3.1409 \(\int \frac{(5-x) (3+2 x)^3}{(2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{7 (2-7 x) (2 x+3)^2}{6 \sqrt{3 x^2+2}}-\frac{2}{9} (51 x+131) \sqrt{3 x^2+2}+\frac{134 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) - (2*(131 + 51*x)*Sqrt[2 + 3*x^2])/9 + (134*ArcSinh[Sqrt[3/2]*x
])/(3*Sqrt[3])

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Rubi [A]  time = 0.0270143, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {819, 780, 215} \[ -\frac{7 (2-7 x) (2 x+3)^2}{6 \sqrt{3 x^2+2}}-\frac{2}{9} (51 x+131) \sqrt{3 x^2+2}+\frac{134 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) - (2*(131 + 51*x)*Sqrt[2 + 3*x^2])/9 + (134*ArcSinh[Sqrt[3/2]*x
])/(3*Sqrt[3])

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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]

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)^3}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)^2}{6 \sqrt{2+3 x^2}}+\frac{1}{6} \int \frac{(44-204 x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^2}{6 \sqrt{2+3 x^2}}-\frac{2}{9} (131+51 x) \sqrt{2+3 x^2}+\frac{134}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^2}{6 \sqrt{2+3 x^2}}-\frac{2}{9} (131+51 x) \sqrt{2+3 x^2}+\frac{134 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.048176, size = 53, normalized size = 0.79 \[ -\frac{24 x^3-24 x^2-268 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-411 x+1426}{18 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1426 - 411*x - 24*x^2 + 24*x^3 - 268*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 = 65, normalized size = 1. \begin{align*} -{\frac{4\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{137\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{134\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{713}{9}{\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/(3*x^2+2)^(3/2),x)

[Out]

-4/3*x^3/(3*x^2+2)^(1/2)+137/6*x/(3*x^2+2)^(1/2)+134/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+4/3*x^2/(3*x^2+2)^(1/2)-
713/9/(3*x^2+2)^(1/2)

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Maxima [A]  time = 1.48426, size = 86, normalized size = 1.28 \begin{align*} -\frac{4 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{4 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{134}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{137 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{713}{9 \, \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/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^3/sqrt(3*x^2 + 2) + 4/3*x^2/sqrt(3*x^2 + 2) + 134/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 137/6*x/sqrt(3*x^2
+ 2) - 713/9/sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.77836, size = 188, normalized size = 2.81 \begin{align*} \frac{134 \, \sqrt{3}{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) -{\left (24 \, x^{3} - 24 \, x^{2} - 411 \, x + 1426\right )} \sqrt{3 \, x^{2} + 2}}{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/(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

1/18*(134*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (24*x^3 - 24*x^2 - 411*x + 1426)*s
qrt(3*x^2 + 2))/(3*x^2 + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{243 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{126 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{4 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{8 x^{4}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{135}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**3/(3*x**2+2)**(3/2),x)

[Out]

-Integral(-243*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-126*x**2/(3*x**2*sqrt(3*x**2 +
2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-4*x**3/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral
(8*x**4/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-135/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3
*x**2 + 2)), x)

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Giac [A]  time = 1.22409, size = 63, normalized size = 0.94 \begin{align*} -\frac{134}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (8 \,{\left (x - 1\right )} x - 137\right )} x + 1426}{18 \, \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/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-134/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*(x - 1)*x - 137)*x + 1426)/sqrt(3*x^2 + 2)