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

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a}{3 x^3}-\frac {b}{x} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {a}{3 x^3}-\frac {b}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^4,x]

[Out]

-1/3*a/x^3 - b/x

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)/x^4,x]

[Out]

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

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fricas [A]  time = 0.80, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}

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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giac [A]  time = 1.11, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}

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

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maple [A]  time = 0.01, size = 14, normalized size = 0.93 \begin {gather*} -\frac {b}{x}-\frac {a}{3 x^{3}} \end {gather*}

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 = 1.29, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3\,b\,x^2+a}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^4,x)

[Out]

-(a + 3*b*x^2)/(3*x^3)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.93 \begin {gather*} \frac {- a - 3 b x^{2}}{3 x^{3}} \end {gather*}

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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