\(\int \frac {a x+b x^3+c x^5}{x^3} \, dx\) [71]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=-\frac {a}{x}+b x+\frac {c x^3}{3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=-\frac {a}{x}+b x+\frac {c x^3}{3} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=-\frac {a}{x}+b x+\frac {c x^3}{3} \]

[In]

Integrate[(a*x + b*x^3 + c*x^5)/x^3,x]

[Out]

-(a/x) + b*x + (c*x^3)/3

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {a}{x}+b x +\frac {c \,x^{3}}{3}\)	\(17\)
risch	\(-\frac {a}{x}+b x +\frac {c \,x^{3}}{3}\)	\(17\)
norman	\(\frac {b \,x^{3}-a x +\frac {1}{3} c \,x^{5}}{x^{2}}\)	\(21\)
parallelrisch	\(\frac {c \,x^{4}+3 b \,x^{2}-3 a}{3 x}\)	\(21\)
gosper	\(-\frac {-c \,x^{4}-3 b \,x^{2}+3 a}{3 x}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=\frac {c x^{4} + 3 \, b x^{2} - 3 \, a}{3 \, x} \]

[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] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=- \frac {a}{x} + b x + \frac {c x^{3}}{3} \]

[In]

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

[Out]

-a/x + b*x + c*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=\frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=\frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {a x+b x^3+c x^5}{x^3} \, dx=b\,x-\frac {a}{x}+\frac {c\,x^3}{3} \]

[In]

int((a*x + b*x^3 + c*x^5)/x^3,x)

[Out]

b*x - a/x + (c*x^3)/3