3.2509 \(\int \frac{5-x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2 (35 x+29)}{\sqrt{3 x^2+5 x+2}} \]

[Out]

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

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Rubi [A]  time = 0.021044, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 (35 x+29)}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.1069, size = 19, normalized size = 0.9 \[ - \frac{70 x + 58}{\sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(70*x + 58)/sqrt(3*x**2 + 5*x + 2)

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Mathematica [A]  time = 0.0192246, size = 21, normalized size = 1. \[ -\frac{2 (35 x+29)}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 1.3 \[ -2\,{\frac{ \left ( 29+35\,x \right ) \left ( 1+x \right ) \left ( 2+3\,x \right ) }{ \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(29+35*x)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(3/2)

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Maxima [A]  time = 0.703384, size = 41, normalized size = 1.95 \[ -\frac{70 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{58}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.274594, size = 26, normalized size = 1.24 \[ -\frac{2 \,{\left (35 \, x + 29\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(35*x + 29)/sqrt(3*x^2 + 5*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt
(3*x**2 + 5*x + 2)), x) - Integral(-5/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(
3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.28092, size = 26, normalized size = 1.24 \[ -\frac{2 \,{\left (35 \, x + 29\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(35*x + 29)/sqrt(3*x^2 + 5*x + 2)