3.1401 \(\int \frac{5-x}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}-\frac{1}{3} \sqrt{3 x^2+2} \]

[Out]

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

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Rubi [A]  time = 0.0071272, antiderivative size = 33, 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, Rules used = {641, 215} \[ \frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}-\frac{1}{3} \sqrt{3 x^2+2} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/Sqrt[2 + 3*x^2],x]

[Out]

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{5-x}{\sqrt{2+3 x^2}} \, dx &=-\frac{1}{3} \sqrt{2+3 x^2}+5 \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{2+3 x^2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0178945, size = 33, normalized size = 1. \[ \frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}-\frac{1}{3} \sqrt{3 x^2+2} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/Sqrt[2 + 3*x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*}{\frac{5\,\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{1}{3}\sqrt{3\,{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3*x^2+2)^(1/2),x)

[Out]

5/3*arcsinh(1/2*x*6^(1/2))*3^(1/2)-1/3*(3*x^2+2)^(1/2)

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Maxima [A]  time = 1.49592, size = 32, normalized size = 0.97 \begin{align*} \frac{5}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{1}{3} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.79864, size = 107, normalized size = 3.24 \begin{align*} \frac{5}{6} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - \frac{1}{3} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.174896, size = 29, normalized size = 0.88 \begin{align*} - \frac{\sqrt{3 x^{2} + 2}}{3} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x**2+2)**(1/2),x)

[Out]

-sqrt(3*x**2 + 2)/3 + 5*sqrt(3)*asinh(sqrt(6)*x/2)/3

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Giac [A]  time = 1.19741, size = 46, normalized size = 1.39 \begin{align*} -\frac{5}{3} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{1}{3} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

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