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

Optimal. Leaf size=24 \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]

[Out]

-(a^2/x) + 2*a*b*x + (b^2*x^3)/3

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Rubi [A]  time = 0.008317, 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} \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0008292, size = 24, normalized size = 1. \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/x) + 2*a*b*x + (b^2*x^3)/3

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Maple [A]  time = 0.044, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{x}}+2\,abx+{\frac{{b}^{2}{x}^{3}}{3}} \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^2,x)

[Out]

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

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Maxima [A]  time = 0.985642, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac{a^{2}}{x} \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^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3 + 2*a*b*x - a^2/x

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Fricas [A]  time = 1.41526, size = 50, normalized size = 2.08 \begin{align*} \frac{b^{2} x^{4} + 6 \, a b x^{2} - 3 \, a^{2}}{3 \, x} \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^2,x, algorithm="fricas")

[Out]

1/3*(b^2*x^4 + 6*a*b*x^2 - 3*a^2)/x

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Sympy [A]  time = 0.24932, size = 19, normalized size = 0.79 \begin{align*} - \frac{a^{2}}{x} + 2 a b x + \frac{b^{2} x^{3}}{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**2,x)

[Out]

-a**2/x + 2*a*b*x + b**2*x**3/3

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Giac [A]  time = 1.15915, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac{a^{2}}{x} \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^2,x, algorithm="giac")

[Out]

1/3*b^2*x^3 + 2*a*b*x - a^2/x