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

Optimal. Leaf size=18 \[ -\frac{a}{x}+b x+\frac{c x^3}{3} \]

[Out]

-(a/x) + b*x + (c*x^3)/3

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Rubi [A]  time = 0.0149666, antiderivative size = 18, 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 \[ -\frac{a}{x}+b x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

-(a/x) + b*x + (c*x^3)/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a}{x} + \frac{c x^{3}}{3} + \int b\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x + c*x**3/3 + Integral(b, x)

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Mathematica [A]  time = 0.00314479, size = 18, normalized size = 1. \[ -\frac{a}{x}+b x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

-(a/x) + b*x + (c*x^3)/3

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \[ -{\frac{a}{x}}+bx+{\frac{c{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^5+b*x^3+a*x)/x^3,x)

[Out]

-a/x+b*x+1/3*c*x^3

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Maxima [A]  time = 0.772295, size = 22, normalized size = 1.22 \[ \frac{1}{3} \, c x^{3} + b x - \frac{a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + b*x - a/x

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Fricas [A]  time = 0.250282, size = 27, normalized size = 1.5 \[ \frac{c x^{4} + 3 \, b x^{2} - 3 \, a}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^4 + 3*b*x^2 - 3*a)/x

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Sympy [A]  time = 0.981781, size = 12, normalized size = 0.67 \[ - \frac{a}{x} + b x + \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x + b*x + c*x**3/3

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GIAC/XCAS [A]  time = 0.252864, size = 22, normalized size = 1.22 \[ \frac{1}{3} \, c x^{3} + b x - \frac{a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + b*x - a/x