3.15.1 \(\int \frac {5-x}{\sqrt {2+3 x^2}} \, dx\) [1401]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {655, 221} \begin {gather*} \frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {3 x^2+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/Sqrt[2 + 3*x^2],x]

[Out]

-1/3*Sqrt[2 + 3*x^2] + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

Rule 221

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]

Rule 655

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]

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*}

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Mathematica [A]
time = 0.08, size = 44, normalized size = 1.33 \begin {gather*} -\frac {1}{3} \sqrt {2+3 x^2}-\frac {5 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/Sqrt[2 + 3*x^2],x]

[Out]

-1/3*Sqrt[2 + 3*x^2] - (5*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/Sqrt[3]

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Maple [A]
time = 0.56, size = 25, normalized size = 0.76

method	result	size
default	\(\frac {5 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{3}-\frac {\sqrt {3 x^{2}+2}}{3}\)	\(25\)
risch	\(\frac {5 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{3}-\frac {\sqrt {3 x^{2}+2}}{3}\)	\(25\)
trager	\(-\frac {\sqrt {3 x^{2}+2}}{3}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{3}\)	\(43\)
meijerg	\(\frac {5 \sqrt {3}\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}-\frac {\sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{6 \sqrt {\pi }}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.48, size = 24, normalized size = 0.73 \begin {gather*} \frac {5}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \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)^(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 = 2.50, size = 40, normalized size = 1.21 \begin {gather*} \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 {gather*}

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 = 0.06, size = 29, normalized size = 0.88 \begin {gather*} - \frac {\sqrt {3 x^{2} + 2}}{3} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{3} \end {gather*}

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 = 1.47, size = 34, normalized size = 1.03 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.76 \begin {gather*} \frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{3}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/(3*x^2 + 2)^(1/2),x)

[Out]

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

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