3.172 \(\int \frac{x^4}{\left (9+x^2\right )^3} \, dx\)

Optimal. Leaf size=37 \[ -\frac{3 x}{8 \left (x^2+9\right )}-\frac{x^3}{4 \left (x^2+9\right )^2}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

-x^3/(4*(9 + x^2)^2) - (3*x)/(8*(9 + x^2)) + ArcTan[x/3]/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0240618, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3 x}{8 \left (x^2+9\right )}-\frac{x^3}{4 \left (x^2+9\right )^2}+\frac{1}{8} \tan ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(9 + x^2)^3,x]

[Out]

-x^3/(4*(9 + x^2)^2) - (3*x)/(8*(9 + x^2)) + ArcTan[x/3]/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.38952, size = 27, normalized size = 0.73 \[ - \frac{x^{3}}{4 \left (x^{2} + 9\right )^{2}} - \frac{3 x}{8 \left (x^{2} + 9\right )} + \frac{\operatorname{atan}{\left (\frac{x}{3} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**2+9)**3,x)

[Out]

-x**3/(4*(x**2 + 9)**2) - 3*x/(8*(x**2 + 9)) + atan(x/3)/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.0199029, size = 28, normalized size = 0.76 \[ \frac{1}{8} \left (\tan ^{-1}\left (\frac{x}{3}\right )-\frac{x \left (5 x^2+27\right )}{\left (x^2+9\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(9 + x^2)^3,x]

[Out]

(-((x*(27 + 5*x^2))/(9 + x^2)^2) + ArcTan[x/3])/8

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 25, normalized size = 0.7 \[{\frac{1}{ \left ({x}^{2}+9 \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8}}-{\frac{27\,x}{8}} \right ) }+{\frac{1}{8}\arctan \left ({\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^2+9)^3,x)

[Out]

(-5/8*x^3-27/8*x)/(x^2+9)^2+1/8*arctan(1/3*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.51219, size = 41, normalized size = 1.11 \[ -\frac{5 \, x^{3} + 27 \, x}{8 \,{\left (x^{4} + 18 \, x^{2} + 81\right )}} + \frac{1}{8} \, \arctan \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^2 + 9)^3,x, algorithm="maxima")

[Out]

-1/8*(5*x^3 + 27*x)/(x^4 + 18*x^2 + 81) + 1/8*arctan(1/3*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.198462, size = 53, normalized size = 1.43 \[ -\frac{5 \, x^{3} -{\left (x^{4} + 18 \, x^{2} + 81\right )} \arctan \left (\frac{1}{3} \, x\right ) + 27 \, x}{8 \,{\left (x^{4} + 18 \, x^{2} + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^2 + 9)^3,x, algorithm="fricas")

[Out]

-1/8*(5*x^3 - (x^4 + 18*x^2 + 81)*arctan(1/3*x) + 27*x)/(x^4 + 18*x^2 + 81)

_______________________________________________________________________________________

Sympy [A]  time = 0.156236, size = 26, normalized size = 0.7 \[ - \frac{5 x^{3} + 27 x}{8 x^{4} + 144 x^{2} + 648} + \frac{\operatorname{atan}{\left (\frac{x}{3} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**2+9)**3,x)

[Out]

-(5*x**3 + 27*x)/(8*x**4 + 144*x**2 + 648) + atan(x/3)/8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213696, size = 34, normalized size = 0.92 \[ -\frac{5 \, x^{3} + 27 \, x}{8 \,{\left (x^{2} + 9\right )}^{2}} + \frac{1}{8} \, \arctan \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^2 + 9)^3,x, algorithm="giac")

[Out]

-1/8*(5*x^3 + 27*x)/(x^2 + 9)^2 + 1/8*arctan(1/3*x)