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

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{3} \sqrt {3 x^2+2} (7-x)+\frac {47 \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]

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

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 \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*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.95 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.18, size = 51, normalized size = 1.28 \begin {gather*} \frac {1}{3} (7-x) \sqrt {3 x^2+2}-\frac {47 \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]

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

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fricas [A]  time = 0.42, size = 43, normalized size = 1.08 \begin {gather*} -\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 {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]

-1/3*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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giac [A]  time = 0.17, size = 37, normalized size = 0.92 \begin {gather*} -\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 {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]

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

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maple [A]  time = 0.05, size = 37, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {3 x^{2}+2}\, x}{3}+\frac {47 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {7 \sqrt {3 x^{2}+2}}{3} \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]

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

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maxima [A]  time = 1.29, size = 36, normalized size = 0.90 \begin {gather*} -\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 {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]

-1/3*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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mupad [B]  time = 0.03, size = 28, normalized size = 0.70 \begin {gather*} \frac {47\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (x-7\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.30, size = 44, normalized size = 1.10 \begin {gather*} - \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 {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]

-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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