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

Optimal. Leaf size=20 \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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Rubi [A]  time = 0.0146872, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*Integral(x, x) + c*x**3/3 + Integral(a, x)

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Mathematica [A]  time = 0.00161879, size = 20, normalized size = 1. \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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Maple [A]  time = 0.001, size = 17, normalized size = 0.9 \[ ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^3+a*x^2)/x^2,x)

[Out]

a*x+1/2*b*x^2+1/3*c*x^3

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Maxima [A]  time = 0.749836, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Fricas [A]  time = 0.254699, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Sympy [A]  time = 0.067583, size = 15, normalized size = 0.75 \[ a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x + b*x**2/2 + c*x**3/3

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GIAC/XCAS [A]  time = 0.263675, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x