3.1872 \(\int \frac{x^5}{\left (a+\frac{b}{x^2}\right )^3} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}+\frac{3 b^2 x^2}{a^5}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.180978, antiderivative size = 87, 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 \[ \frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}+\frac{3 b^2 x^2}{a^5}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{6}}{6 a^{3}} - \frac{3 b \int ^{x^{2}} x\, dx}{2 a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} + \frac{b^{5}}{4 a^{6} \left (a x^{2} + b\right )^{2}} - \frac{5 b^{4}}{2 a^{6} \left (a x^{2} + b\right )} - \frac{5 b^{3} \log{\left (a x^{2} + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(6*a**3) - 3*b*Integral(x, (x, x**2))/(2*a**4) + 3*b**2*x**2/a**5 + b**5/(4
*a**6*(a*x**2 + b)**2) - 5*b**4/(2*a**6*(a*x**2 + b)) - 5*b**3*log(a*x**2 + b)/a
**6

_______________________________________________________________________________________

Mathematica [A]  time = 0.0907792, size = 71, normalized size = 0.82 \[ \frac{2 a^3 x^6-9 a^2 b x^4-\frac{3 b^4 \left (10 a x^2+9 b\right )}{\left (a x^2+b\right )^2}-60 b^3 \log \left (a x^2+b\right )+36 a b^2 x^2}{12 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x^2)^3,x]

[Out]

(36*a*b^2*x^2 - 9*a^2*b*x^4 + 2*a^3*x^6 - (3*b^4*(9*b + 10*a*x^2))/(b + a*x^2)^2
- 60*b^3*Log[b + a*x^2])/(12*a^6)

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 80, normalized size = 0.9 \[ 3\,{\frac{{b}^{2}{x}^{2}}{{a}^{5}}}-{\frac{3\,b{x}^{4}}{4\,{a}^{4}}}+{\frac{{x}^{6}}{6\,{a}^{3}}}+{\frac{{b}^{5}}{4\,{a}^{6} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{5\,{b}^{4}}{2\,{a}^{6} \left ( a{x}^{2}+b \right ) }}-5\,{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x^2)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44732, size = 120, normalized size = 1.38 \[ -\frac{10 \, a b^{4} x^{2} + 9 \, b^{5}}{4 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} - \frac{5 \, b^{3} \log \left (a x^{2} + b\right )}{a^{6}} + \frac{2 \, a^{2} x^{6} - 9 \, a b x^{4} + 36 \, b^{2} x^{2}}{12 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x^2)^3,x, algorithm="maxima")

[Out]

-1/4*(10*a*b^4*x^2 + 9*b^5)/(a^8*x^4 + 2*a^7*b*x^2 + a^6*b^2) - 5*b^3*log(a*x^2
+ b)/a^6 + 1/12*(2*a^2*x^6 - 9*a*b*x^4 + 36*b^2*x^2)/a^5

_______________________________________________________________________________________

Fricas [A]  time = 0.228388, size = 155, normalized size = 1.78 \[ \frac{2 \, a^{5} x^{10} - 5 \, a^{4} b x^{8} + 20 \, a^{3} b^{2} x^{6} + 63 \, a^{2} b^{3} x^{4} + 6 \, a b^{4} x^{2} - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )} \log \left (a x^{2} + b\right )}{12 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*a^5*x^10 - 5*a^4*b*x^8 + 20*a^3*b^2*x^6 + 63*a^2*b^3*x^4 + 6*a*b^4*x^2 -
27*b^5 - 60*(a^2*b^3*x^4 + 2*a*b^4*x^2 + b^5)*log(a*x^2 + b))/(a^8*x^4 + 2*a^7*
b*x^2 + a^6*b^2)

_______________________________________________________________________________________

Sympy [A]  time = 2.3756, size = 90, normalized size = 1.03 \[ - \frac{10 a b^{4} x^{2} + 9 b^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{2} + 4 a^{6} b^{2}} + \frac{x^{6}}{6 a^{3}} - \frac{3 b x^{4}}{4 a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} - \frac{5 b^{3} \log{\left (a x^{2} + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x**2)**3,x)

[Out]

-(10*a*b**4*x**2 + 9*b**5)/(4*a**8*x**4 + 8*a**7*b*x**2 + 4*a**6*b**2) + x**6/(6
*a**3) - 3*b*x**4/(4*a**4) + 3*b**2*x**2/a**5 - 5*b**3*log(a*x**2 + b)/a**6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.243151, size = 124, normalized size = 1.43 \[ -\frac{5 \, b^{3}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{a^{6}} + \frac{30 \, a^{2} b^{3} x^{4} + 50 \, a b^{4} x^{2} + 21 \, b^{5}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x^{6} - 9 \, a^{5} b x^{4} + 36 \, a^{4} b^{2} x^{2}}{12 \, a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b^3*ln(abs(a*x^2 + b))/a^6 + 1/4*(30*a^2*b^3*x^4 + 50*a*b^4*x^2 + 21*b^5)/((a
*x^2 + b)^2*a^6) + 1/12*(2*a^6*x^6 - 9*a^5*b*x^4 + 36*a^4*b^2*x^2)/a^9