Integrand size = 29, antiderivative size = 14 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {(c+d x)^3}{3 d} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 32} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {(c+d x)^3}{3 d} \]
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Rule 32
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (c+d x)^2 \, dx \\ & = \frac {(c+d x)^3}{3 d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {(c+d x)^3}{3 d} \]
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Time = 2.33 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (d x +c \right )^{3}}{3 d}\) | \(13\) |
parallelrisch | \(\frac {1}{3} d^{2} x^{3}+c d \,x^{2}+c^{2} x\) | \(21\) |
gosper | \(\frac {x \left (d^{2} x^{2}+3 c d x +3 c^{2}\right )}{3}\) | \(22\) |
risch | \(\frac {d^{2} x^{3}}{3}+c d \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}\) | \(29\) |
norman | \(\frac {\left (\frac {1}{3} a \,d^{2}+b c d \right ) x^{3}+\left (a c d +b \,c^{2}\right ) x^{2}+a \,c^{2} x +\frac {b \,d^{2} x^{4}}{3}}{b x +a}\) | \(54\) |
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3} \]
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none
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 6.00 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=\frac {{\left (\frac {3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac {3 \, b c d}{b x + a} - \frac {6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac {3 \, a d^{2}}{b x + a} + \frac {3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )} {\left (b x + a\right )}^{3}}{3 \, b^{3}} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx=c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3} \]
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