3.2528 \(\int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right ) \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{24} (2 x+3)^{9/2}+\frac{47}{56} (2 x+3)^{7/2}-\frac{109}{40} (2 x+3)^{5/2}+\frac{65}{24} (2 x+3)^{3/2} \]

[Out]

(65*(3 + 2*x)^(3/2))/24 - (109*(3 + 2*x)^(5/2))/40 + (47*(3 + 2*x)^(7/2))/56 - (
3 + 2*x)^(9/2)/24

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Rubi [A]  time = 0.0527214, 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 \[ -\frac{1}{24} (2 x+3)^{9/2}+\frac{47}{56} (2 x+3)^{7/2}-\frac{109}{40} (2 x+3)^{5/2}+\frac{65}{24} (2 x+3)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(65*(3 + 2*x)^(3/2))/24 - (109*(3 + 2*x)^(5/2))/40 + (47*(3 + 2*x)^(7/2))/56 - (
3 + 2*x)^(9/2)/24

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Rubi in Sympy [A]  time = 9.53922, size = 44, normalized size = 0.83 \[ - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{24} + \frac{47 \left (2 x + 3\right )^{\frac{7}{2}}}{56} - \frac{109 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{65 \left (2 x + 3\right )^{\frac{3}{2}}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(9/2)/24 + 47*(2*x + 3)**(7/2)/56 - 109*(2*x + 3)**(5/2)/40 + 65*(2*
x + 3)**(3/2)/24

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Mathematica [A]  time = 0.0180726, size = 28, normalized size = 0.53 \[ -\frac{1}{105} (2 x+3)^{3/2} \left (35 x^3-195 x^2-249 x-101\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((3 + 2*x)^(3/2)*(-101 - 249*x - 195*x^2 + 35*x^3))/105

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Maple [A]  time = 0.005, size = 25, normalized size = 0.5 \[ -{\frac{35\,{x}^{3}-195\,{x}^{2}-249\,x-101}{105} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(35*x^3-195*x^2-249*x-101)*(3+2*x)^(3/2)

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Maxima [A]  time = 0.711102, size = 50, normalized size = 0.94 \[ -\frac{1}{24} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{47}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{109}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{65}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(2*x + 3)^(9/2) + 47/56*(2*x + 3)^(7/2) - 109/40*(2*x + 3)^(5/2) + 65/24*(
2*x + 3)^(3/2)

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Fricas [A]  time = 0.280466, size = 39, normalized size = 0.74 \[ -\frac{1}{105} \,{\left (70 \, x^{4} - 285 \, x^{3} - 1083 \, x^{2} - 949 \, x - 303\right )} \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(70*x^4 - 285*x^3 - 1083*x^2 - 949*x - 303)*sqrt(2*x + 3)

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Sympy [A]  time = 3.29668, size = 44, normalized size = 0.83 \[ - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{24} + \frac{47 \left (2 x + 3\right )^{\frac{7}{2}}}{56} - \frac{109 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{65 \left (2 x + 3\right )^{\frac{3}{2}}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(9/2)/24 + 47*(2*x + 3)**(7/2)/56 - 109*(2*x + 3)**(5/2)/40 + 65*(2*
x + 3)**(3/2)/24

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GIAC/XCAS [A]  time = 0.267082, size = 50, normalized size = 0.94 \[ -\frac{1}{24} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{47}{56} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{109}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{65}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(2*x + 3)^(9/2) + 47/56*(2*x + 3)^(7/2) - 109/40*(2*x + 3)^(5/2) + 65/24*(
2*x + 3)^(3/2)