Integrand size = 20, antiderivative size = 23 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^4 c x}{d^4}+\frac {b^4 x^2}{2 d^3} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {21} \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^4 c x}{d^4}+\frac {b^4 x^2}{2 d^3} \]
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Rule 21
Rubi steps \begin{align*} \text {integral}& = \frac {b^4 \int (c+d x) \, dx}{d^4} \\ & = \frac {b^4 c x}{d^4}+\frac {b^4 x^2}{2 d^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^4 \left (c x+\frac {d x^2}{2}\right )}{d^4} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74
method | result | size |
gosper | \(\frac {b^{4} x \left (d x +2 c \right )}{2 d^{4}}\) | \(17\) |
default | \(\frac {b^{4} \left (c x +\frac {1}{2} d \,x^{2}\right )}{d^{4}}\) | \(18\) |
risch | \(\frac {b^{4} c x}{d^{4}}+\frac {b^{4} x^{2}}{2 d^{3}}\) | \(22\) |
parallelrisch | \(\frac {x^{2} b^{4} d +2 x \,b^{4} c}{2 d^{4}}\) | \(22\) |
norman | \(\frac {\frac {b^{4} d^{2} x^{4}}{2}-\frac {5 c^{4} b^{4}}{2 d^{2}}+2 b^{4} c d \,x^{3}-\frac {4 c^{3} b^{4} x}{d}}{d^{3} \left (d x +c \right )^{2}}\) | \(57\) |
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^{4} c x}{d^{4}} + \frac {b^{4} x^{2}}{2 d^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^4}{(c+d x)^3} \, dx=\frac {b^4\,x\,\left (2\,c+d\,x\right )}{2\,d^4} \]
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