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

________________________________________________________________________________________

Rubi [A]  time = 0.0068819, 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, Rules used = {14} \[ -\frac{a}{x}+b x+\frac{c x^3}{3} \]

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

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

Mathematica [A]  time = 0.0020158, 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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}+bx+{\frac{c{x}^{3}}{3}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.11163, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{3} \, c x^{3} + b x - \frac{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^3,x, algorithm="maxima")

[Out]

1/3*c*x^3 + b*x - a/x

________________________________________________________________________________________

Fricas [A]  time = 1.19098, size = 42, normalized size = 2.33 \begin{align*} \frac{c x^{4} + 3 \, b x^{2} - 3 \, a}{3 \, x} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.247622, size = 12, normalized size = 0.67 \begin{align*} - \frac{a}{x} + b x + \frac{c x^{3}}{3} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.09081, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{3} \, c x^{3} + b x - \frac{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^3,x, algorithm="giac")

[Out]

1/3*c*x^3 + b*x - a/x