3.1.4 \(\int \frac {a x^2+b x^3+c x^4}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {14} \begin {gather*} \frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)/x,x]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

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

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)/x,x]

[Out]

(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4

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IntegrateAlgebraic [A]  time = 0.02, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{12} x^2 \left (6 a+4 b x+3 c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x^2 + b*x^3 + c*x^4)/x,x]

[Out]

(x^2*(6*a + 4*b*x + 3*c*x^2))/12

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fricas [A]  time = 0.53, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x,x, algorithm="fricas")

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

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giac [A]  time = 0.36, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x,x, algorithm="giac")

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{4} c \,x^{4}+\frac {1}{3} b \,x^{3}+\frac {1}{2} a \,x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)/x,x)

[Out]

1/2*a*x^2+1/3*b*x^3+1/4*c*x^4

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maxima [A]  time = 0.45, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x,x, algorithm="maxima")

[Out]

1/4*c*x^4 + 1/3*b*x^3 + 1/2*a*x^2

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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} \frac {x^2\,\left (3\,c\,x^2+4\,b\,x+6\,a\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x^3 + c*x^4)/x,x)

[Out]

(x^2*(6*a + 4*b*x + 3*c*x^2))/12

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sympy [A]  time = 0.06, size = 19, normalized size = 0.76 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{3}}{3} + \frac {c x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**3+a*x**2)/x,x)

[Out]

a*x**2/2 + b*x**3/3 + c*x**4/4

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