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

Optimal. Leaf size=10 \[ c x-\frac {b}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} c x-\frac {b}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^4,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} c x-\frac {b}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^4,x]

[Out]

-(b/x) + c*x

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 13, normalized size = 1.30 \begin {gather*} \frac {c x^{2} - b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 - b)/x

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giac [A]  time = 0.16, size = 10, normalized size = 1.00 \begin {gather*} c x - \frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - b/x

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maple [A]  time = 0.00, size = 11, normalized size = 1.10 \begin {gather*} c x -\frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/x+c*x

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maxima [A]  time = 1.33, size = 10, normalized size = 1.00 \begin {gather*} c x - \frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - b/x

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mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} c\,x-\frac {b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^4,x)

[Out]

c*x - b/x

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sympy [A]  time = 0.10, size = 5, normalized size = 0.50 \begin {gather*} - \frac {b}{x} + c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/x + c*x

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