3.2538 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^2}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]

[Out]

-325/(32*Sqrt[3 + 2*x]) - (1065*Sqrt[3 + 2*x])/32 + (217*(3 + 2*x)^(3/2))/16 - (
359*(3 + 2*x)^(5/2))/80 + (165*(3 + 2*x)^(7/2))/224 - (3 + 2*x)^(9/2)/32

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Rubi [A]  time = 0.0749355, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-325/(32*Sqrt[3 + 2*x]) - (1065*Sqrt[3 + 2*x])/32 + (217*(3 + 2*x)^(3/2))/16 - (
359*(3 + 2*x)^(5/2))/80 + (165*(3 + 2*x)^(7/2))/224 - (3 + 2*x)^(9/2)/32

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Rubi in Sympy [A]  time = 14.1377, size = 68, normalized size = 0.86 \[ - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{32} + \frac{165 \left (2 x + 3\right )^{\frac{7}{2}}}{224} - \frac{359 \left (2 x + 3\right )^{\frac{5}{2}}}{80} + \frac{217 \left (2 x + 3\right )^{\frac{3}{2}}}{16} - \frac{1065 \sqrt{2 x + 3}}{32} - \frac{325}{32 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(9/2)/32 + 165*(2*x + 3)**(7/2)/224 - 359*(2*x + 3)**(5/2)/80 + 217*
(2*x + 3)**(3/2)/16 - 1065*sqrt(2*x + 3)/32 - 325/(32*sqrt(2*x + 3))

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Mathematica [A]  time = 0.0259753, size = 38, normalized size = 0.48 \[ -\frac{35 x^5-150 x^4-431 x^3-632 x^2+432 x+1996}{35 \sqrt{2 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(1996 + 432*x - 632*x^2 - 431*x^3 - 150*x^4 + 35*x^5)/(35*Sqrt[3 + 2*x])

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Maple [A]  time = 0.007, size = 35, normalized size = 0.4 \[ -{\frac{35\,{x}^{5}-150\,{x}^{4}-431\,{x}^{3}-632\,{x}^{2}+432\,x+1996}{35}{\frac{1}{\sqrt{3+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(35*x^5-150*x^4-431*x^3-632*x^2+432*x+1996)/(3+2*x)^(1/2)

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Maxima [A]  time = 0.706459, size = 74, normalized size = 0.94 \[ -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/1
6*(2*x + 3)^(3/2) - 1065/32*sqrt(2*x + 3) - 325/32/sqrt(2*x + 3)

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Fricas [A]  time = 0.276237, size = 46, normalized size = 0.58 \[ -\frac{35 \, x^{5} - 150 \, x^{4} - 431 \, x^{3} - 632 \, x^{2} + 432 \, x + 1996}{35 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(35*x^5 - 150*x^4 - 431*x^3 - 632*x^2 + 432*x + 1996)/sqrt(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{96 x}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{165 x^{2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{113 x^{3}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{15 x^{4}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \frac{9 x^{5}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{20}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-96*x/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x) - Integral(-165*x**2/(
2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x) - Integral(-113*x**3/(2*x*sqrt(2*x + 3)
+ 3*sqrt(2*x + 3)), x) - Integral(-15*x**4/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)
), x) - Integral(9*x**5/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x) - Integral(-20
/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.271176, size = 74, normalized size = 0.94 \[ -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(2*x + 3)^(9/2) + 165/224*(2*x + 3)^(7/2) - 359/80*(2*x + 3)^(5/2) + 217/1
6*(2*x + 3)^(3/2) - 1065/32*sqrt(2*x + 3) - 325/32/sqrt(2*x + 3)