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

Optimal. Leaf size=23 \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Rubi [A]  time = 0.0089646, antiderivative size = 23, 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} \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0009226, size = 23, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Maple [A]  time = 0.048, size = 22, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}}+{b}^{2}x \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^4,x)

[Out]

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

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Maxima [A]  time = 0.986414, size = 30, normalized size = 1.3 \begin{align*} b^{2} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3

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Fricas [A]  time = 1.37206, size = 53, normalized size = 2.3 \begin{align*} \frac{3 \, b^{2} x^{4} - 6 \, a b x^{2} - a^{2}}{3 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/3*(3*b^2*x^4 - 6*a*b*x^2 - a^2)/x^3

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Sympy [A]  time = 0.285569, size = 20, normalized size = 0.87 \begin{align*} b^{2} x - \frac{a^{2} + 6 a b x^{2}}{3 x^{3}} \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**4,x)

[Out]

b**2*x - (a**2 + 6*a*b*x**2)/(3*x**3)

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Giac [A]  time = 1.15332, size = 30, normalized size = 1.3 \begin{align*} b^{2} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3