3.1410 \(\int \frac{(5-x) (3+2 x)^2}{(2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{7 (2-7 x) (2 x+3)}{6 \sqrt{3 x^2+2}}-\frac{53}{9} \sqrt{3 x^2+2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x))/(6*Sqrt[2 + 3*x^2]) - (53*Sqrt[2 + 3*x^2])/9 + (8*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0222036, antiderivative size = 60, 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, 641, 215} \[ -\frac{7 (2-7 x) (2 x+3)}{6 \sqrt{3 x^2+2}}-\frac{53}{9} \sqrt{3 x^2+2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x))/(6*Sqrt[2 + 3*x^2]) - (53*Sqrt[2 + 3*x^2])/9 + (8*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 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]

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)^2}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)}{6 \sqrt{2+3 x^2}}+\frac{1}{6} \int \frac{16-106 x}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)}{6 \sqrt{2+3 x^2}}-\frac{53}{9} \sqrt{2+3 x^2}+\frac{8}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)}{6 \sqrt{2+3 x^2}}-\frac{53}{9} \sqrt{2+3 x^2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0433055, size = 48, normalized size = 0.8 \[ -\frac{24 x^2-16 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-357 x+338}{18 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(338 - 357*x + 24*x^2 - 16*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 51, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{169}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{119\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{8\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*x^2/(3*x^2+2)^(1/2)-169/9/(3*x^2+2)^(1/2)+119/6*x/(3*x^2+2)^(1/2)+8/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.50315, size = 68, normalized size = 1.13 \begin{align*} -\frac{4 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{8}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{119 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{169}{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)^2/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^2/sqrt(3*x^2 + 2) + 8/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 119/6*x/sqrt(3*x^2 + 2) - 169/9/sqrt(3*x^2 + 2
)

________________________________________________________________________________________

Fricas [A]  time = 1.57743, size = 171, normalized size = 2.85 \begin{align*} \frac{8 \, \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^{2} - 357 \, x + 338\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)^2/(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

1/18*(8*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (24*x^2 - 357*x + 338)*sqrt(3*x^2 +
2))/(3*x^2 + 2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{51 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{8 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{45}{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)**2/(3*x**2+2)**(3/2),x)

[Out]

-Integral(-51*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-8*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(-45
/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x)

________________________________________________________________________________________

Giac [A]  time = 1.20012, size = 59, normalized size = 0.98 \begin{align*} -\frac{8}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (8 \, x - 119\right )} x + 338}{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)^2/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-8/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*x - 119)*x + 338)/sqrt(3*x^2 + 2)