3.1842 \(\int \frac{x^5}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

[Out]

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

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Rubi [A]  time = 0.102726, antiderivative size = 53, 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 \[ -\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} \int ^{x^{2}} \frac{1}{a^{3}}\, dx}{2} + \frac{x^{6}}{6 a} - \frac{b \int ^{x^{2}} x\, dx}{2 a^{2}} - \frac{b^{3} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00945838, size = 53, normalized size = 1. \[ -\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 46, normalized size = 0.9 \[{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{6}}{6\,a}}-{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x^2),x)

[Out]

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

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Maxima [A]  time = 1.45781, size = 62, normalized size = 1.17 \[ -\frac{b^{3} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223901, size = 61, normalized size = 1.15 \[ \frac{2 \, a^{3} x^{6} - 3 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 6 \, b^{3} \log \left (a x^{2} + b\right )}{12 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.32283, size = 44, normalized size = 0.83 \[ \frac{x^{6}}{6 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.230714, size = 63, normalized size = 1.19 \[ -\frac{b^{3}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^3*ln(abs(a*x^2 + b))/a^4 + 1/12*(2*a^2*x^6 - 3*a*b*x^4 + 6*b^2*x^2)/a^3