3.2510 \(\int \frac{5-x}{(2+5 x+3 x^2)^{3/2}} \, dx\)

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 = 0.0049068, 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, Rules used = {636} \[ -\frac{2 (35 x+29)}{\sqrt{3 x^2+5 x+2}} \]

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]

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{5-x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (29+35 x)}{\sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.052172, 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 = 0.002, size = 28, normalized size = 1.3 \begin{align*} -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}}} \end{align*}

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 = 1.03045, size = 41, normalized size = 1.95 \begin{align*} -\frac{70 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{58}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(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 = 1.87837, size = 51, normalized size = 2.43 \begin{align*} -\frac{2 \,{\left (35 \, x + 29\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(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. \begin{align*} - \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 - \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}}\, dx \end{align*}

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) - Inte
gral(-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 [A]  time = 1.09593, size = 26, normalized size = 1.24 \begin{align*} -\frac{2 \,{\left (35 \, x + 29\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(35*x + 29)/sqrt(3*x^2 + 5*x + 2)