3.2540 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^2}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{9}{160} (2 x+3)^{5/2}+\frac{55}{32} (2 x+3)^{3/2}-\frac{359}{16} \sqrt{2 x+3}-\frac{651}{16 \sqrt{2 x+3}}+\frac{355}{32 (2 x+3)^{3/2}}-\frac{65}{32 (2 x+3)^{5/2}} \]

[Out]

-65/(32*(3 + 2*x)^(5/2)) + 355/(32*(3 + 2*x)^(3/2)) - 651/(16*Sqrt[3 + 2*x]) - (
359*Sqrt[3 + 2*x])/16 + (55*(3 + 2*x)^(3/2))/32 - (9*(3 + 2*x)^(5/2))/160

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Rubi [A]  time = 0.0759172, 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 \[ -\frac{9}{160} (2 x+3)^{5/2}+\frac{55}{32} (2 x+3)^{3/2}-\frac{359}{16} \sqrt{2 x+3}-\frac{651}{16 \sqrt{2 x+3}}+\frac{355}{32 (2 x+3)^{3/2}}-\frac{65}{32 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-65/(32*(3 + 2*x)^(5/2)) + 355/(32*(3 + 2*x)^(3/2)) - 651/(16*Sqrt[3 + 2*x]) - (
359*Sqrt[3 + 2*x])/16 + (55*(3 + 2*x)^(3/2))/32 - (9*(3 + 2*x)^(5/2))/160

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Rubi in Sympy [A]  time = 13.9076, size = 70, normalized size = 0.89 \[ - \frac{9 \left (2 x + 3\right )^{\frac{5}{2}}}{160} + \frac{55 \left (2 x + 3\right )^{\frac{3}{2}}}{32} - \frac{359 \sqrt{2 x + 3}}{16} - \frac{651}{16 \sqrt{2 x + 3}} + \frac{355}{32 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{65}{32 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(2*x + 3)**(5/2)/160 + 55*(2*x + 3)**(3/2)/32 - 359*sqrt(2*x + 3)/16 - 651/(1
6*sqrt(2*x + 3)) + 355/(32*(2*x + 3)**(3/2)) - 65/(32*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.027955, size = 38, normalized size = 0.48 \[ -\frac{9 x^5-70 x^4+275 x^3+3300 x^2+6760 x+4076}{5 (2 x+3)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4076 + 6760*x + 3300*x^2 + 275*x^3 - 70*x^4 + 9*x^5)/(5*(3 + 2*x)^(5/2))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.4 \[ -{\frac{9\,{x}^{5}-70\,{x}^{4}+275\,{x}^{3}+3300\,{x}^{2}+6760\,x+4076}{5} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(9*x^5-70*x^4+275*x^3+3300*x^2+6760*x+4076)/(3+2*x)^(5/2)

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Maxima [A]  time = 0.70928, size = 69, normalized size = 0.87 \[ -\frac{9}{160} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{55}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{359}{16} \, \sqrt{2 \, x + 3} - \frac{651 \,{\left (2 \, x + 3\right )}^{2} - 355 \, x - 500}{16 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/160*(2*x + 3)^(5/2) + 55/32*(2*x + 3)^(3/2) - 359/16*sqrt(2*x + 3) - 1/16*(65
1*(2*x + 3)^2 - 355*x - 500)/(2*x + 3)^(5/2)

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Fricas [A]  time = 0.276185, size = 62, normalized size = 0.78 \[ -\frac{9 \, x^{5} - 70 \, x^{4} + 275 \, x^{3} + 3300 \, x^{2} + 6760 \, x + 4076}{5 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(9*x^5 - 70*x^4 + 275*x^3 + 3300*x^2 + 6760*x + 4076)/((4*x^2 + 12*x + 9)*s
qrt(2*x + 3))

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Sympy [A]  time = 5.26751, size = 238, normalized size = 3.01 \[ - \frac{9 x^{5}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} + \frac{70 x^{4}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{275 x^{3}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{3300 x^{2}}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{6760 x}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} - \frac{4076}{20 x^{2} \sqrt{2 x + 3} + 60 x \sqrt{2 x + 3} + 45 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*x**5/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) + 70*x**
4/(20*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 275*x**3/(20
*x**2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 3300*x**2/(20*x**
2*sqrt(2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 6760*x/(20*x**2*sqrt(
2*x + 3) + 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3)) - 4076/(20*x**2*sqrt(2*x + 3)
+ 60*x*sqrt(2*x + 3) + 45*sqrt(2*x + 3))

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GIAC/XCAS [A]  time = 0.267435, size = 69, normalized size = 0.87 \[ -\frac{9}{160} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{55}{32} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{359}{16} \, \sqrt{2 \, x + 3} - \frac{651 \,{\left (2 \, x + 3\right )}^{2} - 355 \, x - 500}{16 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/160*(2*x + 3)^(5/2) + 55/32*(2*x + 3)^(3/2) - 359/16*sqrt(2*x + 3) - 1/16*(65
1*(2*x + 3)^2 - 355*x - 500)/(2*x + 3)^(5/2)