Integrand size = 24, antiderivative size = 84 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {\left (a+b x^2\right )^7}{20 a x^{20}}+\frac {b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac {b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac {b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}} \]
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Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 272, 47, 37} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=\frac {b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}}-\frac {b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac {b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac {\left (a+b x^2\right )^7}{20 a x^{20}} \]
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Rule 28
Rule 37
Rule 47
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{21}} \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^{11}} \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {\left (a+b x^2\right )^7}{20 a x^{20}}-\frac {3 \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^{10}} \, dx,x,x^2\right )}{20 a b^5} \\ & = -\frac {\left (a+b x^2\right )^7}{20 a x^{20}}+\frac {b \left (a+b x^2\right )^7}{60 a^2 x^{18}}+\frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{30 a^2 b^4} \\ & = -\frac {\left (a+b x^2\right )^7}{20 a x^{20}}+\frac {b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac {b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}-\frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{240 a^3 b^3} \\ & = -\frac {\left (a+b x^2\right )^7}{20 a x^{20}}+\frac {b \left (a+b x^2\right )^7}{60 a^2 x^{18}}-\frac {b^2 \left (a+b x^2\right )^7}{240 a^3 x^{16}}+\frac {b^3 \left (a+b x^2\right )^7}{1680 a^4 x^{14}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {a^6}{20 x^{20}}-\frac {a^5 b}{3 x^{18}}-\frac {15 a^4 b^2}{16 x^{16}}-\frac {10 a^3 b^3}{7 x^{14}}-\frac {5 a^2 b^4}{4 x^{12}}-\frac {3 a b^5}{5 x^{10}}-\frac {b^6}{8 x^8} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {5 a^{2} b^{4}}{4 x^{12}}-\frac {10 a^{3} b^{3}}{7 x^{14}}-\frac {b^{6}}{8 x^{8}}-\frac {a^{6}}{20 x^{20}}-\frac {a^{5} b}{3 x^{18}}-\frac {15 a^{4} b^{2}}{16 x^{16}}-\frac {3 b^{5} a}{5 x^{10}}\) | \(69\) |
norman | \(\frac {-\frac {1}{20} a^{6}-\frac {1}{3} a^{5} b \,x^{2}-\frac {15}{16} a^{4} b^{2} x^{4}-\frac {10}{7} a^{3} b^{3} x^{6}-\frac {5}{4} a^{2} b^{4} x^{8}-\frac {3}{5} a \,b^{5} x^{10}-\frac {1}{8} b^{6} x^{12}}{x^{20}}\) | \(70\) |
risch | \(\frac {-\frac {1}{20} a^{6}-\frac {1}{3} a^{5} b \,x^{2}-\frac {15}{16} a^{4} b^{2} x^{4}-\frac {10}{7} a^{3} b^{3} x^{6}-\frac {5}{4} a^{2} b^{4} x^{8}-\frac {3}{5} a \,b^{5} x^{10}-\frac {1}{8} b^{6} x^{12}}{x^{20}}\) | \(70\) |
gosper | \(-\frac {210 b^{6} x^{12}+1008 a \,b^{5} x^{10}+2100 a^{2} b^{4} x^{8}+2400 a^{3} b^{3} x^{6}+1575 a^{4} b^{2} x^{4}+560 a^{5} b \,x^{2}+84 a^{6}}{1680 x^{20}}\) | \(71\) |
parallelrisch | \(\frac {-210 b^{6} x^{12}-1008 a \,b^{5} x^{10}-2100 a^{2} b^{4} x^{8}-2400 a^{3} b^{3} x^{6}-1575 a^{4} b^{2} x^{4}-560 a^{5} b \,x^{2}-84 a^{6}}{1680 x^{20}}\) | \(71\) |
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Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \]
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Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=\frac {- 84 a^{6} - 560 a^{5} b x^{2} - 1575 a^{4} b^{2} x^{4} - 2400 a^{3} b^{3} x^{6} - 2100 a^{2} b^{4} x^{8} - 1008 a b^{5} x^{10} - 210 b^{6} x^{12}}{1680 x^{20}} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \]
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Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {210 \, b^{6} x^{12} + 1008 \, a b^{5} x^{10} + 2100 \, a^{2} b^{4} x^{8} + 2400 \, a^{3} b^{3} x^{6} + 1575 \, a^{4} b^{2} x^{4} + 560 \, a^{5} b x^{2} + 84 \, a^{6}}{1680 \, x^{20}} \]
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Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{21}} \, dx=-\frac {\frac {a^6}{20}+\frac {a^5\,b\,x^2}{3}+\frac {15\,a^4\,b^2\,x^4}{16}+\frac {10\,a^3\,b^3\,x^6}{7}+\frac {5\,a^2\,b^4\,x^8}{4}+\frac {3\,a\,b^5\,x^{10}}{5}+\frac {b^6\,x^{12}}{8}}{x^{20}} \]
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