3.839 \(\int \frac{(a+b x^2+c x^4)^2}{x^{12}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2}{11 x^{11}}-\frac{2 a c+b^2}{7 x^7}-\frac{2 a b}{9 x^9}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

[Out]

-a^2/(11*x^11) - (2*a*b)/(9*x^9) - (b^2 + 2*a*c)/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)

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Rubi [A]  time = 0.0248372, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1108} \[ -\frac{a^2}{11 x^{11}}-\frac{2 a c+b^2}{7 x^7}-\frac{2 a b}{9 x^9}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^12,x]

[Out]

-a^2/(11*x^11) - (2*a*b)/(9*x^9) - (b^2 + 2*a*c)/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{x^{12}} \, dx &=\int \left (\frac{a^2}{x^{12}}+\frac{2 a b}{x^{10}}+\frac{b^2+2 a c}{x^8}+\frac{2 b c}{x^6}+\frac{c^2}{x^4}\right ) \, dx\\ &=-\frac{a^2}{11 x^{11}}-\frac{2 a b}{9 x^9}-\frac{b^2+2 a c}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.024197, size = 56, normalized size = 1.04 \[ -\frac{a^2}{11 x^{11}}+\frac{-2 a c-b^2}{7 x^7}-\frac{2 a b}{9 x^9}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^12,x]

[Out]

-a^2/(11*x^11) - (2*a*b)/(9*x^9) + (-b^2 - 2*a*c)/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)

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Maple [A]  time = 0.048, size = 45, normalized size = 0.8 \begin{align*} -{\frac{{c}^{2}}{3\,{x}^{3}}}-{\frac{2\,bc}{5\,{x}^{5}}}-{\frac{{a}^{2}}{11\,{x}^{11}}}-{\frac{2\,ac+{b}^{2}}{7\,{x}^{7}}}-{\frac{2\,ab}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^12,x)

[Out]

-1/3*c^2/x^3-2/5*b*c/x^5-1/11*a^2/x^11-1/7*(2*a*c+b^2)/x^7-2/9*a*b/x^9

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Maxima [A]  time = 0.960844, size = 62, normalized size = 1.15 \begin{align*} -\frac{1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^12,x, algorithm="maxima")

[Out]

-1/3465*(1155*c^2*x^8 + 1386*b*c*x^6 + 495*(b^2 + 2*a*c)*x^4 + 770*a*b*x^2 + 315*a^2)/x^11

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Fricas [A]  time = 1.56134, size = 124, normalized size = 2.3 \begin{align*} -\frac{1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^12,x, algorithm="fricas")

[Out]

-1/3465*(1155*c^2*x^8 + 1386*b*c*x^6 + 495*(b^2 + 2*a*c)*x^4 + 770*a*b*x^2 + 315*a^2)/x^11

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Sympy [A]  time = 2.04834, size = 49, normalized size = 0.91 \begin{align*} - \frac{315 a^{2} + 770 a b x^{2} + 1386 b c x^{6} + 1155 c^{2} x^{8} + x^{4} \left (990 a c + 495 b^{2}\right )}{3465 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**12,x)

[Out]

-(315*a**2 + 770*a*b*x**2 + 1386*b*c*x**6 + 1155*c**2*x**8 + x**4*(990*a*c + 495*b**2))/(3465*x**11)

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Giac [A]  time = 1.11809, size = 65, normalized size = 1.2 \begin{align*} -\frac{1155 \, c^{2} x^{8} + 1386 \, b c x^{6} + 495 \, b^{2} x^{4} + 990 \, a c x^{4} + 770 \, a b x^{2} + 315 \, a^{2}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^12,x, algorithm="giac")

[Out]

-1/3465*(1155*c^2*x^8 + 1386*b*c*x^6 + 495*b^2*x^4 + 990*a*c*x^4 + 770*a*b*x^2 + 315*a^2)/x^11