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

Optimal. Leaf size=20 \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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Rubi [A]  time = 0.0056522, antiderivative size = 20, 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, Rules used = {14} \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3 + c*x^5)/x,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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+b x^3+c x^5}{x} \, dx &=\int \left (a+b x^2+c x^4\right ) \, dx\\ &=a x+\frac{b x^3}{3}+\frac{c x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0011684, size = 20, normalized size = 1. \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^3 + c*x^5)/x,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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Maple [A]  time = 0., size = 17, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{3}}{3}}+{\frac{c{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^5+b*x^3+a*x)/x,x)

[Out]

a*x+1/3*b*x^3+1/5*c*x^5

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Maxima [A]  time = 1.13096, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

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Fricas [A]  time = 1.22959, size = 39, normalized size = 1.95 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

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Sympy [A]  time = 0.055914, size = 15, normalized size = 0.75 \begin{align*} a x + \frac{b x^{3}}{3} + \frac{c x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**3/3 + c*x**5/5

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Giac [A]  time = 1.09995, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x