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 = 0.0063249, 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, Rules used = {14} \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a x^2+b x^3+c x^4}{x^2} \, dx &=\int \left (a+b x+c x^2\right ) \, dx\\ &=a x+\frac{b x^2}{2}+\frac{c x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.001181, 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 = 0.002, size = 17, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \end{align*}

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 = 1.11766, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

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 = 1.46768, size = 39, normalized size = 1.95 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

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 = 0.056357, size = 15, normalized size = 0.75 \begin{align*} a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3} \end{align*}

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 [A]  time = 1.07503, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

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