3.231 \(\int \frac{1}{\left (9+3 x-5 x^2+x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{15 (x+1) (3-x)^3}{256 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{5 (x+1) (3-x)^2}{64 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{(x+1) (3-x)}{8 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{15 (x+1)^{3/2} (3-x)^3 \tanh ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )}{512 \left (x^3-5 x^2+3 x+9\right )^{3/2}} \]

[Out]

((3 - x)*(1 + x))/(8*(9 + 3*x - 5*x^2 + x^3)^(3/2)) + (5*(3 - x)^2*(1 + x))/(64*
(9 + 3*x - 5*x^2 + x^3)^(3/2)) - (15*(3 - x)^3*(1 + x))/(256*(9 + 3*x - 5*x^2 +
x^3)^(3/2)) + (15*(3 - x)^3*(1 + x)^(3/2)*ArcTanh[Sqrt[1 + x]/2])/(512*(9 + 3*x
- 5*x^2 + x^3)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.235337, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{15 (x+1) (3-x)^3}{256 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{5 (x+1) (3-x)^2}{64 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{(x+1) (3-x)}{8 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac{15 (x+1)^{3/2} (3-x)^3 \tanh ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )}{512 \left (x^3-5 x^2+3 x+9\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(9 + 3*x - 5*x^2 + x^3)^(-3/2),x]

[Out]

((3 - x)*(1 + x))/(8*(9 + 3*x - 5*x^2 + x^3)^(3/2)) + (5*(3 - x)^2*(1 + x))/(64*
(9 + 3*x - 5*x^2 + x^3)^(3/2)) - (15*(3 - x)^3*(1 + x))/(256*(9 + 3*x - 5*x^2 +
x^3)^(3/2)) + (15*(3 - x)^3*(1 + x)^(3/2)*ArcTanh[Sqrt[1 + x]/2])/(512*(9 + 3*x
- 5*x^2 + x^3)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.47437, size = 128, normalized size = 0.92 \[ \frac{15 \left (- x + 3\right )^{3} \left (x + 1\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{x + 1}}{2} \right )}}{512 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{3}{2}}} + \frac{15 \left (- x + 3\right )^{2} \left (x + 1\right )^{2}}{256 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{3}{2}}} + \frac{5 \left (- x + 3\right ) \left (x + 1\right )^{2}}{32 \left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{3}{2}}} - \frac{\left (- \frac{x}{2} + \frac{3}{2}\right ) \left (x + 1\right )}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-5*x**2+3*x+9)**(3/2),x)

[Out]

15*(-x + 3)**3*(x + 1)**(3/2)*atanh(sqrt(x + 1)/2)/(512*(x**3 - 5*x**2 + 3*x + 9
)**(3/2)) + 15*(-x + 3)**2*(x + 1)**2/(256*(x**3 - 5*x**2 + 3*x + 9)**(3/2)) + 5
*(-x + 3)*(x + 1)**2/(32*(x**3 - 5*x**2 + 3*x + 9)**(3/2)) - (-x/2 + 3/2)*(x + 1
)/(x**3 - 5*x**2 + 3*x + 9)**(3/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0406292, size = 58, normalized size = 0.42 \[ \frac{30 x^2-140 x-15 \sqrt{x+1} (x-3)^2 \tanh ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )+86}{512 (x-3) \sqrt{(x-3)^2 (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(9 + 3*x - 5*x^2 + x^3)^(-3/2),x]

[Out]

(86 - 140*x + 30*x^2 - 15*(-3 + x)^2*Sqrt[1 + x]*ArcTanh[Sqrt[1 + x]/2])/(512*(-
3 + x)*Sqrt[(-3 + x)^2*(1 + x)])

