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

Optimal. Leaf size=25 \[ \frac{a x^2}{2}+\frac{b x^3}{3}+\frac{c x^4}{4} \]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

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Rubi [A]  time = 0.0061748, antiderivative size = 25, 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} \[ \frac{a x^2}{2}+\frac{b x^3}{3}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)/x,x]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

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

Mathematica [A]  time = 0.0014316, size = 25, normalized size = 1. \[ \frac{a x^2}{2}+\frac{b x^3}{3}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

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Maple [A]  time = 0.001, size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{3}}{3}}+{\frac{c{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)/x,x)

[Out]

1/2*a*x^2+1/3*b*x^3+1/4*c*x^4

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Maxima [A]  time = 1.07064, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

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Fricas [A]  time = 1.44192, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

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Sympy [A]  time = 0.057205, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{3}}{3} + \frac{c x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**3/3 + c*x**4/4

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Giac [A]  time = 1.0827, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2