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

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

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

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Rubi [A]  time = 0.009459, antiderivative size = 24, 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} \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

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*}

Mathematica [A]  time = 0.0010922, size = 24, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.047, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{ab}{{x}^{2}}}+{b}^{2}\ln \left ( x \right ) \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^5,x)

[Out]

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

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Maxima [A]  time = 0.970607, size = 35, normalized size = 1.46 \begin{align*} \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \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^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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Fricas [A]  time = 1.44448, size = 62, normalized size = 2.58 \begin{align*} \frac{4 \, b^{2} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \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^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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Sympy [A]  time = 0.319698, size = 22, normalized size = 0.92 \begin{align*} b^{2} \log{\left (x \right )} - \frac{a^{2} + 4 a b x^{2}}{4 x^{4}} \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**5,x)

[Out]

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

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Giac [A]  time = 1.17739, size = 46, normalized size = 1.92 \begin{align*} \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{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^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