3.819 \(\int \frac{a+b x^2+c x^4}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2} \]

[Out]

-a/(2*x^2) + (c*x^2)/2 + b*Log[x]

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Rubi [A]  time = 0.0189481, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

-a/(2*x^2) + (c*x^2)/2 + b*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.00349613, size = 21, normalized size = 1. \[ -\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

-a/(2*x^2) + (c*x^2)/2 + b*Log[x]

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Maple [A]  time = 0.005, size = 18, normalized size = 0.9 \[ -{\frac{a}{2\,{x}^{2}}}+{\frac{c{x}^{2}}{2}}+b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.698306, size = 27, normalized size = 1.29 \[ \frac{1}{2} \, c x^{2} + \frac{1}{2} \, b \log \left (x^{2}\right ) - \frac{a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254102, size = 30, normalized size = 1.43 \[ \frac{c x^{4} + 2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^3,x, algorithm="fricas")

[Out]

1/2*(c*x^4 + 2*b*x^2*log(x) - a)/x^2

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Sympy [A]  time = 1.01701, size = 17, normalized size = 0.81 \[ - \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/x**3,x)

[Out]

-a/(2*x**2) + b*log(x) + c*x**2/2

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GIAC/XCAS [A]  time = 0.263205, size = 35, normalized size = 1.67 \[ \frac{1}{2} \, c x^{2} + \frac{1}{2} \, b{\rm ln}\left (x^{2}\right ) - \frac{b x^{2} + a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^3,x, algorithm="giac")

[Out]

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