1001} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24*x**6*sqrt(2*x + 3)/13 + 68*x**5*sqrt(2*x + 3)/143 + 53606*x**4*sqrt(2*x + 3)
/1287 + 372019*x**3*sqrt(2*x + 3)/3003 + 151563*x**2*sqrt(2*x + 3)/1001 + 84261*
x*sqrt(2*x + 3)/1001 + 17487*sqrt(2*x + 3)/1001

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GIAC/XCAS [A]  time = 0.277922, size = 50, normalized size = 0.94 \[ -\frac{3}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{47}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{109}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{65}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/104*(2*x + 3)^(13/2) + 47/88*(2*x + 3)^(11/2) - 109/72*(2*x + 3)^(9/2) + 65/5
6*(2*x + 3)^(7/2)