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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\left (b+c x^2\right )^3}{6 b x^6} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.58 \begin {gather*} -\frac {b^2}{6 x^6}-\frac {b c}{2 x^4}-\frac {c^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^11,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x^2+c x^4\right )^2}{x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^11,x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^11, x]

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fricas [A]  time = 0.56, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end {gather*}

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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giac [A]  time = 0.17, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end {gather*}

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

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maple [A]  time = 0.01, size = 25, normalized size = 1.32 \begin {gather*} -\frac {c^{2}}{2 x^{2}}-\frac {b c}{2 x^{4}}-\frac {b^{2}}{6 x^{6}} \end {gather*}

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 = 1.23, size = 24, normalized size = 1.26 \begin {gather*} -\frac {3 \, c^{2} x^{4} + 3 \, b c x^{2} + b^{2}}{6 \, x^{6}} \end {gather*}

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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mupad [B]  time = 0.04, size = 26, normalized size = 1.37 \begin {gather*} -\frac {\frac {b^2}{6}+\frac {b\,c\,x^2}{2}+\frac {c^2\,x^4}{2}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^11,x)

[Out]

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

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sympy [A]  time = 0.21, size = 26, normalized size = 1.37 \begin {gather*} \frac {- b^{2} - 3 b c x^{2} - 3 c^{2} x^{4}}{6 x^{6}} \end {gather*}

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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