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

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \[ -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \]

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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fricas [A]  time = 0.86, size = 25, normalized size = 1.04 \[ \frac {b^{2} x^{4} + 6 \, a b x^{2} - 3 \, a^{2}}{3 \, x} \]

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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giac [A]  time = 1.07, size = 22, normalized size = 0.92 \[ \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]

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

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maple [A]  time = 0.00, size = 23, normalized size = 0.96 \[ \frac {b^{2} x^{3}}{3}+2 a b x -\frac {a^{2}}{x} \]

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 = 1.36, size = 22, normalized size = 0.92 \[ \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]

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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mupad [B]  time = 0.03, size = 22, normalized size = 0.92 \[ \frac {b^2\,x^3}{3}-\frac {a^2}{x}+2\,a\,b\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^2,x)

[Out]

(b^2*x^3)/3 - a^2/x + 2*a*b*x

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sympy [A]  time = 0.10, size = 19, normalized size = 0.79 \[ - \frac {a^{2}}{x} + 2 a b x + \frac {b^{2} x^{3}}{3} \]

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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