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

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

[Out]

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

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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} \[ -\frac {a}{3 x^3}-\frac {b}{x}+c x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

-a/(3*x^3) - b/x + c*x

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

-1/3*a/x^3 - b/x + c*x

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fricas [A]  time = 0.63, size = 21, normalized size = 1.17 \[ \frac {3 \, c x^{4} - 3 \, b x^{2} - a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*c*x^4 - 3*b*x^2 - a)/x^3

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giac [A]  time = 0.15, size = 17, normalized size = 0.94 \[ c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3

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maple [A]  time = 0.01, size = 17, normalized size = 0.94 \[ c x -\frac {b}{x}-\frac {a}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 17, normalized size = 0.94 \[ c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3

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mupad [B]  time = 0.02, size = 18, normalized size = 1.00 \[ c\,x-\frac {b\,x^2+\frac {a}{3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^4,x)

[Out]

c*x - (a/3 + b*x^2)/x^3

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sympy [A]  time = 0.13, size = 17, normalized size = 0.94 \[ c x + \frac {- a - 3 b x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + (-a - 3*b*x**2)/(3*x**3)

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