Integrand size = 20, antiderivative size = 23 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \]
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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} \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \]
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Rule 21
Rubi steps \begin{align*} \text {integral}& = \frac {b^3 \int (a+b x) \, dx}{d^3} \\ & = \frac {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3 \left (a x+\frac {b x^2}{2}\right )}{d^3} \]
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Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74
method | result | size |
gosper | \(\frac {b^{3} x \left (b x +2 a \right )}{2 d^{3}}\) | \(17\) |
default | \(\frac {b^{3} \left (a x +\frac {1}{2} b \,x^{2}\right )}{d^{3}}\) | \(18\) |
parallelrisch | \(\frac {x^{2} b^{4}+2 x a \,b^{3}}{2 d^{3}}\) | \(21\) |
risch | \(\frac {a \,b^{3} x}{d^{3}}+\frac {b^{4} x^{2}}{2 d^{3}}\) | \(22\) |
norman | \(\frac {\frac {b^{6} x^{4}}{2 d}+\frac {2 a \,b^{5} x^{3}}{d}-\frac {5 a^{4} b^{2}}{2 d}-\frac {4 a^{3} b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\) | \(59\) |
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Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \]
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Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {a b^{3} x}{d^{3}} + \frac {b^{4} x^{2}}{2 d^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3\,x\,\left (2\,a+b\,x\right )}{2\,d^3} \]
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