Integrand size = 18, antiderivative size = 54 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {a^2}{10 x^{10}}-\frac {a b}{4 x^8}-\frac {b^2+2 a c}{6 x^6}-\frac {b c}{2 x^4}-\frac {c^2}{2 x^2} \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1128, 712} \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {a^2}{10 x^{10}}-\frac {2 a c+b^2}{6 x^6}-\frac {a b}{4 x^8}-\frac {b c}{2 x^4}-\frac {c^2}{2 x^2} \]
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Rule 712
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^6} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{x^6}+\frac {2 a b}{x^5}+\frac {b^2+2 a c}{x^4}+\frac {2 b c}{x^3}+\frac {c^2}{x^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{10 x^{10}}-\frac {a b}{4 x^8}-\frac {b^2+2 a c}{6 x^6}-\frac {b c}{2 x^4}-\frac {c^2}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {6 a^2+5 a \left (3 b x^2+4 c x^4\right )+10 x^4 \left (b^2+3 b c x^2+3 c^2 x^4\right )}{60 x^{10}} \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {c^{2}}{2 x^{2}}-\frac {a b}{4 x^{8}}-\frac {2 a c +b^{2}}{6 x^{6}}-\frac {b c}{2 x^{4}}-\frac {a^{2}}{10 x^{10}}\) | \(45\) |
norman | \(\frac {-\frac {c^{2} x^{8}}{2}-\frac {b c \,x^{6}}{2}+\left (-\frac {a c}{3}-\frac {b^{2}}{6}\right ) x^{4}-\frac {a b \,x^{2}}{4}-\frac {a^{2}}{10}}{x^{10}}\) | \(47\) |
risch | \(\frac {-\frac {c^{2} x^{8}}{2}-\frac {b c \,x^{6}}{2}+\left (-\frac {a c}{3}-\frac {b^{2}}{6}\right ) x^{4}-\frac {a b \,x^{2}}{4}-\frac {a^{2}}{10}}{x^{10}}\) | \(47\) |
gosper | \(-\frac {30 c^{2} x^{8}+30 b c \,x^{6}+20 a c \,x^{4}+10 b^{2} x^{4}+15 a b \,x^{2}+6 a^{2}}{60 x^{10}}\) | \(49\) |
parallelrisch | \(\frac {-30 c^{2} x^{8}-30 b c \,x^{6}-20 a c \,x^{4}-10 b^{2} x^{4}-15 a b \,x^{2}-6 a^{2}}{60 x^{10}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]
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Time = 0.96 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=\frac {- 6 a^{2} - 15 a b x^{2} - 30 b c x^{6} - 30 c^{2} x^{8} + x^{4} \left (- 20 a c - 10 b^{2}\right )}{60 x^{10}} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {30 \, c^{2} x^{8} + 30 \, b c x^{6} + 10 \, b^{2} x^{4} + 20 \, a c x^{4} + 15 \, a b x^{2} + 6 \, a^{2}}{60 \, x^{10}} \]
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Time = 12.90 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{11}} \, dx=-\frac {\frac {a^2}{10}+x^4\,\left (\frac {b^2}{6}+\frac {a\,c}{3}\right )+\frac {c^2\,x^8}{2}+\frac {a\,b\,x^2}{4}+\frac {b\,c\,x^6}{2}}{x^{10}} \]
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