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)/(3*x^2+5*x+2)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 1.00 \[ -\frac {2 (35 x+29)}{\sqrt {3 x^2+5 x+2}} \]

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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fricas [A]  time = 0.49, size = 19, normalized size = 0.90 \[ -\frac {2 \, {\left (35 \, x + 29\right )}}{\sqrt {3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.23, size = 19, normalized size = 0.90 \[ -\frac {2 \, {\left (35 \, x + 29\right )}}{\sqrt {3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [A]  time = 0.00, size = 28, normalized size = 1.33 \[ -\frac {2 \left (35 x +29\right ) \left (x +1\right ) \left (3 x +2\right )}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}} \]

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 = 0.51, size = 30, normalized size = 1.43 \[ -\frac {70 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {58}{\sqrt {3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [B]  time = 0.07, size = 19, normalized size = 0.90 \[ -\frac {2\,\left (35\,x+29\right )}{\sqrt {3\,x^2+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

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) - 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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