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

Optimal. Leaf size=15 \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

[Out]

-a/(3*x^3) - b/x

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Rubi [A]  time = 0.0047591, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0017611, size = 15, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^4,x]

[Out]

-a/(3*x^3) - b/x

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Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.57621, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b*x^2 + a)/x^3

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Fricas [A]  time = 1.38826, size = 32, normalized size = 2.13 \begin{align*} -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b*x^2 + a)/x^3

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Sympy [A]  time = 0.258209, size = 14, normalized size = 0.93 \begin{align*} - \frac{a + 3 b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + 3*b*x**2)/(3*x**3)

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Giac [A]  time = 2.51309, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b*x^2 + a)/x^3