3.2548 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{3}{128} (2 x+3)^{9/2}+\frac{81}{128} (2 x+3)^{7/2}-\frac{3519}{640} (2 x+3)^{5/2}+\frac{10475}{384} (2 x+3)^{3/2}-\frac{17201}{128} \sqrt{2 x+3}-\frac{16005}{128 \sqrt{2 x+3}}+\frac{7925}{384 (2 x+3)^{3/2}}-\frac{325}{128 (2 x+3)^{5/2}} \]

[Out]

-325/(128*(3 + 2*x)^(5/2)) + 7925/(384*(3 + 2*x)^(3/2)) - 16005/(128*Sqrt[3 + 2*
x]) - (17201*Sqrt[3 + 2*x])/128 + (10475*(3 + 2*x)^(3/2))/384 - (3519*(3 + 2*x)^
(5/2))/640 + (81*(3 + 2*x)^(7/2))/128 - (3*(3 + 2*x)^(9/2))/128

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Rubi [A]  time = 0.0897821, antiderivative size = 105, 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 \[ -\frac{3}{128} (2 x+3)^{9/2}+\frac{81}{128} (2 x+3)^{7/2}-\frac{3519}{640} (2 x+3)^{5/2}+\frac{10475}{384} (2 x+3)^{3/2}-\frac{17201}{128} \sqrt{2 x+3}-\frac{16005}{128 \sqrt{2 x+3}}+\frac{7925}{384 (2 x+3)^{3/2}}-\frac{325}{128 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^3)/(3 + 2*x)^(7/2),x]

[Out]

-325/(128*(3 + 2*x)^(5/2)) + 7925/(384*(3 + 2*x)^(3/2)) - 16005/(128*Sqrt[3 + 2*
x]) - (17201*Sqrt[3 + 2*x])/128 + (10475*(3 + 2*x)^(3/2))/384 - (3519*(3 + 2*x)^
(5/2))/640 + (81*(3 + 2*x)^(7/2))/128 - (3*(3 + 2*x)^(9/2))/128

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Rubi in Sympy [A]  time = 15.6679, size = 94, normalized size = 0.9 \[ - \frac{3 \left (2 x + 3\right )^{\frac{9}{2}}}{128} + \frac{81 \left (2 x + 3\right )^{\frac{7}{2}}}{128} - \frac{3519 \left (2 x + 3\right )^{\frac{5}{2}}}{640} + \frac{10475 \left (2 x + 3\right )^{\frac{3}{2}}}{384} - \frac{17201 \sqrt{2 x + 3}}{128} - \frac{16005}{128 \sqrt{2 x + 3}} + \frac{7925}{384 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{325}{128 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**3/(3+2*x)**(7/2),x)

[Out]

-3*(2*x + 3)**(9/2)/128 + 81*(2*x + 3)**(7/2)/128 - 3519*(2*x + 3)**(5/2)/640 +
10475*(2*x + 3)**(3/2)/384 - 17201*sqrt(2*x + 3)/128 - 16005/(128*sqrt(2*x + 3))
+ 7925/(384*(2*x + 3)**(3/2)) - 325/(128*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.0460942, size = 48, normalized size = 0.46 \[ -\frac{45 x^7-135 x^6-702 x^5-1940 x^4+3195 x^3+41805 x^2+85070 x+51162}{15 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^3)/(3 + 2*x)^(7/2),x]

[Out]

-(51162 + 85070*x + 41805*x^2 + 3195*x^3 - 1940*x^4 - 702*x^5 - 135*x^6 + 45*x^7
)/(15*(3 + 2*x)^(5/2))

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Maple [A]  time = 0.007, size = 45, normalized size = 0.4 \[ -{\frac{45\,{x}^{7}-135\,{x}^{6}-702\,{x}^{5}-1940\,{x}^{4}+3195\,{x}^{3}+41805\,{x}^{2}+85070\,x+51162}{15} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^3/(3+2*x)^(7/2),x)

[Out]

-1/15*(45*x^7-135*x^6-702*x^5-1940*x^4+3195*x^3+41805*x^2+85070*x+51162)/(3+2*x)
^(5/2)

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Maxima [A]  time = 0.704176, size = 93, normalized size = 0.89 \[ -\frac{3}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{81}{128} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{3519}{640} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{10475}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{17201}{128} \, \sqrt{2 \, x + 3} - \frac{5 \,{\left (9603 \,{\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^3*(x - 5)/(2*x + 3)^(7/2),x, algorithm="maxima")

[Out]

-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 104
75/384*(2*x + 3)^(3/2) - 17201/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 317
0*x - 4560)/(2*x + 3)^(5/2)

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Fricas [A]  time = 0.277456, size = 76, normalized size = 0.72 \[ -\frac{45 \, x^{7} - 135 \, x^{6} - 702 \, x^{5} - 1940 \, x^{4} + 3195 \, x^{3} + 41805 \, x^{2} + 85070 \, x + 51162}{15 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^3*(x - 5)/(2*x + 3)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(45*x^7 - 135*x^6 - 702*x^5 - 1940*x^4 + 3195*x^3 + 41805*x^2 + 85070*x +
51162)/((4*x^2 + 12*x + 9)*sqrt(2*x + 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{292 x}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{870 x^{2}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{1339 x^{3}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{1090 x^{4}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{396 x^{5}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \frac{27 x^{7}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{40}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**3/(3+2*x)**(7/2),x)

[Out]

-Integral(-292*x/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x +
3) + 27*sqrt(2*x + 3)), x) - Integral(-870*x**2/(8*x**3*sqrt(2*x + 3) + 36*x**2
*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(-1339*x**
3/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2
*x + 3)), x) - Integral(-1090*x**4/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3)
+ 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(-396*x**5/(8*x**3*sqrt(
2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) -
Integral(27*x**7/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x +
3) + 27*sqrt(2*x + 3)), x) - Integral(-40/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(
2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.271032, size = 93, normalized size = 0.89 \[ -\frac{3}{128} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{81}{128} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{3519}{640} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{10475}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{17201}{128} \, \sqrt{2 \, x + 3} - \frac{5 \,{\left (9603 \,{\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^3*(x - 5)/(2*x + 3)^(7/2),x, algorithm="giac")

[Out]

-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 104
75/384*(2*x + 3)^(3/2) - 17201/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 317
0*x - 4560)/(2*x + 3)^(5/2)