3.1818 \(\int \left (a+\frac{b}{x^2}\right )^2 x^3 \, dx\)

Optimal. Leaf size=23 \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]

[Out]

a*b*x^2 + (a^2*x^4)/4 + b^2*Log[x]

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Rubi [A]  time = 0.0457592, antiderivative size = 23, 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 \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*x^3,x]

[Out]

a*b*x^2 + (a^2*x^4)/4 + b^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00210901, size = 23, normalized size = 1. \[ \frac{a^2 x^4}{4}+a b x^2+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*x^3,x]

[Out]

a*b*x^2 + (a^2*x^4)/4 + b^2*Log[x]

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Maple [A]  time = 0.003, size = 22, normalized size = 1. \[ ab{x}^{2}+{\frac{{x}^{4}{a}^{2}}{4}}+{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*x^3,x)

[Out]

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

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Maxima [A]  time = 1.44281, size = 32, normalized size = 1.39 \[ \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231294, size = 28, normalized size = 1.22 \[ \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + b^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x^4 + a*b*x^2 + b^2*log(x)

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Sympy [A]  time = 1.07473, size = 20, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*x**3,x)

[Out]

a**2*x**4/4 + a*b*x**2 + b**2*log(x)

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GIAC/XCAS [A]  time = 0.228774, size = 30, normalized size = 1.3 \[ \frac{1}{4} \, a^{2} x^{4} + a b x^{2} + b^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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