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

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \[ a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

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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fricas [A]  time = 0.56, size = 16, normalized size = 0.80 \[ \frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

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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giac [A]  time = 0.36, size = 16, normalized size = 0.80 \[ \frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

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

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maple [A]  time = 0.00, size = 17, normalized size = 0.85 \[ \frac {1}{5} c \,x^{5}+\frac {1}{3} b \,x^{3}+a x \]

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 = 0.42, size = 16, normalized size = 0.80 \[ \frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \]

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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mupad [B]  time = 0.03, size = 16, normalized size = 0.80 \[ \frac {c\,x^5}{5}+\frac {b\,x^3}{3}+a\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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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sympy [A]  time = 0.07, size = 15, normalized size = 0.75 \[ a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5} \]

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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