3.2539 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^2}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{224} (2 x+3)^{7/2}+\frac{33}{32} (2 x+3)^{5/2}-\frac{359}{48} (2 x+3)^{3/2}+\frac{651}{16} \sqrt{2 x+3}+\frac{1065}{32 \sqrt{2 x+3}}-\frac{325}{96 (2 x+3)^{3/2}} \]

[Out]

-325/(96*(3 + 2*x)^(3/2)) + 1065/(32*Sqrt[3 + 2*x]) + (651*Sqrt[3 + 2*x])/16 - (
359*(3 + 2*x)^(3/2))/48 + (33*(3 + 2*x)^(5/2))/32 - (9*(3 + 2*x)^(7/2))/224

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Rubi [A]  time = 0.0744194, 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 \[ -\frac{9}{224} (2 x+3)^{7/2}+\frac{33}{32} (2 x+3)^{5/2}-\frac{359}{48} (2 x+3)^{3/2}+\frac{651}{16} \sqrt{2 x+3}+\frac{1065}{32 \sqrt{2 x+3}}-\frac{325}{96 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-325/(96*(3 + 2*x)^(3/2)) + 1065/(32*Sqrt[3 + 2*x]) + (651*Sqrt[3 + 2*x])/16 - (
359*(3 + 2*x)^(3/2))/48 + (33*(3 + 2*x)^(5/2))/32 - (9*(3 + 2*x)^(7/2))/224

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Rubi in Sympy [A]  time = 13.9747, size = 70, normalized size = 0.89 \[ - \frac{9 \left (2 x + 3\right )^{\frac{7}{2}}}{224} + \frac{33 \left (2 x + 3\right )^{\frac{5}{2}}}{32} - \frac{359 \left (2 x + 3\right )^{\frac{3}{2}}}{48} + \frac{651 \sqrt{2 x + 3}}{16} + \frac{1065}{32 \sqrt{2 x + 3}} - \frac{325}{96 \left (2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(2*x + 3)**(7/2)/224 + 33*(2*x + 3)**(5/2)/32 - 359*(2*x + 3)**(3/2)/48 + 651
*sqrt(2*x + 3)/16 + 1065/(32*sqrt(2*x + 3)) - 325/(96*(2*x + 3)**(3/2))

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Mathematica [A]  time = 0.0281092, size = 38, normalized size = 0.48 \[ -\frac{27 x^5-144 x^4-215 x^3-1530 x^2-7164 x-7024}{21 (2 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(-7024 - 7164*x - 1530*x^2 - 215*x^3 - 144*x^4 + 27*x^5)/(21*(3 + 2*x)^(3/2))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.4 \[ -{\frac{27\,{x}^{5}-144\,{x}^{4}-215\,{x}^{3}-1530\,{x}^{2}-7164\,x-7024}{21} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(27*x^5-144*x^4-215*x^3-1530*x^2-7164*x-7024)/(3+2*x)^(3/2)

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Maxima [A]  time = 0.709298, size = 69, normalized size = 0.87 \[ -\frac{9}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{33}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{359}{48} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{651}{16} \, \sqrt{2 \, x + 3} + \frac{5 \,{\left (639 \, x + 926\right )}}{48 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/224*(2*x + 3)^(7/2) + 33/32*(2*x + 3)^(5/2) - 359/48*(2*x + 3)^(3/2) + 651/16
*sqrt(2*x + 3) + 5/48*(639*x + 926)/(2*x + 3)^(3/2)

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Fricas [A]  time = 0.276485, size = 46, normalized size = 0.58 \[ -\frac{27 \, x^{5} - 144 \, x^{4} - 215 \, x^{3} - 1530 \, x^{2} - 7164 \, x - 7024}{21 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(27*x^5 - 144*x^4 - 215*x^3 - 1530*x^2 - 7164*x - 7024)/(2*x + 3)^(3/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{96 x}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{165 x^{2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{113 x^{3}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{15 x^{4}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \frac{9 x^{5}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{20}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-96*x/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x
) - Integral(-165*x**2/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x +
3)), x) - Integral(-113*x**3/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqr
t(2*x + 3)), x) - Integral(-15*x**4/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) +
9*sqrt(2*x + 3)), x) - Integral(9*x**5/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x +
3) + 9*sqrt(2*x + 3)), x) - Integral(-20/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x +
3) + 9*sqrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.274153, size = 69, normalized size = 0.87 \[ -\frac{9}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{33}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{359}{48} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{651}{16} \, \sqrt{2 \, x + 3} + \frac{5 \,{\left (639 \, x + 926\right )}}{48 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/224*(2*x + 3)^(7/2) + 33/32*(2*x + 3)^(5/2) - 359/48*(2*x + 3)^(3/2) + 651/16
*sqrt(2*x + 3) + 5/48*(639*x + 926)/(2*x + 3)^(3/2)