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]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, 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} \[ -\frac {a^2}{3 x^3}-\frac {2 a b}{x}+b^2 x \]

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*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {a^2}{3 x^3}-\frac {2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^4,x]

[Out]

-1/3*a^2/x^3 - (2*a*b)/x + b^2*x

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fricas [A]  time = 0.86, size = 26, normalized size = 1.13 \[ \frac {3 \, b^{2} x^{4} - 6 \, a b x^{2} - a^{2}}{3 \, x^{3}} \]

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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giac [A]  time = 0.88, size = 22, normalized size = 0.96 \[ b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]

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

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maple [A]  time = 0.01, size = 22, normalized size = 0.96 \[ b^{2} x -\frac {2 a b}{x}-\frac {a^{2}}{3 x^{3}} \]

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 = 1.32, size = 22, normalized size = 0.96 \[ b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]

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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mupad [B]  time = 0.03, size = 24, normalized size = 1.04 \[ b^2\,x-\frac {\frac {a^2}{3}+2\,b\,a\,x^2}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^4,x)

[Out]

b^2*x - (a^2/3 + 2*a*b*x^2)/x^3

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sympy [A]  time = 0.14, size = 22, normalized size = 0.96 \[ b^{2} x + \frac {- a^{2} - 6 a b x^{2}}{3 x^{3}} \]

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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