Integrand size = 24, antiderivative size = 72 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=-\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \]
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Time = 0.02 (sec) , antiderivative size = 72, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=-\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \]
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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 )^6}{x^8} \, dx}{b^6} \\ & = \frac {\int \left (15 a^2 b^{10}+\frac {a^6 b^6}{x^8}+\frac {6 a^5 b^7}{x^6}+\frac {15 a^4 b^8}{x^4}+\frac {20 a^3 b^9}{x^2}+6 a b^{11} x^2+b^{12} x^4\right ) \, dx}{b^6} \\ & = -\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=-\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {a^{6}}{7 x^{7}}-\frac {6 a^{5} b}{5 x^{5}}-\frac {5 a^{4} b^{2}}{x^{3}}-\frac {20 a^{3} b^{3}}{x}+15 a^{2} b^{4} x +2 a \,b^{5} x^{3}+\frac {b^{6} x^{5}}{5}\) | \(67\) |
risch | \(\frac {b^{6} x^{5}}{5}+2 a \,b^{5} x^{3}+15 a^{2} b^{4} x +\frac {-20 a^{3} b^{3} x^{6}-5 a^{4} b^{2} x^{4}-\frac {6}{5} a^{5} b \,x^{2}-\frac {1}{7} a^{6}}{x^{7}}\) | \(69\) |
norman | \(\frac {\frac {1}{5} b^{6} x^{12}+2 a \,b^{5} x^{10}+15 a^{2} b^{4} x^{8}-20 a^{3} b^{3} x^{6}-5 a^{4} b^{2} x^{4}-\frac {6}{5} a^{5} b \,x^{2}-\frac {1}{7} a^{6}}{x^{7}}\) | \(70\) |
gosper | \(-\frac {-7 b^{6} x^{12}-70 a \,b^{5} x^{10}-525 a^{2} b^{4} x^{8}+700 a^{3} b^{3} x^{6}+175 a^{4} b^{2} x^{4}+42 a^{5} b \,x^{2}+5 a^{6}}{35 x^{7}}\) | \(71\) |
parallelrisch | \(\frac {7 b^{6} x^{12}+70 a \,b^{5} x^{10}+525 a^{2} b^{4} x^{8}-700 a^{3} b^{3} x^{6}-175 a^{4} b^{2} x^{4}-42 a^{5} b \,x^{2}-5 a^{6}}{35 x^{7}}\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=\frac {7 \, b^{6} x^{12} + 70 \, a b^{5} x^{10} + 525 \, a^{2} b^{4} x^{8} - 700 \, a^{3} b^{3} x^{6} - 175 \, a^{4} b^{2} x^{4} - 42 \, a^{5} b x^{2} - 5 \, a^{6}}{35 \, x^{7}} \]
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Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=15 a^{2} b^{4} x + 2 a b^{5} x^{3} + \frac {b^{6} x^{5}}{5} + \frac {- 5 a^{6} - 42 a^{5} b x^{2} - 175 a^{4} b^{2} x^{4} - 700 a^{3} b^{3} x^{6}}{35 x^{7}} \]
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Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=\frac {1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac {700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{6}}{35 \, x^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=\frac {1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac {700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{6}}{35 \, x^{7}} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^8} \, dx=\frac {b^6\,x^5}{5}-\frac {\frac {a^6}{7}+\frac {6\,a^5\,b\,x^2}{5}+5\,a^4\,b^2\,x^4+20\,a^3\,b^3\,x^6}{x^7}+15\,a^2\,b^4\,x+2\,a\,b^5\,x^3 \]
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