3.3.45 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{x^5} \, dx\)

Optimal. Leaf size=24 \[ -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 24, 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}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5,x]

[Out]

-1/4*a^2/x^4 - (a*b)/x^2 + b^2*Log[x]

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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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5, x]

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fricas [A]  time = 0.85, size = 28, normalized size = 1.17 \begin {gather*} \frac {4 \, b^{2} x^{4} \log \relax (x) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/4*(4*b^2*x^4*log(x) - 4*a*b*x^2 - a^2)/x^4

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giac [A]  time = 0.15, size = 34, normalized size = 1.42 \begin {gather*} \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} b^{2} \ln \relax (x )-\frac {a b}{x^{2}}-\frac {a^{2}}{4 x^{4}} \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^5,x)

[Out]

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

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maxima [A]  time = 1.37, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} b^2\,\ln \relax (x)-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4} \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^5,x)

[Out]

b^2*log(x) - (a^2/4 + a*b*x^2)/x^4

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sympy [A]  time = 0.17, size = 24, normalized size = 1.00 \begin {gather*} b^{2} \log {\relax (x )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \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**5,x)

[Out]

b**2*log(x) + (-a**2 - 4*a*b*x**2)/(4*x**4)

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