3.418 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x^6} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x} \]

[Out]

-a^2/(5*x^5) - (2*a*b)/(3*x^3) - b^2/x

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Rubi [A]  time = 0.0096647, antiderivative size = 28, 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} \[ -\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0008508, size = 28, normalized size = 1. \[ -\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^6,x]

[Out]

-a^2/(5*x^5) - (2*a*b)/(3*x^3) - b^2/x

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Maple [A]  time = 0.047, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{5\,{x}^{5}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{b}^{2}}{x}} \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^6,x)

[Out]

-1/5*a^2/x^5-2/3*a*b/x^3-b^2/x

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Maxima [A]  time = 1.35489, size = 35, normalized size = 1.25 \begin{align*} -\frac{15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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Fricas [A]  time = 1.38187, size = 61, normalized size = 2.18 \begin{align*} -\frac{15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5

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Sympy [A]  time = 0.323813, size = 27, normalized size = 0.96 \begin{align*} - \frac{3 a^{2} + 10 a b x^{2} + 15 b^{2} x^{4}}{15 x^{5}} \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**6,x)

[Out]

-(3*a**2 + 10*a*b*x**2 + 15*b**2*x**4)/(15*x**5)

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Giac [A]  time = 1.12716, size = 35, normalized size = 1.25 \begin{align*} -\frac{15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

-1/15*(15*b^2*x^4 + 10*a*b*x^2 + 3*a^2)/x^5