3.1874 \(\int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^3} \, dx\)

Optimal. Leaf size=74 \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]

[Out]

(-3*b*x^2)/(2*a^4) + x^4/(4*a^3) - b^4/(4*a^5*(b + a*x^2)^2) + (2*b^3)/(a^5*(b +
a*x^2)) + (3*b^2*Log[b + a*x^2])/a^5

_______________________________________________________________________________________

Rubi [A]  time = 0.14951, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^2)^3,x]

[Out]

(-3*b*x^2)/(2*a^4) + x^4/(4*a^3) - b^4/(4*a^5*(b + a*x^2)^2) + (2*b^3)/(a^5*(b +
a*x^2)) + (3*b^2*Log[b + a*x^2])/a^5

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} x\, dx}{2 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} - \frac{b^{4}}{4 a^{5} \left (a x^{2} + b\right )^{2}} + \frac{2 b^{3}}{a^{5} \left (a x^{2} + b\right )} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x**2)**3,x)

[Out]

Integral(x, (x, x**2))/(2*a**3) - 3*b*x**2/(2*a**4) - b**4/(4*a**5*(a*x**2 + b)*
*2) + 2*b**3/(a**5*(a*x**2 + b)) + 3*b**2*log(a*x**2 + b)/a**5

_______________________________________________________________________________________

Mathematica [A]  time = 0.08738, size = 58, normalized size = 0.78 \[ \frac{a^2 x^4+\frac{b^3 \left (8 a x^2+7 b\right )}{\left (a x^2+b\right )^2}+12 b^2 \log \left (a x^2+b\right )-6 a b x^2}{4 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^2)^3,x]

[Out]

(-6*a*b*x^2 + a^2*x^4 + (b^3*(7*b + 8*a*x^2))/(b + a*x^2)^2 + 12*b^2*Log[b + a*x
^2])/(4*a^5)

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 69, normalized size = 0.9 \[ -{\frac{3\,b{x}^{2}}{2\,{a}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{3}}}-{\frac{{b}^{4}}{4\,{a}^{5} \left ( a{x}^{2}+b \right ) ^{2}}}+2\,{\frac{{b}^{3}}{{a}^{5} \left ( a{x}^{2}+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2)^3,x)

[Out]

-3/2*b*x^2/a^4+1/4*x^4/a^3-1/4*b^4/a^5/(a*x^2+b)^2+2*b^3/a^5/(a*x^2+b)+3*b^2*ln(
a*x^2+b)/a^5

_______________________________________________________________________________________

Maxima [A]  time = 1.4149, size = 104, normalized size = 1.41 \[ \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{a^{5}} + \frac{a x^{4} - 6 \, b x^{2}}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(8*a*b^3*x^2 + 7*b^4)/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2) + 3*b^2*log(a*x^2 +
b)/a^5 + 1/4*(a*x^4 - 6*b*x^2)/a^4

_______________________________________________________________________________________

Fricas [A]  time = 0.227276, size = 139, normalized size = 1.88 \[ \frac{a^{4} x^{8} - 4 \, a^{3} b x^{6} - 11 \, a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + 7 \, b^{4} + 12 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(a^4*x^8 - 4*a^3*b*x^6 - 11*a^2*b^2*x^4 + 2*a*b^3*x^2 + 7*b^4 + 12*(a^2*b^2*
x^4 + 2*a*b^3*x^2 + b^4)*log(a*x^2 + b))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2)

_______________________________________________________________________________________

Sympy [A]  time = 2.18154, size = 78, normalized size = 1.05 \[ \frac{8 a b^{3} x^{2} + 7 b^{4}}{4 a^{7} x^{4} + 8 a^{6} b x^{2} + 4 a^{5} b^{2}} + \frac{x^{4}}{4 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**2)**3,x)

[Out]

(8*a*b**3*x**2 + 7*b**4)/(4*a**7*x**4 + 8*a**6*b*x**2 + 4*a**5*b**2) + x**4/(4*a
**3) - 3*b*x**2/(2*a**4) + 3*b**2*log(a*x**2 + b)/a**5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223261, size = 93, normalized size = 1.26 \[ \frac{3 \, b^{2}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac{a^{3} x^{4} - 6 \, a^{2} b x^{2}}{4 \, a^{6}} + \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^2*ln(abs(a*x^2 + b))/a^5 + 1/4*(a^3*x^4 - 6*a^2*b*x^2)/a^6 + 1/4*(8*a*b^3*x^
2 + 7*b^4)/((a*x^2 + b)^2*a^5)