3.1424 \(\int \frac{5-x}{\left (2+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

[Out]

(2 + 15*x)/(18*(2 + 3*x^2)^(3/2)) + (5*x)/(6*Sqrt[2 + 3*x^2])

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Rubi [A]  time = 0.0232922, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{5 x}{6 \sqrt{3 x^2+2}}+\frac{15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2 + 15*x)/(18*(2 + 3*x^2)^(3/2)) + (5*x)/(6*Sqrt[2 + 3*x^2])

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Rubi in Sympy [A]  time = 3.28255, size = 31, normalized size = 0.84 \[ \frac{5 x}{6 \sqrt{3 x^{2} + 2}} + \frac{15 x + 2}{18 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x/(6*sqrt(3*x**2 + 2)) + (15*x + 2)/(18*(3*x**2 + 2)**(3/2))

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Mathematica [A]  time = 0.0167617, size = 25, normalized size = 0.68 \[ \frac{45 x^3+45 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2 + 45*x + 45*x^3)/(18*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.6 \[{\frac{45\,{x}^{3}+45\,x+2}{18} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+2)^(5/2),x)

[Out]

1/18*(45*x^3+45*x+2)/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.677444, size = 49, normalized size = 1.32 \[ \frac{5 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} + \frac{5 \, x}{6 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{1}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*x/sqrt(3*x^2 + 2) + 5/6*x/(3*x^2 + 2)^(3/2) + 1/9/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.269364, size = 47, normalized size = 1.27 \[ \frac{{\left (45 \, x^{3} + 45 \, x + 2\right )} \sqrt{3 \, x^{2} + 2}}{18 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(45*x^3 + 45*x + 2)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.309827, size = 28, normalized size = 0.76 \[ \frac{45 \,{\left (x^{2} + 1\right )} x + 2}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(45*(x^2 + 1)*x + 2)/(3*x^2 + 2)^(3/2)