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

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 = 0.0090147, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{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 = 0.0007299, 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 + b*x^2)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Maple [A]  time = 0.003, 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*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 = 2.32474, 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*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 = 1.38166, 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*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 = 0.278253, 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*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 = 1.91432, 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*x^2+a)^2/x^4,x, algorithm="giac")

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3