Integrand size = 20, antiderivative size = 41 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {c^6 (a-b x)^7}{8 x^8}-\frac {9 b c^6 (a-b x)^7}{56 a x^7} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {c^6 (a-b x)^7}{8 x^8}-\frac {9 b c^6 (a-b x)^7}{56 a x^7} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {c^6 (a-b x)^7}{8 x^8}+\frac {1}{8} (9 b) \int \frac {(a c-b c x)^6}{x^8} \, dx \\ & = -\frac {c^6 (a-b x)^7}{8 x^8}-\frac {9 b c^6 (a-b x)^7}{56 a x^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(41)=82\).
Time = 0.01 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.73 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {a^7 c^6}{8 x^8}+\frac {5 a^6 b c^6}{7 x^7}-\frac {3 a^5 b^2 c^6}{2 x^6}+\frac {a^4 b^3 c^6}{x^5}+\frac {5 a^3 b^4 c^6}{4 x^4}-\frac {3 a^2 b^5 c^6}{x^3}+\frac {5 a b^6 c^6}{2 x^2}-\frac {b^7 c^6}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(37)=74\).
Time = 0.37 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.02
method | result | size |
gosper | \(-\frac {c^{6} \left (56 b^{7} x^{7}-140 a \,b^{6} x^{6}+168 a^{2} b^{5} x^{5}-70 a^{3} b^{4} x^{4}-56 a^{4} b^{3} x^{3}+84 a^{5} b^{2} x^{2}-40 a^{6} b x +7 a^{7}\right )}{56 x^{8}}\) | \(83\) |
default | \(c^{6} \left (-\frac {3 a^{5} b^{2}}{2 x^{6}}+\frac {5 a^{6} b}{7 x^{7}}-\frac {a^{7}}{8 x^{8}}-\frac {3 a^{2} b^{5}}{x^{3}}-\frac {b^{7}}{x}+\frac {5 a \,b^{6}}{2 x^{2}}+\frac {5 a^{3} b^{4}}{4 x^{4}}+\frac {a^{4} b^{3}}{x^{5}}\right )\) | \(83\) |
norman | \(\frac {a^{4} b^{3} c^{6} x^{3}-\frac {1}{8} a^{7} c^{6}-b^{7} c^{6} x^{7}+\frac {5}{2} a \,b^{6} c^{6} x^{6}-3 a^{2} b^{5} c^{6} x^{5}+\frac {5}{4} a^{3} b^{4} c^{6} x^{4}-\frac {3}{2} a^{5} b^{2} c^{6} x^{2}+\frac {5}{7} a^{6} b \,c^{6} x}{x^{8}}\) | \(102\) |
risch | \(\frac {a^{4} b^{3} c^{6} x^{3}-\frac {1}{8} a^{7} c^{6}-b^{7} c^{6} x^{7}+\frac {5}{2} a \,b^{6} c^{6} x^{6}-3 a^{2} b^{5} c^{6} x^{5}+\frac {5}{4} a^{3} b^{4} c^{6} x^{4}-\frac {3}{2} a^{5} b^{2} c^{6} x^{2}+\frac {5}{7} a^{6} b \,c^{6} x}{x^{8}}\) | \(102\) |
parallelrisch | \(\frac {-56 b^{7} c^{6} x^{7}+140 a \,b^{6} c^{6} x^{6}-168 a^{2} b^{5} c^{6} x^{5}+70 a^{3} b^{4} c^{6} x^{4}+56 a^{4} b^{3} c^{6} x^{3}-84 a^{5} b^{2} c^{6} x^{2}+40 a^{6} b \,c^{6} x -7 a^{7} c^{6}}{56 x^{8}}\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (39) = 78\).
Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (36) = 72\).
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.68 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=\frac {- 7 a^{7} c^{6} + 40 a^{6} b c^{6} x - 84 a^{5} b^{2} c^{6} x^{2} + 56 a^{4} b^{3} c^{6} x^{3} + 70 a^{3} b^{4} c^{6} x^{4} - 168 a^{2} b^{5} c^{6} x^{5} + 140 a b^{6} c^{6} x^{6} - 56 b^{7} c^{6} x^{7}}{56 x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (39) = 78\).
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (39) = 78\).
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]
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Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.49 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx=-\frac {\frac {a^7\,c^6}{8}-\frac {5\,a^6\,b\,c^6\,x}{7}+\frac {3\,a^5\,b^2\,c^6\,x^2}{2}-a^4\,b^3\,c^6\,x^3-\frac {5\,a^3\,b^4\,c^6\,x^4}{4}+3\,a^2\,b^5\,c^6\,x^5-\frac {5\,a\,b^6\,c^6\,x^6}{2}+b^7\,c^6\,x^7}{x^8} \]
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