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

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])

_______________________________________________________________________________________

Rubi [A]  time = 0.0964272, 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 \[ -\frac{7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac{556-1461 x}{54 \sqrt{3 x^2+2}}-\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.0755, size = 60, normalized size = 0.9 \[ - \frac{- 5844 x + 2224}{216 \sqrt{3 x^{2} + 2}} - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{2}}{36 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} - \frac{8 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**3/(3*x**2+2)**(5/2),x)

[Out]

-(-5844*x + 2224)/(216*sqrt(3*x**2 + 2)) - (-98*x + 28)*(2*x + 3)**2/(36*(3*x**2
+ 2)**(3/2)) - 8*sqrt(3)*asinh(sqrt(6)*x/2)/27

_______________________________________________________________________________________

Mathematica [A]  time = 0.10808, size = 50, normalized size = 0.75 \[ \frac{1}{54} \left (-\frac{-4971 x^3+72 x^2-3741 x+1490}{\left (3 x^2+2\right )^{3/2}}-16 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

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)/(2 + 3*x^2)^(3/2)) - 16*Sqrt[3]*ArcSinh[S
qrt[3/2]*x])/54

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 77, normalized size = 1.2 \[{\frac{17\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{547\,x}{18}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{745}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{x}^{2}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/2*x/(3*x^2+2)^(3/2)+547/18*x/(3*x^2+2)^(1/2)-745/27/(3*x^2+2)^(3/2)-4/3*x^2/(
3*x^2+2)^(3/2)+8/9*x^3/(3*x^2+2)^(3/2)-8/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.766688, size = 123, normalized size = 1.84 \[ \frac{8}{27} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} - \frac{8}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{1609 \, x}{54 \, \sqrt{3 \, x^{2} + 2}} - \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{17 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{745}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^3*(x - 5)/(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)

_______________________________________________________________________________________

Fricas [A]  time = 0.276268, size = 119, normalized size = 1.78 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (4971 \, x^{3} - 72 \, x^{2} + 3741 \, x - 1490\right )} \sqrt{3 \, x^{2} + 2} + 24 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} + 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{162 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^3*(x - 5)/(3*x^2 + 2)^(5/2),x, algorithm="fricas")

[Out]

1/162*sqrt(3)*(sqrt(3)*(4971*x^3 - 72*x^2 + 3741*x - 1490)*sqrt(3*x^2 + 2) + 24*
(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*(3*x^2 + 1) + 3*sqrt(3*x^2 + 2)*x))/(9*x^4 + 1
2*x^2 + 4)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.315811, size = 65, normalized size = 0.97 \[ \frac{8}{27} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{3 \,{\left ({\left (1657 \, x - 24\right )} x + 1247\right )} x - 1490}{54 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^3*(x - 5)/(3*x^2 + 2)^(5/2),x, algorithm="giac")

[Out]

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