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

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 = 0.009647, antiderivative size = 24, 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} \[ -\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^2,x]

[Out]

-(a^2/x) + 2*a*b*x + (b^2*x^3)/3

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^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 = 0.0006078, 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 + b*x^2)^2/x^2,x]

[Out]

-(a^2/x) + 2*a*b*x + (b^2*x^3)/3

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Maple [A]  time = 0.003, 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*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 = 1.54354, 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*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 = 1.41159, 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*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 = 0.242517, 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*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 = 1.6803, 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*x^2+a)^2/x^2,x, algorithm="giac")

[Out]

1/3*b^2*x^3 + 2*a*b*x - a^2/x