3.25 \(\int \frac{(a+b x^2)^2}{x^7} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^2\right )^3}{6 a x^6} \]

[Out]

-(a + b*x^2)^3/(6*a*x^6)

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Rubi [A]  time = 0.0034544, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\left (a+b x^2\right )^3}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^7,x]

[Out]

-(a + b*x^2)^3/(6*a*x^6)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^7} \, dx &=-\frac{\left (a+b x^2\right )^3}{6 a x^6}\\ \end{align*}

Mathematica [A]  time = 0.0007695, size = 30, normalized size = 1.58 \[ -\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^7,x]

[Out]

-a^2/(6*x^6) - (a*b)/(2*x^4) - b^2/(2*x^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.3 \begin{align*} -{\frac{ab}{2\,{x}^{4}}}-{\frac{{b}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^7,x)

[Out]

-1/2*a*b/x^4-1/2*b^2/x^2-1/6*a^2/x^6

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Maxima [A]  time = 1.73741, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="maxima")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Fricas [A]  time = 1.42218, size = 54, normalized size = 2.84 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="fricas")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Sympy [A]  time = 0.353072, size = 26, normalized size = 1.37 \begin{align*} - \frac{a^{2} + 3 a b x^{2} + 3 b^{2} x^{4}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**7,x)

[Out]

-(a**2 + 3*a*b*x**2 + 3*b**2*x**4)/(6*x**6)

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Giac [A]  time = 2.44744, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7,x, algorithm="giac")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6