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]

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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \[ -\frac {a^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {b^2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 26, normalized size = 0.93 \[ -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \]

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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giac [A]  time = 0.17, size = 26, normalized size = 0.93 \[ -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \]

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

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maple [A]  time = 0.00, size = 25, normalized size = 0.89 \[ -\frac {b^{2}}{x}-\frac {2 a b}{3 x^{3}}-\frac {a^{2}}{5 x^{5}} \]

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 = 1.34, size = 26, normalized size = 0.93 \[ -\frac {15 \, b^{2} x^{4} + 10 \, a b x^{2} + 3 \, a^{2}}{15 \, x^{5}} \]

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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mupad [B]  time = 0.04, size = 25, normalized size = 0.89 \[ -\frac {\frac {a^2}{5}+\frac {2\,a\,b\,x^2}{3}+b^2\,x^4}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)/x^6,x)

[Out]

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

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sympy [A]  time = 0.18, size = 27, normalized size = 0.96 \[ \frac {- 3 a^{2} - 10 a b x^{2} - 15 b^{2} x^{4}}{15 x^{5}} \]

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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