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

Optimal. Leaf size=18 \[ -\frac {a}{x}+b x+\frac {c x^3}{3} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \begin {gather*} -\frac {a}{x}+b x+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^2,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+b x^2+c x^4}{x^2} \, 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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {a}{x}+b x+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^2,x]

[Out]

-(a/x) + b*x + (c*x^3)/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2+c x^4}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.46, size = 20, normalized size = 1.11 \begin {gather*} \frac {c x^{4} + 3 \, b x^{2} - 3 \, a}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*x^4 + 3*b*x^2 - 3*a)/x

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giac [A]  time = 0.15, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + b*x - a/x

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maple [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} \frac {c \,x^{3}}{3}+b x -\frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + b*x - a/x

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mupad [B]  time = 0.03, size = 16, normalized size = 0.89 \begin {gather*} b\,x-\frac {a}{x}+\frac {c\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^2,x)

[Out]

b*x - a/x + (c*x^3)/3

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sympy [A]  time = 0.10, size = 12, normalized size = 0.67 \begin {gather*} - \frac {a}{x} + b x + \frac {c x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x + b*x + c*x**3/3

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