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

Optimal. Leaf size=56 \[ \frac {1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac {23}{3} x \sqrt {3 x^2+2}+\frac {46 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {780, 195, 215} \begin {gather*} \frac {1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac {23}{3} x \sqrt {3 x^2+2}+\frac {46 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(23*x*Sqrt[2 + 3*x^2])/3 + ((14 - 3*x)*(2 + 3*x^2)^(3/2))/18 + (46*ArcSinh[Sqrt[3/2]*x])/(3*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 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]

Rubi steps

\begin {align*} \int (5-x) (3+2 x) \sqrt {2+3 x^2} \, dx &=\frac {1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac {46}{3} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {23}{3} x \sqrt {2+3 x^2}+\frac {1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac {46}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {23}{3} x \sqrt {2+3 x^2}+\frac {1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac {46 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{18} \left (92 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (9 x^3-42 x^2-132 x-28\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-28 - 132*x - 42*x^2 + 9*x^3)) + 92*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/18

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IntegrateAlgebraic [A]  time = 0.18, size = 61, normalized size = 1.09 \begin {gather*} \frac {1}{18} \sqrt {3 x^2+2} \left (-9 x^3+42 x^2+132 x+28\right )-\frac {46 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(28 + 132*x + 42*x^2 - 9*x^3))/18 - (46*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])

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fricas [A]  time = 0.42, size = 55, normalized size = 0.98 \begin {gather*} -\frac {1}{18} \, {\left (9 \, x^{3} - 42 \, x^{2} - 132 \, x - 28\right )} \sqrt {3 \, x^{2} + 2} + \frac {23}{9} \, \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+2*x)*(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/18*(9*x^3 - 42*x^2 - 132*x - 28)*sqrt(3*x^2 + 2) + 23/9*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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giac [A]  time = 0.22, size = 48, normalized size = 0.86 \begin {gather*} -\frac {1}{18} \, {\left (3 \, {\left ({\left (3 \, x - 14\right )} x - 44\right )} x - 28\right )} \sqrt {3 \, x^{2} + 2} - \frac {46}{9} \, \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+2*x)*(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-1/18*(3*((3*x - 14)*x - 44)*x - 28)*sqrt(3*x^2 + 2) - 46/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))

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maple [A]  time = 0.06, size = 49, normalized size = 0.88 \begin {gather*} -\frac {\left (3 x^{2}+2\right )^{\frac {3}{2}} x}{6}+\frac {23 \sqrt {3 x^{2}+2}\, x}{3}+\frac {46 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {7 \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)*(2*x+3)*(3*x^2+2)^(1/2),x)

[Out]

-1/6*(3*x^2+2)^(3/2)*x+23/3*(3*x^2+2)^(1/2)*x+46/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+7/9*(3*x^2+2)^(3/2)

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maxima [A]  time = 1.46, size = 48, normalized size = 0.86 \begin {gather*} -\frac {1}{6} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {7}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {23}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {46}{9} \, \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+2*x)*(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-1/6*(3*x^2 + 2)^(3/2)*x + 7/9*(3*x^2 + 2)^(3/2) + 23/3*sqrt(3*x^2 + 2)*x + 46/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x
)

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mupad [B]  time = 0.03, size = 40, normalized size = 0.71 \begin {gather*} \frac {46\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {3\,x^3}{2}+7\,x^2+22\,x+\frac {14}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(46*3^(1/2)*asinh((6^(1/2)*x)/2))/9 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(22*x + 7*x^2 - (3*x^3)/2 + 14/3))/3

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sympy [A]  time = 8.08, size = 94, normalized size = 1.68 \begin {gather*} - \frac {3 x^{5}}{2 \sqrt {3 x^{2} + 2}} - \frac {3 x^{3}}{2 \sqrt {3 x^{2} + 2}} + \frac {15 x \sqrt {3 x^{2} + 2}}{2} - \frac {x}{3 \sqrt {3 x^{2} + 2}} + \frac {7 \left (3 x^{2} + 2\right )^{\frac {3}{2}}}{9} + \frac {46 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \end {gather*}

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]

-3*x**5/(2*sqrt(3*x**2 + 2)) - 3*x**3/(2*sqrt(3*x**2 + 2)) + 15*x*sqrt(3*x**2 + 2)/2 - x/(3*sqrt(3*x**2 + 2))
+ 7*(3*x**2 + 2)**(3/2)/9 + 46*sqrt(3)*asinh(sqrt(6)*x/2)/9

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