Integrand size = 17, antiderivative size = 38 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 a}{c^5 (a-b x)^5}-\frac {1}{c^5 (a-b x)^4}\right ) \, dx \\ & = \frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {a+2 b x}{6 b c^5 (a-b x)^4} \]
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Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {2 b x +a}{6 \left (-b x +a \right )^{4} c^{5} b}\) | \(23\) |
risch | \(\frac {\frac {x}{3}+\frac {a}{6 b}}{c^{5} \left (-b x +a \right )^{4}}\) | \(23\) |
norman | \(\frac {\frac {a}{6 b c}+\frac {x}{3 c}}{c^{4} \left (-b x +a \right )^{4}}\) | \(29\) |
parallelrisch | \(\frac {2 x \,b^{4}+a \,b^{3}}{6 b^{4} c^{5} \left (b x -a \right )^{4}}\) | \(30\) |
default | \(\frac {\frac {a}{2 b \left (-b x +a \right )^{4}}-\frac {1}{3 b \left (-b x +a \right )^{3}}}{c^{5}}\) | \(33\) |
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Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=- \frac {- a - 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {a}{2 \, {\left (b c x - a c\right )}^{4} b c} + \frac {1}{3 \, {\left (b c x - a c\right )}^{3} b c^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {a+b x}{(a c-b c x)^5} \, dx=\frac {\frac {x}{3}+\frac {a}{6\,b}}{a^4\,c^5-4\,a^3\,b\,c^5\,x+6\,a^2\,b^2\,c^5\,x^2-4\,a\,b^3\,c^5\,x^3+b^4\,c^5\,x^4} \]
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