3.1421 \(\int \frac{(5-x) (3+2 x)^3}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac{556-1461 x}{54 \sqrt{3 x^2+2}}-\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(18*(2 + 3*x^2)^(3/2)) - (556 - 1461*x)/(54*Sqrt[2 + 3*x^2]) - (8*ArcSinh[Sqrt[3/2]
*x])/(9*Sqrt[3])

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Rubi [A]  time = 0.0270609, 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, 778, 215} \[ -\frac{7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac{556-1461 x}{54 \sqrt{3 x^2+2}}-\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(18*(2 + 3*x^2)^(3/2)) - (556 - 1461*x)/(54*Sqrt[2 + 3*x^2]) - (8*ArcSinh[Sqrt[3/2]
*x])/(9*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 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)^3}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}+\frac{1}{18} \int \frac{(314-24 x) (3+2 x)}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac{556-1461 x}{54 \sqrt{2+3 x^2}}-\frac{8}{9} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac{556-1461 x}{54 \sqrt{2+3 x^2}}-\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0630992, size = 58, normalized size = 0.87 \[ -\frac{-4971 x^3+72 x^2+16 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-3741 x+1490}{54 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1490 - 3741*x + 72*x^2 - 4971*x^3 + 16*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(54*(2 + 3*x^2)^(3/2)
)

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Maple [A]  time = 0.006, size = 77, normalized size = 1.2 \begin{align*}{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{547\,x}{18}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{8\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{4\,{x}^{2}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{745}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{17\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{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)^(5/2),x)

[Out]

8/9*x^3/(3*x^2+2)^(3/2)+547/18*x/(3*x^2+2)^(1/2)-8/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)-4/3*x^2/(3*x^2+2)^(3/2)-7
45/27/(3*x^2+2)^(3/2)+17/2*x/(3*x^2+2)^(3/2)

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Maxima [A]  time = 1.50585, size = 123, normalized size = 1.84 \begin{align*} \frac{8}{27} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} - \frac{8}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{1609 \, x}{54 \, \sqrt{3 \, x^{2} + 2}} - \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{17 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{745}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="maxima")

[Out]

8/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) - 8/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1609/54*x/sqrt(
3*x^2 + 2) - 4/3*x^2/(3*x^2 + 2)^(3/2) + 17/2*x/(3*x^2 + 2)^(3/2) - 745/27/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.5691, size = 212, normalized size = 3.16 \begin{align*} \frac{8 \, \sqrt{3}{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) +{\left (4971 \, x^{3} - 72 \, x^{2} + 3741 \, x - 1490\right )} \sqrt{3 \, x^{2} + 2}}{54 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\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)^(5/2),x, algorithm="fricas")

[Out]

1/54*(8*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + (4971*x^3 - 72*x^2 + 3741*x
- 1490)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{243 x}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{126 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{4 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{8 x^{4}}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{135}{9 x^{4} \sqrt{3 x^{2} + 2} + 12 x^{2} \sqrt{3 x^{2} + 2} + 4 \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)**(5/2),x)

[Out]

-Integral(-243*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-126
*x**2/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-4*x**3/(9*x**4
*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(8*x**4/(9*x**4*sqrt(3*x**2 +
2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-135/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sq
rt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)

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Giac [A]  time = 1.12946, size = 65, normalized size = 0.97 \begin{align*} \frac{8}{27} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{3 \,{\left ({\left (1657 \, x - 24\right )} x + 1247\right )} x - 1490}{54 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

8/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/54*(3*((1657*x - 24)*x + 1247)*x - 1490)/(3*x^2 + 2)^(3/2)