3.415 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]

[Out]

-a^2/(2*x^2) + (b^2*x^2)/2 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0091011, antiderivative size = 27, 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} \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0012949, size = 27, normalized size = 1. \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]

[Out]

-a^2/(2*x^2) + (b^2*x^2)/2 + 2*a*b*Log[x]

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Maple [A]  time = 0.056, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}+{\frac{{b}^{2}{x}^{2}}{2}}+2\,ab\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^3,x)

[Out]

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

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Maxima [A]  time = 0.984704, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{a^{2}}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45842, size = 59, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - a^{2}}{2 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.274006, size = 24, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2} \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**3,x)

[Out]

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

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Giac [A]  time = 1.13649, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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