Integrand size = 13, antiderivative size = 38 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b c-a d}{b (a+b x)^5}+\frac {d}{b (a+b x)^4}\right ) \, dx \\ & = -\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {3 b c+a d+4 b d x}{12 b^2 (a+b x)^4} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {4 b d x +a d +3 b c}{12 b^{2} \left (b x +a \right )^{4}}\) | \(26\) |
risch | \(\frac {-\frac {d x}{3 b}-\frac {a d +3 b c}{12 b^{2}}}{\left (b x +a \right )^{4}}\) | \(30\) |
parallelrisch | \(\frac {-4 d x \,b^{3}-a \,b^{2} d -3 b^{3} c}{12 b^{4} \left (b x +a \right )^{4}}\) | \(34\) |
default | \(-\frac {d}{3 b^{2} \left (b x +a \right )^{3}}-\frac {-a d +b c}{4 b^{2} \left (b x +a \right )^{4}}\) | \(35\) |
norman | \(\frac {-\frac {d x}{3 b}+\frac {-a \,b^{2} d -3 b^{3} c}{12 b^{4}}}{\left (b x +a \right )^{4}}\) | \(36\) |
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Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32) = 64\).
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=\frac {- a d - 3 b c - 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {c}{4 \, {\left (b x + a\right )}^{4} b} - \frac {d}{3 \, {\left (b x + a\right )}^{3} b^{2}} + \frac {a d}{4 \, {\left (b x + a\right )}^{4} b^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int \frac {c+d x}{(a+b x)^5} \, dx=-\frac {\frac {a\,d+3\,b\,c}{12\,b^2}+\frac {d\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]
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