\(\int \frac {a x+b x^3+c x^5}{x} \, dx\) [69]

Optimal result

Integrand size = 18, antiderivative size = 20 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, 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} \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

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

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

[In]

Integrate[(a*x + b*x^3 + c*x^5)/x,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
default	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
norman	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
risch	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
parallelrisch	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
parts	\(a x +\frac {1}{3} b \,x^{3}+\frac {1}{5} c \,x^{5}\)	\(17\)
gosper	\(\frac {x \left (3 c \,x^{4}+5 b \,x^{2}+15 a \right )}{15}\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5} \]

[In]

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

[Out]

a*x + b*x**3/3 + c*x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=\frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

[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

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {a x+b x^3+c x^5}{x} \, dx=\frac {c\,x^5}{5}+\frac {b\,x^3}{3}+a\,x \]

[In]

int((a*x + b*x^3 + c*x^5)/x,x)

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5