3.1411 \(\int \frac{(5-x) (3+2 x)}{(2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{7 (2-7 x)}{6 \sqrt{3 x^2+2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.01125, 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 = {778, 215} \[ -\frac{7 (2-7 x)}{6 \sqrt{3 x^2+2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[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)}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{7 (2-7 x)}{6 \sqrt{2+3 x^2}}-\frac{2}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x)}{6 \sqrt{2+3 x^2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0297375, size = 43, normalized size = 1.08 \[ -\frac{4 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-147 x+42}{18 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(42 - 147*x + 4*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])

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Maple [A]  time = 0.006, size = 37, normalized size = 0.9 \begin{align*}{\frac{49\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{2\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{7}{3}{\frac{1}{\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)^(3/2),x)

[Out]

49/6*x/(3*x^2+2)^(1/2)-2/9*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 = 1.49588, size = 49, normalized size = 1.22 \begin{align*} -\frac{2}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{49 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \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)^(3/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 49/6*x/sqrt(3*x^2 + 2) - 7/3/sqrt(3*x^2 + 2)

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Fricas [B]  time = 1.52625, size = 157, normalized size = 3.92 \begin{align*} \frac{2 \, \sqrt{3}{\left (3 \, x^{2} + 2\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 21 \, \sqrt{3 \, x^{2} + 2}{\left (7 \, x - 2\right )}}{18 \,{\left (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)^(3/2),x, algorithm="fricas")

[Out]

1/18*(2*sqrt(3)*(3*x^2 + 2)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 21*sqrt(3*x^2 + 2)*(7*x - 2))/(3*x^2
+ 2)

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Sympy [B]  time = 13.0461, size = 99, normalized size = 2.48 \begin{align*} - \frac{6 \sqrt{3} x^{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} + \frac{6 x \sqrt{3 x^{2} + 2}}{27 x^{2} + 18} + \frac{15 x}{2 \sqrt{3 x^{2} + 2}} - \frac{4 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} - \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)**(3/2),x)

[Out]

-6*sqrt(3)*x**2*asinh(sqrt(6)*x/2)/(27*x**2 + 18) + 6*x*sqrt(3*x**2 + 2)/(27*x**2 + 18) + 15*x/(2*sqrt(3*x**2
+ 2)) - 4*sqrt(3)*asinh(sqrt(6)*x/2)/(27*x**2 + 18) - 7/(3*sqrt(3*x**2 + 2))

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Giac [A]  time = 1.27343, size = 53, normalized size = 1.32 \begin{align*} \frac{2}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{7 \,{\left (7 \, x - 2\right )}}{6 \, \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)^(3/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 7/6*(7*x - 2)/sqrt(3*x^2 + 2)