3.1360 \(\int (5-x) \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{9} \left (3 x^2+2\right )^{3/2}+\frac{5}{2} x \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]

[Out]

(5*x*Sqrt[2 + 3*x^2])/2 - (2 + 3*x^2)^(3/2)/9 + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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Rubi [A]  time = 0.0104546, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {641, 195, 215} \[ -\frac{1}{9} \left (3 x^2+2\right )^{3/2}+\frac{5}{2} x \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*Sqrt[2 + 3*x^2],x]

[Out]

(5*x*Sqrt[2 + 3*x^2])/2 - (2 + 3*x^2)^(3/2)/9 + (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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 (5-x) \sqrt{2+3 x^2} \, dx &=-\frac{1}{9} \left (2+3 x^2\right )^{3/2}+5 \int \sqrt{2+3 x^2} \, dx\\ &=\frac{5}{2} x \sqrt{2+3 x^2}-\frac{1}{9} \left (2+3 x^2\right )^{3/2}+5 \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{5}{2} x \sqrt{2+3 x^2}-\frac{1}{9} \left (2+3 x^2\right )^{3/2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0219029, size = 43, normalized size = 0.88 \[ \frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}-\frac{1}{18} \sqrt{3 x^2+2} \left (6 x^2-45 x+4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*Sqrt[2 + 3*x^2],x]

[Out]

-(Sqrt[2 + 3*x^2]*(4 - 45*x + 6*x^2))/18 + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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Maple [A]  time = 0.004, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{5\,x}{2}\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]

-1/9*(3*x^2+2)^(3/2)+5/3*arcsinh(1/2*x*6^(1/2))*3^(1/2)+5/2*x*(3*x^2+2)^(1/2)

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Maxima [A]  time = 1.46528, size = 49, normalized size = 1. \begin{align*} -\frac{1}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{5}{2} \, \sqrt{3 \, x^{2} + 2} x + \frac{5}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \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]

-1/9*(3*x^2 + 2)^(3/2) + 5/2*sqrt(3*x^2 + 2)*x + 5/3*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.2878, size = 135, normalized size = 2.76 \begin{align*} -\frac{1}{18} \,{\left (6 \, x^{2} - 45 \, x + 4\right )} \sqrt{3 \, x^{2} + 2} + \frac{5}{6} \, \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*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/18*(6*x^2 - 45*x + 4)*sqrt(3*x^2 + 2) + 5/6*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 0.360853, size = 61, normalized size = 1.24 \begin{align*} - \frac{x^{2} \sqrt{3 x^{2} + 2}}{3} + \frac{5 x \sqrt{3 x^{2} + 2}}{2} - \frac{2 \sqrt{3 x^{2} + 2}}{9} + \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]

-x**2*sqrt(3*x**2 + 2)/3 + 5*x*sqrt(3*x**2 + 2)/2 - 2*sqrt(3*x**2 + 2)/9 + 5*sqrt(3)*asinh(sqrt(6)*x/2)/3

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Giac [A]  time = 1.16903, size = 59, normalized size = 1.2 \begin{align*} -\frac{1}{18} \,{\left (3 \,{\left (2 \, x - 15\right )} x + 4\right )} \sqrt{3 \, x^{2} + 2} - \frac{5}{3} \, \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*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-1/18*(3*(2*x - 15)*x + 4)*sqrt(3*x^2 + 2) - 5/3*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))