Integrand size = 11, antiderivative size = 36 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {(a+b x)^8}{9 a x^9}+\frac {b (a+b x)^8}{72 a^2 x^8} \]
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Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=\frac {b (a+b x)^8}{72 a^2 x^8}-\frac {(a+b x)^8}{9 a x^9} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{9 a x^9}-\frac {b \int \frac {(a+b x)^7}{x^9} \, dx}{9 a} \\ & = -\frac {(a+b x)^8}{9 a x^9}+\frac {b (a+b x)^8}{72 a^2 x^8} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(36)=72\).
Time = 0.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {a^7}{9 x^9}-\frac {7 a^6 b}{8 x^8}-\frac {3 a^5 b^2}{x^7}-\frac {35 a^4 b^3}{6 x^6}-\frac {7 a^3 b^4}{x^5}-\frac {21 a^2 b^5}{4 x^4}-\frac {7 a b^6}{3 x^3}-\frac {b^7}{2 x^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19
method | result | size |
norman | \(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\) | \(79\) |
risch | \(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\) | \(79\) |
gosper | \(-\frac {36 b^{7} x^{7}+168 a \,b^{6} x^{6}+378 a^{2} b^{5} x^{5}+504 a^{3} b^{4} x^{4}+420 a^{4} b^{3} x^{3}+216 a^{5} b^{2} x^{2}+63 a^{6} b x +8 a^{7}}{72 x^{9}}\) | \(80\) |
default | \(-\frac {35 a^{4} b^{3}}{6 x^{6}}-\frac {3 a^{5} b^{2}}{x^{7}}-\frac {a^{7}}{9 x^{9}}-\frac {7 a \,b^{6}}{3 x^{3}}-\frac {b^{7}}{2 x^{2}}-\frac {21 a^{2} b^{5}}{4 x^{4}}-\frac {7 a^{3} b^{4}}{x^{5}}-\frac {7 a^{6} b}{8 x^{8}}\) | \(80\) |
parallelrisch | \(\frac {-36 b^{7} x^{7}-168 a \,b^{6} x^{6}-378 a^{2} b^{5} x^{5}-504 a^{3} b^{4} x^{4}-420 a^{4} b^{3} x^{3}-216 a^{5} b^{2} x^{2}-63 a^{6} b x -8 a^{7}}{72 x^{9}}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=\frac {- 8 a^{7} - 63 a^{6} b x - 216 a^{5} b^{2} x^{2} - 420 a^{4} b^{3} x^{3} - 504 a^{3} b^{4} x^{4} - 378 a^{2} b^{5} x^{5} - 168 a b^{6} x^{6} - 36 b^{7} x^{7}}{72 x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {\left (8\,a-b\,x\right )\,{\left (a+b\,x\right )}^8}{72\,a^2\,x^9} \]
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