3.420 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x^8} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

[Out]

-a^2/(7*x^7) - (2*a*b)/(5*x^5) - b^2/(3*x^3)

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Rubi [A]  time = 0.0094775, 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} \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8,x]

[Out]

-a^2/(7*x^7) - (2*a*b)/(5*x^5) - b^2/(3*x^3)

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^8} \, dx &=\int \left (\frac{a^2}{x^8}+\frac{2 a b}{x^6}+\frac{b^2}{x^4}\right ) \, dx\\ &=-\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0008715, size = 30, normalized size = 1. \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8,x]

[Out]

-a^2/(7*x^7) - (2*a*b)/(5*x^5) - b^2/(3*x^3)

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Maple [A]  time = 0.047, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{7\,{x}^{7}}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{b}^{2}}{3\,{x}^{3}}} \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^8,x)

[Out]

-1/7*a^2/x^7-2/5*a*b/x^5-1/3*b^2/x^3

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Maxima [A]  time = 1.00139, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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Fricas [A]  time = 1.41524, size = 63, normalized size = 2.1 \begin{align*} -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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Sympy [A]  time = 0.348299, size = 27, normalized size = 0.9 \begin{align*} - \frac{15 a^{2} + 42 a b x^{2} + 35 b^{2} x^{4}}{105 x^{7}} \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**8,x)

[Out]

-(15*a**2 + 42*a*b*x**2 + 35*b**2*x**4)/(105*x**7)

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Giac [A]  time = 1.13422, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7