3.70 \(\int \frac{a x+b x^3+c x^5}{x^2} \, dx\)

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

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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Rubi [A]  time = 0.0161841, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3 + c*x^5)/x^2,x]

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0026517, size = 21, normalized size = 1. \[ a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^3 + c*x^5)/x^2,x]

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \[{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{4}}{4}}+a\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^5+b*x^3+a*x)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.765729, size = 23, normalized size = 1.1 \[ \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*x^4 + 1/2*b*x^2 + a*log(x)

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Fricas [A]  time = 0.254858, size = 23, normalized size = 1.1 \[ \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*x^4 + 1/2*b*x^2 + a*log(x)

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Sympy [A]  time = 0.169227, size = 17, normalized size = 0.81 \[ a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x) + b*x**2/2 + c*x**4/4

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GIAC/XCAS [A]  time = 0.278529, size = 27, normalized size = 1.29 \[ \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + \frac{1}{2} \, a{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*x^4 + 1/2*b*x^2 + 1/2*a*ln(x^2)