Integrand size = 24, antiderivative size = 56 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {a^4}{15 x^{15}}-\frac {4 a^3 b}{13 x^{13}}-\frac {6 a^2 b^2}{11 x^{11}}-\frac {4 a b^3}{9 x^9}-\frac {b^4}{7 x^7} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {a^4}{15 x^{15}}-\frac {4 a^3 b}{13 x^{13}}-\frac {6 a^2 b^2}{11 x^{11}}-\frac {4 a b^3}{9 x^9}-\frac {b^4}{7 x^7} \]
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Rule 28
Rule 276
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^{16}} \, dx}{b^4} \\ & = \frac {\int \left (\frac {a^4 b^4}{x^{16}}+\frac {4 a^3 b^5}{x^{14}}+\frac {6 a^2 b^6}{x^{12}}+\frac {4 a b^7}{x^{10}}+\frac {b^8}{x^8}\right ) \, dx}{b^4} \\ & = -\frac {a^4}{15 x^{15}}-\frac {4 a^3 b}{13 x^{13}}-\frac {6 a^2 b^2}{11 x^{11}}-\frac {4 a b^3}{9 x^9}-\frac {b^4}{7 x^7} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {a^4}{15 x^{15}}-\frac {4 a^3 b}{13 x^{13}}-\frac {6 a^2 b^2}{11 x^{11}}-\frac {4 a b^3}{9 x^9}-\frac {b^4}{7 x^7} \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {a^{4}}{15 x^{15}}-\frac {4 a^{3} b}{13 x^{13}}-\frac {6 a^{2} b^{2}}{11 x^{11}}-\frac {4 a \,b^{3}}{9 x^{9}}-\frac {b^{4}}{7 x^{7}}\) | \(47\) |
norman | \(\frac {-\frac {1}{15} a^{4}-\frac {4}{13} a^{3} b \,x^{2}-\frac {6}{11} a^{2} b^{2} x^{4}-\frac {4}{9} a \,b^{3} x^{6}-\frac {1}{7} b^{4} x^{8}}{x^{15}}\) | \(48\) |
risch | \(\frac {-\frac {1}{15} a^{4}-\frac {4}{13} a^{3} b \,x^{2}-\frac {6}{11} a^{2} b^{2} x^{4}-\frac {4}{9} a \,b^{3} x^{6}-\frac {1}{7} b^{4} x^{8}}{x^{15}}\) | \(48\) |
gosper | \(-\frac {6435 b^{4} x^{8}+20020 a \,b^{3} x^{6}+24570 a^{2} b^{2} x^{4}+13860 a^{3} b \,x^{2}+3003 a^{4}}{45045 x^{15}}\) | \(49\) |
parallelrisch | \(\frac {-6435 b^{4} x^{8}-20020 a \,b^{3} x^{6}-24570 a^{2} b^{2} x^{4}-13860 a^{3} b \,x^{2}-3003 a^{4}}{45045 x^{15}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=\frac {- 3003 a^{4} - 13860 a^{3} b x^{2} - 24570 a^{2} b^{2} x^{4} - 20020 a b^{3} x^{6} - 6435 b^{4} x^{8}}{45045 x^{15}} \]
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Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]
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Time = 13.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx=-\frac {\frac {a^4}{15}+\frac {4\,a^3\,b\,x^2}{13}+\frac {6\,a^2\,b^2\,x^4}{11}+\frac {4\,a\,b^3\,x^6}{9}+\frac {b^4\,x^8}{7}}{x^{15}} \]
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