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 144, normalized size = 1. \[ -{\frac{ \left ( -3+x \right ) ^{3} \left ( 1+x \right ) }{1024} \left ( 15\, \left ( 1+x \right ) ^{5/2}\ln \left ( \sqrt{1+x}+2 \right ) -15\, \left ( 1+x \right ) ^{5/2}\ln \left ( \sqrt{1+x}-2 \right ) -120\, \left ( 1+x \right ) ^{3/2}\ln \left ( \sqrt{1+x}+2 \right ) +120\, \left ( 1+x \right ) ^{3/2}\ln \left ( \sqrt{1+x}-2 \right ) +240\,\ln \left ( \sqrt{1+x}+2 \right ) \sqrt{1+x}-240\,\ln \left ( \sqrt{1+x}-2 \right ) \sqrt{1+x}-60\,{x}^{2}+280\,x-172 \right ) \left ({x}^{3}-5\,{x}^{2}+3\,x+9 \right ) ^{-{\frac{3}{2}}} \left ( \sqrt{1+x}+2 \right ) ^{-2} \left ( \sqrt{1+x}-2 \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-5*x^2+3*x+9)^(3/2),x)

[Out]

-1/1024*(-3+x)^3*(1+x)*(15*(1+x)^(5/2)*ln((1+x)^(1/2)+2)-15*(1+x)^(5/2)*ln((1+x)
^(1/2)-2)-120*(1+x)^(3/2)*ln((1+x)^(1/2)+2)+120*(1+x)^(3/2)*ln((1+x)^(1/2)-2)+24
0*ln((1+x)^(1/2)+2)*(1+x)^(1/2)-240*ln((1+x)^(1/2)-2)*(1+x)^(1/2)-60*x^2+280*x-1
72)/(x^3-5*x^2+3*x+9)^(3/2)/((1+x)^(1/2)+2)^2/((1+x)^(1/2)-2)^2

_______________________________________________________________________________________

Maxima [A]  time = 1.59453, size = 77, normalized size = 0.55 \[ \frac{15 \,{\left (x + 1\right )}^{2} - 100 \, x + 28}{256 \,{\left ({\left (x + 1\right )}^{\frac{5}{2}} - 8 \,{\left (x + 1\right )}^{\frac{3}{2}} + 16 \, \sqrt{x + 1}\right )}} - \frac{15}{1024} \, \log \left (\sqrt{x + 1} + 2\right ) + \frac{15}{1024} \, \log \left (\sqrt{x + 1} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-3/2),x, algorithm="maxima")

[Out]

1/256*(15*(x + 1)^2 - 100*x + 28)/((x + 1)^(5/2) - 8*(x + 1)^(3/2) + 16*sqrt(x +
1)) - 15/1024*log(sqrt(x + 1) + 2) + 15/1024*log(sqrt(x + 1) - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.218368, size = 186, normalized size = 1.34 \[ -\frac{15 \,{\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )} \log \left (\frac{2 \, x + \sqrt{x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) - 15 \,{\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )} \log \left (-\frac{2 \, x - \sqrt{x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) - 4 \, \sqrt{x^{3} - 5 \, x^{2} + 3 \, x + 9}{\left (15 \, x^{2} - 70 \, x + 43\right )}}{1024 \,{\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-3/2),x, algorithm="fricas")

[Out]

-1/1024*(15*(x^4 - 8*x^3 + 18*x^2 - 27)*log((2*x + sqrt(x^3 - 5*x^2 + 3*x + 9) -
6)/(x - 3)) - 15*(x^4 - 8*x^3 + 18*x^2 - 27)*log(-(2*x - sqrt(x^3 - 5*x^2 + 3*x
+ 9) - 6)/(x - 3)) - 4*sqrt(x^3 - 5*x^2 + 3*x + 9)*(15*x^2 - 70*x + 43))/(x^4 -
8*x^3 + 18*x^2 - 27)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-5*x**2+3*x+9)**(3/2),x)

[Out]

Integral((x**3 - 5*x**2 + 3*x + 9)**(-3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.527824, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 5*x^2 + 3*x + 9)^(-3/2),x, algorithm="giac")

[Out]

sage0*x