Integrand size = 20, antiderivative size = 65 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=-\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}-\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6} \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=-\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {c^5 (a-b x)^6}{8 x^8}+\frac {1}{4} (5 b) \int \frac {(a c-b c x)^5}{x^8} \, dx \\ & = -\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}+\frac {\left (5 b^2\right ) \int \frac {(a c-b c x)^5}{x^7} \, dx}{28 a} \\ & = -\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}-\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=c^5 \left (-\frac {a^6}{8 x^8}+\frac {4 a^5 b}{7 x^7}-\frac {5 a^4 b^2}{6 x^6}+\frac {5 a^2 b^4}{4 x^4}-\frac {4 a b^5}{3 x^3}+\frac {b^6}{2 x^2}\right ) \]
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Time = 0.86 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {c^{5} \left (-84 b^{6} x^{6}+224 a \,x^{5} b^{5}-210 a^{2} x^{4} b^{4}+140 a^{4} x^{2} b^{2}-96 a^{5} x b +21 a^{6}\right )}{168 x^{8}}\) | \(61\) |
default | \(c^{5} \left (-\frac {5 a^{4} b^{2}}{6 x^{6}}+\frac {4 a^{5} b}{7 x^{7}}-\frac {a^{6}}{8 x^{8}}-\frac {4 a \,b^{5}}{3 x^{3}}+\frac {b^{6}}{2 x^{2}}+\frac {5 a^{2} b^{4}}{4 x^{4}}\right )\) | \(62\) |
norman | \(\frac {-\frac {1}{8} a^{6} c^{5}+\frac {1}{2} b^{6} c^{5} x^{6}-\frac {4}{3} a \,b^{5} c^{5} x^{5}+\frac {5}{4} a^{2} b^{4} c^{5} x^{4}-\frac {5}{6} a^{4} b^{2} c^{5} x^{2}+\frac {4}{7} a^{5} b \,c^{5} x}{x^{8}}\) | \(75\) |
risch | \(\frac {-\frac {1}{8} a^{6} c^{5}+\frac {1}{2} b^{6} c^{5} x^{6}-\frac {4}{3} a \,b^{5} c^{5} x^{5}+\frac {5}{4} a^{2} b^{4} c^{5} x^{4}-\frac {5}{6} a^{4} b^{2} c^{5} x^{2}+\frac {4}{7} a^{5} b \,c^{5} x}{x^{8}}\) | \(75\) |
parallelrisch | \(\frac {84 b^{6} c^{5} x^{6}-224 a \,b^{5} c^{5} x^{5}+210 a^{2} b^{4} c^{5} x^{4}-140 a^{4} b^{2} c^{5} x^{2}+96 a^{5} b \,c^{5} x -21 a^{6} c^{5}}{168 x^{8}}\) | \(76\) |
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Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=\frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=- \frac {21 a^{6} c^{5} - 96 a^{5} b c^{5} x + 140 a^{4} b^{2} c^{5} x^{2} - 210 a^{2} b^{4} c^{5} x^{4} + 224 a b^{5} c^{5} x^{5} - 84 b^{6} c^{5} x^{6}}{168 x^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=\frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=\frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx=-\frac {\frac {a^6\,c^5}{8}-\frac {4\,a^5\,b\,c^5\,x}{7}+\frac {5\,a^4\,b^2\,c^5\,x^2}{6}-\frac {5\,a^2\,b^4\,c^5\,x^4}{4}+\frac {4\,a\,b^5\,c^5\,x^5}{3}-\frac {b^6\,c^5\,x^6}{2}}{x^8} \]
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