3.1859 \(\int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]

[Out]

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

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Rubi [A]  time = 0.115351, antiderivative size = 57, 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 \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^2)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0299552, size = 49, normalized size = 0.86 \[ \frac{a^2 x^4+\frac{2 b^3}{a x^2+b}+6 b^2 \log \left (a x^2+b\right )-4 a b x^2}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.013, size = 52, normalized size = 0.9 \[ -{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{3}}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}+{\frac{3\,{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.44641, size = 73, normalized size = 1.28 \[ \frac{b^{3}}{2 \,{\left (a^{5} x^{2} + a^{4} b\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{a x^{4} - 4 \, b x^{2}}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229068, size = 95, normalized size = 1.67 \[ \frac{a^{3} x^{6} - 3 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65252, size = 53, normalized size = 0.93 \[ \frac{b^{3}}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{x^{4}}{4 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.234726, size = 74, normalized size = 1.3 \[ \frac{3 \, b^{2}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{b^{3}}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{a^{2} x^{4} - 4 \, a b x^{2}}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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