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

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

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.88 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*Sqrt[2 + 3*x^2],x]

[Out]

-1/18*(Sqrt[2 + 3*x^2]*(4 - 45*x + 6*x^2)) + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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IntegrateAlgebraic [A]  time = 0.13, size = 54, normalized size = 1.10 \begin {gather*} \frac {1}{18} \left (-6 x^2+45 x-4\right ) \sqrt {3 x^2+2}-\frac {5 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 50, normalized size = 1.02 \begin {gather*} -\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 {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]

-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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giac [A]  time = 0.16, size = 44, normalized size = 0.90 \begin {gather*} -\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 {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]

-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))

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maple [A]  time = 0.05, size = 37, normalized size = 0.76 \begin {gather*} \frac {5 \sqrt {3 x^{2}+2}\, x}{2}+\frac {5 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{3}-\frac {\left (3 x^{2}+2\right )^{\frac {3}{2}}}{9} \end {gather*}

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

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maxima [A]  time = 1.38, size = 36, normalized size = 0.73 \begin {gather*} -\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 {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]

-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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mupad [B]  time = 0.03, size = 33, normalized size = 0.67 \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}}\,\left (x^2-\frac {15\,x}{2}+\frac {2}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 + 2)^(1/2)*(x - 5),x)

[Out]

(5*3^(1/2)*asinh((6^(1/2)*x)/2))/3 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(x^2 - (15*x)/2 + 2/3))/3

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sympy [A]  time = 0.26, size = 61, normalized size = 1.24 \begin {gather*} - \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 {gather*}

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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