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

Optimal. Leaf size=20 \[ \frac {15 x+2}{6 \sqrt {3 x^2+2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {637} \begin {gather*} \frac {15 x+2}{6 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/(2 + 3*x^2)^(3/2),x]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

Rule 637

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

Rubi steps

\begin {align*} \int \frac {5-x}{\left (2+3 x^2\right )^{3/2}} \, dx &=\frac {2+15 x}{6 \sqrt {2+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {15 x+2}{6 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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IntegrateAlgebraic [A]  time = 0.20, size = 20, normalized size = 1.00 \begin {gather*} \frac {15 x+2}{6 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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fricas [A]  time = 0.41, size = 16, normalized size = 0.80 \begin {gather*} \frac {15 \, x + 2}{6 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)

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giac [A]  time = 0.17, size = 16, normalized size = 0.80 \begin {gather*} \frac {15 \, x + 2}{6 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)

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maple [A]  time = 0.04, size = 17, normalized size = 0.85 \begin {gather*} \frac {15 x +2}{6 \sqrt {3 x^{2}+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2+15*x)/(3*x^2+2)^(1/2)

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maxima [A]  time = 0.54, size = 24, normalized size = 1.20 \begin {gather*} \frac {5 \, x}{2 \, \sqrt {3 \, x^{2} + 2}} + \frac {1}{3 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 15, normalized size = 0.75 \begin {gather*} \frac {\frac {5\,x}{2}+\frac {1}{3}}{\sqrt {3\,x^2+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/(3*x^2 + 2)^(3/2),x)

[Out]

((5*x)/2 + 1/3)/(3*x^2 + 2)^(1/2)

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sympy [A]  time = 14.37, size = 27, normalized size = 1.35 \begin {gather*} \frac {5 x}{2 \sqrt {3 x^{2} + 2}} + \frac {1}{3 \sqrt {3 x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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