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

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

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Rubi [A]  time = 0.01, antiderivative size = 30, 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} \begin {gather*} -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^2+2 a b x^2+b^2 x^4}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

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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giac [A]  time = 0.17, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

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

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} -\frac {b^{2}}{3 x^{3}}-\frac {2 a b}{5 x^{5}}-\frac {a^{2}}{7 x^{7}} \end {gather*}

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 = 1.35, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

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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mupad [B]  time = 0.03, size = 26, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{7}+\frac {2\,a\,b\,x^2}{5}+\frac {b^2\,x^4}{3}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)/x^8,x)

[Out]

-(a^2/7 + (b^2*x^4)/3 + (2*a*b*x^2)/5)/x^7

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sympy [A]  time = 0.21, size = 27, normalized size = 0.90 \begin {gather*} \frac {- 15 a^{2} - 42 a b x^{2} - 35 b^{2} x^{4}}{105 x^{7}} \end {gather*}

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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