3.1829 \(\int \left (a+\frac{b}{x^2}\right )^3 x^5 \, dx\)

Optimal. Leaf size=39 \[ \frac{a^3 x^6}{6}+\frac{3}{4} a^2 b x^4+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0620908, antiderivative size = 39, 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 \[ \frac{a^3 x^6}{6}+\frac{3}{4} a^2 b x^4+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3*x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00776631, size = 39, normalized size = 1. \[ \frac{a^3 x^6}{6}+\frac{3}{4} a^2 b x^4+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3*x^5,x]

[Out]

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

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Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \[{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{6}}{6}}+{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3*x^5,x)

[Out]

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

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Maxima [A]  time = 1.43471, size = 49, normalized size = 1.26 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233753, size = 45, normalized size = 1.15 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + b^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.11896, size = 37, normalized size = 0.95 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{2}}{2} + b^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3*x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.225501, size = 49, normalized size = 1.26 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{1}{2} \, b^{3}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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