Integrand size = 17, antiderivative size = 75 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d x^4+\frac {1}{5} b^2 (3 c d+b e) x^5+\frac {1}{2} b c (c d+b e) x^6+\frac {1}{7} c^2 (c d+3 b e) x^7+\frac {1}{8} c^3 e x^8 \]
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Time = 0.04 (sec) , antiderivative size = 75, 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 = {645} \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d x^4+\frac {1}{5} b^2 x^5 (b e+3 c d)+\frac {1}{7} c^2 x^7 (3 b e+c d)+\frac {1}{2} b c x^6 (b e+c d)+\frac {1}{8} c^3 e x^8 \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (b^3 d x^3+b^2 (3 c d+b e) x^4+3 b c (c d+b e) x^5+c^2 (c d+3 b e) x^6+c^3 e x^7\right ) \, dx \\ & = \frac {1}{4} b^3 d x^4+\frac {1}{5} b^2 (3 c d+b e) x^5+\frac {1}{2} b c (c d+b e) x^6+\frac {1}{7} c^2 (c d+3 b e) x^7+\frac {1}{8} c^3 e x^8 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d x^4+\frac {1}{5} b^2 (3 c d+b e) x^5+\frac {1}{2} b c (c d+b e) x^6+\frac {1}{7} c^2 (c d+3 b e) x^7+\frac {1}{8} c^3 e x^8 \]
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Time = 1.97 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {c^{3} e \,x^{8}}{8}+\left (\frac {3}{7} b \,c^{2} e +\frac {1}{7} c^{3} d \right ) x^{7}+\left (\frac {1}{2} b^{2} c e +\frac {1}{2} b \,c^{2} d \right ) x^{6}+\left (\frac {1}{5} b^{3} e +\frac {3}{5} b^{2} c d \right ) x^{5}+\frac {d \,x^{4} b^{3}}{4}\) | \(75\) |
gosper | \(\frac {x^{4} \left (35 c^{3} x^{4} e +120 b \,c^{2} x^{3} e +40 c^{3} d \,x^{3}+140 b^{2} c e \,x^{2}+140 b \,c^{2} d \,x^{2}+56 b^{3} e x +168 x \,b^{2} c d +70 b^{3} d \right )}{280}\) | \(76\) |
default | \(\frac {c^{3} e \,x^{8}}{8}+\frac {\left (3 b \,c^{2} e +c^{3} d \right ) x^{7}}{7}+\frac {\left (3 b^{2} c e +3 b \,c^{2} d \right ) x^{6}}{6}+\frac {\left (b^{3} e +3 b^{2} c d \right ) x^{5}}{5}+\frac {d \,x^{4} b^{3}}{4}\) | \(76\) |
risch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {3}{7} x^{7} b \,c^{2} e +\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} x^{6} b^{2} c e +\frac {1}{2} x^{6} b \,c^{2} d +\frac {1}{5} b^{3} e \,x^{5}+\frac {3}{5} b^{2} c d \,x^{5}+\frac {1}{4} d \,x^{4} b^{3}\) | \(78\) |
parallelrisch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {3}{7} x^{7} b \,c^{2} e +\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} x^{6} b^{2} c e +\frac {1}{2} x^{6} b \,c^{2} d +\frac {1}{5} b^{3} e \,x^{5}+\frac {3}{5} b^{2} c d \,x^{5}+\frac {1}{4} d \,x^{4} b^{3}\) | \(78\) |
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Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{4} \, b^{3} d x^{4} + \frac {1}{7} \, {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d + b^{2} c e\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d + b^{3} e\right )} x^{5} \]
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Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {b^{3} d x^{4}}{4} + \frac {c^{3} e x^{8}}{8} + x^{7} \cdot \left (\frac {3 b c^{2} e}{7} + \frac {c^{3} d}{7}\right ) + x^{6} \left (\frac {b^{2} c e}{2} + \frac {b c^{2} d}{2}\right ) + x^{5} \left (\frac {b^{3} e}{5} + \frac {3 b^{2} c d}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{4} \, b^{3} d x^{4} + \frac {1}{7} \, {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d + b^{2} c e\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d + b^{3} e\right )} x^{5} \]
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {3}{7} \, b c^{2} e x^{7} + \frac {1}{2} \, b c^{2} d x^{6} + \frac {1}{2} \, b^{2} c e x^{6} + \frac {3}{5} \, b^{2} c d x^{5} + \frac {1}{5} \, b^{3} e x^{5} + \frac {1}{4} \, b^{3} d x^{4} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int (d+e x) \left (b x+c x^2\right )^3 \, dx=x^5\,\left (\frac {e\,b^3}{5}+\frac {3\,c\,d\,b^2}{5}\right )+x^7\,\left (\frac {d\,c^3}{7}+\frac {3\,b\,e\,c^2}{7}\right )+\frac {b^3\,d\,x^4}{4}+\frac {c^3\,e\,x^8}{8}+\frac {b\,c\,x^6\,\left (b\,e+c\,d\right )}{2} \]
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