Integrand size = 9, antiderivative size = 30 \[ \int \frac {x}{(a+b x)^7} \, dx=\frac {a}{6 b^2 (a+b x)^6}-\frac {1}{5 b^2 (a+b x)^5} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{(a+b x)^7} \, dx=\frac {a}{6 b^2 (a+b x)^6}-\frac {1}{5 b^2 (a+b x)^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^7}+\frac {1}{b (a+b x)^6}\right ) \, dx \\ & = \frac {a}{6 b^2 (a+b x)^6}-\frac {1}{5 b^2 (a+b x)^5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {x}{(a+b x)^7} \, dx=-\frac {a+6 b x}{30 b^2 (a+b x)^6} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {6 b x +a}{30 \left (b x +a \right )^{6} b^{2}}\) | \(19\) |
norman | \(\frac {-\frac {x}{5 b}-\frac {a}{30 b^{2}}}{\left (b x +a \right )^{6}}\) | \(22\) |
risch | \(\frac {-\frac {x}{5 b}-\frac {a}{30 b^{2}}}{\left (b x +a \right )^{6}}\) | \(22\) |
parallelrisch | \(\frac {-6 b^{5} x -a \,b^{4}}{30 b^{6} \left (b x +a \right )^{6}}\) | \(26\) |
default | \(\frac {a}{6 b^{2} \left (b x +a \right )^{6}}-\frac {1}{5 b^{2} \left (b x +a \right )^{5}}\) | \(27\) |
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {x}{(a+b x)^7} \, dx=-\frac {6 \, b x + a}{30 \, {\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {x}{(a+b x)^7} \, dx=\frac {- a - 6 b x}{30 a^{6} b^{2} + 180 a^{5} b^{3} x + 450 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{3} + 450 a^{2} b^{6} x^{4} + 180 a b^{7} x^{5} + 30 b^{8} x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {x}{(a+b x)^7} \, dx=-\frac {6 \, b x + a}{30 \, {\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int \frac {x}{(a+b x)^7} \, dx=-\frac {6 \, b x + a}{30 \, {\left (b x + a\right )}^{6} b^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int \frac {x}{(a+b x)^7} \, dx=-\frac {a+6\,b\,x}{30\,b^2\,{\left (a+b\,x\right )}^6} \]
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