3.156 \(\int \frac{(b x^2+c x^4)^2}{x^{11}} \, dx\)

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

[Out]

-(b + c*x^2)^3/(6*b*x^6)

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Rubi [A]  time = 0.009546, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 264} \[ -\frac{\left (b+c x^2\right )^3}{6 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^11,x]

[Out]

-(b + c*x^2)^3/(6*b*x^6)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 (b x^2+c x^4\right )^2}{x^{11}} \, dx &=\int \frac{\left (b+c x^2\right )^2}{x^7} \, dx\\ &=-\frac{\left (b+c x^2\right )^3}{6 b x^6}\\ \end{align*}

Mathematica [A]  time = 0.0008909, size = 30, normalized size = 1.58 \[ -\frac{b^2}{6 x^6}-\frac{b c}{2 x^4}-\frac{c^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^11,x]

[Out]

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

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Maple [A]  time = 0.049, size = 25, normalized size = 1.3 \begin{align*} -{\frac{bc}{2\,{x}^{4}}}-{\frac{{c}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^11,x)

[Out]

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

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Maxima [A]  time = 0.995802, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^11,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.13117, size = 54, normalized size = 2.84 \begin{align*} -\frac{3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^11,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.385201, size = 26, normalized size = 1.37 \begin{align*} - \frac{b^{2} + 3 b c x^{2} + 3 c^{2} x^{4}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**11,x)

[Out]

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

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Giac [A]  time = 1.29452, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^11,x, algorithm="giac")

[Out]

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