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

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

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 56, normalized size = 1.04 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/11*a^2/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 46, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.19, size = 48, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \begin {gather*} -\frac {c^{2}}{3 x^{3}}-\frac {2 b c}{5 x^{5}}-\frac {2 a b}{9 x^{9}}-\frac {2 a c +b^{2}}{7 x^{7}}-\frac {a^{2}}{11 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 46, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 4.16, size = 47, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{11}+x^4\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {c^2\,x^8}{3}+\frac {2\,a\,b\,x^2}{9}+\frac {2\,b\,c\,x^6}{5}}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^12,x)

[Out]

-(a^2/11 + x^4*((2*a*c)/7 + b^2/7) + (c^2*x^8)/3 + (2*a*b*x^2)/9 + (2*b*c*x^6)/5)/x^11

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sympy [A]  time = 1.96, size = 49, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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