Optimal. Leaf size=54 \[ -\frac {a^2}{12 x^{12}}-\frac {2 a c+b^2}{8 x^8}-\frac {a b}{5 x^{10}}-\frac {b c}{3 x^6}-\frac {c^2}{4 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1114, 698} \begin {gather*} -\frac {a^2}{12 x^{12}}-\frac {2 a c+b^2}{8 x^8}-\frac {a b}{5 x^{10}}-\frac {b c}{3 x^6}-\frac {c^2}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rule 1114
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^7} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2+2 a c}{x^5}+\frac {2 b c}{x^4}+\frac {c^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^{12}}-\frac {a b}{5 x^{10}}-\frac {b^2+2 a c}{8 x^8}-\frac {b c}{3 x^6}-\frac {c^2}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.93 \begin {gather*} -\frac {10 a^2+24 a b x^2+30 a c x^4+15 b^2 x^4+40 b c x^6+30 c^2 x^8}{120 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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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^{13}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 46, normalized size = 0.85 \begin {gather*} -\frac {30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 0.89 \begin {gather*} -\frac {30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, b^{2} x^{4} + 30 \, a c x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.83 \begin {gather*} -\frac {c^{2}}{4 x^{4}}-\frac {b c}{3 x^{6}}-\frac {a b}{5 x^{10}}-\frac {2 a c +b^{2}}{8 x^{8}}-\frac {a^{2}}{12 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 46, normalized size = 0.85 \begin {gather*} -\frac {30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.16, size = 47, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{12}+x^4\,\left (\frac {b^2}{8}+\frac {a\,c}{4}\right )+\frac {c^2\,x^8}{4}+\frac {a\,b\,x^2}{5}+\frac {b\,c\,x^6}{3}}{x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.70, size = 49, normalized size = 0.91 \begin {gather*} \frac {- 10 a^{2} - 24 a b x^{2} - 40 b c x^{6} - 30 c^{2} x^{8} + x^{4} \left (- 30 a c - 15 b^{2}\right )}{120 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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