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

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

[Out]

-(b/x) + c*x

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Rubi [A]  time = 0.0052681, antiderivative size = 10, normalized size of antiderivative = 1., 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} \[ c x-\frac{b}{x} \]

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*}

Mathematica [A]  time = 0.0008532, size = 10, normalized size = 1. \[ c x-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^4,x]

[Out]

-(b/x) + c*x

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Maple [A]  time = 0.045, size = 11, normalized size = 1.1 \begin{align*} -{\frac{b}{x}}+cx \end{align*}

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 = 1.00607, size = 14, normalized size = 1.4 \begin{align*} c x - \frac{b}{x} \end{align*}

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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Fricas [A]  time = 1.23142, size = 20, normalized size = 2. \begin{align*} \frac{c x^{2} - b}{x} \end{align*}

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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Sympy [A]  time = 0.253475, size = 5, normalized size = 0.5 \begin{align*} - \frac{b}{x} + c x \end{align*}

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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Giac [A]  time = 1.12271, size = 14, normalized size = 1.4 \begin{align*} c x - \frac{b}{x} \end{align*}

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