3.2519 \(\int \frac{5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{280 (6 x+5)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(-2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (280*(5 + 6*x))/(3*Sqrt[2 + 5*x +
3*x^2])

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Rubi [A]  time = 0.0319071, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{280 (6 x+5)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (280*(5 + 6*x))/(3*Sqrt[2 + 5*x +
3*x^2])

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Rubi in Sympy [A]  time = 5.3281, size = 39, normalized size = 0.83 \[ \frac{140 \left (12 x + 10\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \left (35 x + 29\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

140*(12*x + 10)/(3*sqrt(3*x**2 + 5*x + 2)) - 2*(35*x + 29)/(3*(3*x**2 + 5*x + 2)
**(3/2))

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Mathematica [A]  time = 0.0249398, size = 31, normalized size = 0.66 \[ \frac{2 \left (840 x^3+2100 x^2+1715 x+457\right )}{\left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(457 + 1715*x + 2100*x^2 + 840*x^3))/(2 + 5*x + 3*x^2)^(3/2)

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \[ 2\,{\frac{ \left ( 840\,{x}^{3}+2100\,{x}^{2}+1715\,x+457 \right ) \left ( 1+x \right ) \left ( 2+3\,x \right ) }{ \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(840*x^3+2100*x^2+1715*x+457)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.712843, size = 80, normalized size = 1.7 \[ \frac{560 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{1400}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{70 \, x}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{58}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

560*x/sqrt(3*x^2 + 5*x + 2) + 1400/3/sqrt(3*x^2 + 5*x + 2) - 70/3*x/(3*x^2 + 5*x
+ 2)^(3/2) - 58/3/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.271774, size = 69, normalized size = 1.47 \[ \frac{2 \,{\left (840 \, x^{3} + 2100 \, x^{2} + 1715 \, x + 457\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(840*x^3 + 2100*x^2 + 1715*x + 457)*sqrt(3*x^2 + 5*x + 2)/(9*x^4 + 30*x^3 + 37
*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37
*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x
+ 2)), x) - Integral(-5/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 +
5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt
(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.274848, size = 39, normalized size = 0.83 \[ \frac{2 \,{\left (35 \,{\left (12 \,{\left (2 \, x + 5\right )} x + 49\right )} x + 457\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(35*(12*(2*x + 5)*x + 49)*x + 457)/(3*x^2 + 5*x + 2)^(3/2)