3.1400 \(\int \frac{(5-x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{3} \sqrt{3 x^2+2} (7-x)+\frac{47 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

((7 - x)*Sqrt[2 + 3*x^2])/3 + (47*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0122139, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {780, 215} \[ \frac{1}{3} \sqrt{3 x^2+2} (7-x)+\frac{47 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7 - x)*Sqrt[2 + 3*x^2])/3 + (47*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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) (3+2 x)}{\sqrt{2+3 x^2}} \, dx &=\frac{1}{3} (7-x) \sqrt{2+3 x^2}+\frac{47}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{1}{3} (7-x) \sqrt{2+3 x^2}+\frac{47 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0229712, size = 38, normalized size = 0.95 \[ \frac{1}{9} \left (47 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-3 (x-7) \sqrt{3 x^2+2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-7 + x)*Sqrt[2 + 3*x^2] + 47*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/9

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Maple [A]  time = 0.006, size = 37, normalized size = 0.9 \begin{align*} -{\frac{x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{47\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{3}\sqrt{3\,{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x*(3*x^2+2)^(1/2)+47/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+7/3*(3*x^2+2)^(1/2)

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Maxima [A]  time = 1.4684, size = 49, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{47}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{7}{3} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3*x^2 + 2)*x + 47/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 7/3*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.63284, size = 122, normalized size = 3.05 \begin{align*} -\frac{1}{3} \, \sqrt{3 \, x^{2} + 2}{\left (x - 7\right )} + \frac{47}{18} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3*x^2 + 2)*(x - 7) + 47/18*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 0.318595, size = 44, normalized size = 1.1 \begin{align*} - \frac{x \sqrt{3 x^{2} + 2}}{3} + \frac{7 \sqrt{3 x^{2} + 2}}{3} + \frac{47 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt(3*x**2 + 2)/3 + 7*sqrt(3*x**2 + 2)/3 + 47*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Giac [A]  time = 1.17074, size = 50, normalized size = 1.25 \begin{align*} -\frac{1}{3} \, \sqrt{3 \, x^{2} + 2}{\left (x - 7\right )} - \frac{47}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3*x^2 + 2)*(x - 7) - 47/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))