3.419 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x^7} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2} \]

[Out]

-a^2/(6*x^6) - (a*b)/(2*x^4) - b^2/(2*x^2)

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Rubi [A]  time = 0.0089832, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {14} \[ -\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7,x]

[Out]

-a^2/(6*x^6) - (a*b)/(2*x^4) - b^2/(2*x^2)

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^2+2 a b x^2+b^2 x^4}{x^7} \, dx &=\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^5}+\frac{b^2}{x^3}\right ) \, dx\\ &=-\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0010388, size = 30, normalized size = 1. \[ -\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7,x]

[Out]

-a^2/(6*x^6) - (a*b)/(2*x^4) - b^2/(2*x^2)

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Maple [A]  time = 0.048, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{6\,{x}^{6}}}-{\frac{ab}{2\,{x}^{4}}}-{\frac{{b}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)/x^7,x)

[Out]

-1/6*a^2/x^6-1/2*a*b/x^4-1/2*b^2/x^2

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Maxima [A]  time = 0.994061, size = 32, normalized size = 1.07 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="maxima")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Fricas [A]  time = 1.3988, size = 54, normalized size = 1.8 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Sympy [A]  time = 0.338559, size = 26, normalized size = 0.87 \begin{align*} - \frac{a^{2} + 3 a b x^{2} + 3 b^{2} x^{4}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)/x**7,x)

[Out]

-(a**2 + 3*a*b*x**2 + 3*b**2*x**4)/(6*x**6)

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Giac [A]  time = 1.14528, size = 32, normalized size = 1.07 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="giac")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6