3.821 \(\int \frac{a+b x^2+c x^4}{x^5} \, dx\)

Optimal. Leaf size=21 \[ -\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x) \]

[Out]

-a/(4*x^4) - b/(2*x^2) + c*Log[x]

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Rubi [A]  time = 0.0192931, 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 \[ -\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

-a/(4*x^4) - b/(2*x^2) + c*Log[x]

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Rubi in Sympy [A]  time = 6.7375, size = 20, normalized size = 0.95 \[ - \frac{a}{4 x^{4}} - \frac{b}{2 x^{2}} + \frac{c \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.00479111, size = 21, normalized size = 1. \[ -\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

-a/(4*x^4) - b/(2*x^2) + c*Log[x]

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Maple [A]  time = 0.008, size = 18, normalized size = 0.9 \[ -{\frac{a}{4\,{x}^{4}}}-{\frac{b}{2\,{x}^{2}}}+c\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^5,x)

[Out]

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

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Maxima [A]  time = 0.683969, size = 28, normalized size = 1.33 \[ \frac{1}{2} \, c \log \left (x^{2}\right ) - \frac{2 \, b x^{2} + a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254707, size = 31, normalized size = 1.48 \[ \frac{4 \, c x^{4} \log \left (x\right ) - 2 \, b x^{2} - a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.32537, size = 17, normalized size = 0.81 \[ c \log{\left (x \right )} - \frac{a + 2 b x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/x**5,x)

[Out]

c*log(x) - (a + 2*b*x**2)/(4*x**4)

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GIAC/XCAS [A]  time = 0.261654, size = 36, normalized size = 1.71 \[ \frac{1}{2} \, c{\rm ln}\left (x^{2}\right ) - \frac{3 \, c x^{4} + 2 \, b x^{2} + a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^5,x, algorithm="giac")

[Out]

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