Integrand size = 19, antiderivative size = 225 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (3 c d+4 b e) x^5+\frac {1}{2} b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^6+\frac {1}{7} d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^7+\frac {1}{8} e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^8+\frac {1}{3} c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^9+\frac {1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac {1}{11} c^3 e^4 x^{11} \]
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Time = 0.14 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^4 x^4+\frac {1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac {1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac {1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac {1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac {1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac {1}{11} c^3 e^4 x^{11} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (b^3 d^4 x^3+b^2 d^3 (3 c d+4 b e) x^4+3 b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^5+d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^6+e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^7+3 c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^8+c^2 e^3 (4 c d+3 b e) x^9+c^3 e^4 x^{10}\right ) \, dx \\ & = \frac {1}{4} b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (3 c d+4 b e) x^5+\frac {1}{2} b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^6+\frac {1}{7} d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^7+\frac {1}{8} e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^8+\frac {1}{3} c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^9+\frac {1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac {1}{11} c^3 e^4 x^{11} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (3 c d+4 b e) x^5+\frac {1}{2} b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^6+\frac {1}{7} d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^7+\frac {1}{8} e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^8+\frac {1}{3} c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^9+\frac {1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac {1}{11} c^3 e^4 x^{11} \]
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Time = 2.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.01
method | result | size |
norman | \(\frac {c^{3} e^{4} x^{11}}{11}+\left (\frac {3}{10} e^{4} b \,c^{2}+\frac {2}{5} d \,e^{3} c^{3}\right ) x^{10}+\left (\frac {1}{3} e^{4} b^{2} c +\frac {4}{3} d \,e^{3} b \,c^{2}+\frac {2}{3} d^{2} e^{2} c^{3}\right ) x^{9}+\left (\frac {1}{8} e^{4} b^{3}+\frac {3}{2} b^{2} d \,e^{3} c +\frac {9}{4} d^{2} e^{2} b \,c^{2}+\frac {1}{2} d^{3} e \,c^{3}\right ) x^{8}+\left (\frac {4}{7} b^{3} d \,e^{3}+\frac {18}{7} d^{2} e^{2} b^{2} c +\frac {12}{7} d^{3} e b \,c^{2}+\frac {1}{7} d^{4} c^{3}\right ) x^{7}+\left (d^{2} e^{2} b^{3}+2 d^{3} e \,b^{2} c +\frac {1}{2} d^{4} b \,c^{2}\right ) x^{6}+\left (\frac {4}{5} d^{3} e \,b^{3}+\frac {3}{5} d^{4} b^{2} c \right ) x^{5}+\frac {b^{3} d^{4} x^{4}}{4}\) | \(227\) |
default | \(\frac {c^{3} e^{4} x^{11}}{11}+\frac {\left (3 e^{4} b \,c^{2}+4 d \,e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (3 e^{4} b^{2} c +12 d \,e^{3} b \,c^{2}+6 d^{2} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (e^{4} b^{3}+12 b^{2} d \,e^{3} c +18 d^{2} e^{2} b \,c^{2}+4 d^{3} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (4 b^{3} d \,e^{3}+18 d^{2} e^{2} b^{2} c +12 d^{3} e b \,c^{2}+d^{4} c^{3}\right ) x^{7}}{7}+\frac {\left (6 d^{2} e^{2} b^{3}+12 d^{3} e \,b^{2} c +3 d^{4} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (4 d^{3} e \,b^{3}+3 d^{4} b^{2} c \right ) x^{5}}{5}+\frac {b^{3} d^{4} x^{4}}{4}\) | \(232\) |
gosper | \(\frac {x^{4} \left (840 c^{3} e^{4} x^{7}+2772 x^{6} e^{4} b \,c^{2}+3696 x^{6} d \,e^{3} c^{3}+3080 x^{5} e^{4} b^{2} c +12320 x^{5} d \,e^{3} b \,c^{2}+6160 x^{5} d^{2} e^{2} c^{3}+1155 x^{4} e^{4} b^{3}+13860 x^{4} b^{2} d \,e^{3} c +20790 x^{4} d^{2} e^{2} b \,c^{2}+4620 x^{4} d^{3} e \,c^{3}+5280 x^{3} b^{3} d \,e^{3}+23760 x^{3} d^{2} e^{2} b^{2} c +15840 x^{3} d^{3} e b \,c^{2}+1320 x^{3} d^{4} c^{3}+9240 x^{2} d^{2} e^{2} b^{3}+18480 x^{2} d^{3} e \,b^{2} c +4620 x^{2} d^{4} b \,c^{2}+7392 x \,d^{3} e \,b^{3}+5544 x \,d^{4} b^{2} c +2310 b^{3} d^{4}\right )}{9240}\) | \(250\) |
risch | \(\frac {1}{11} c^{3} e^{4} x^{11}+\frac {3}{10} x^{10} e^{4} b \,c^{2}+\frac {2}{5} x^{10} d \,e^{3} c^{3}+\frac {1}{3} x^{9} e^{4} b^{2} c +\frac {4}{3} x^{9} d \,e^{3} b \,c^{2}+\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {1}{8} x^{8} e^{4} b^{3}+\frac {3}{2} x^{8} b^{2} d \,e^{3} c +\frac {9}{4} x^{8} d^{2} e^{2} b \,c^{2}+\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {4}{7} x^{7} b^{3} d \,e^{3}+\frac {18}{7} x^{7} d^{2} e^{2} b^{2} c +\frac {12}{7} x^{7} d^{3} e b \,c^{2}+\frac {1}{7} x^{7} d^{4} c^{3}+x^{6} d^{2} e^{2} b^{3}+2 x^{6} d^{3} e \,b^{2} c +\frac {1}{2} x^{6} d^{4} b \,c^{2}+\frac {4}{5} x^{5} d^{3} e \,b^{3}+\frac {3}{5} x^{5} d^{4} b^{2} c +\frac {1}{4} b^{3} d^{4} x^{4}\) | \(251\) |
parallelrisch | \(\frac {1}{11} c^{3} e^{4} x^{11}+\frac {3}{10} x^{10} e^{4} b \,c^{2}+\frac {2}{5} x^{10} d \,e^{3} c^{3}+\frac {1}{3} x^{9} e^{4} b^{2} c +\frac {4}{3} x^{9} d \,e^{3} b \,c^{2}+\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {1}{8} x^{8} e^{4} b^{3}+\frac {3}{2} x^{8} b^{2} d \,e^{3} c +\frac {9}{4} x^{8} d^{2} e^{2} b \,c^{2}+\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {4}{7} x^{7} b^{3} d \,e^{3}+\frac {18}{7} x^{7} d^{2} e^{2} b^{2} c +\frac {12}{7} x^{7} d^{3} e b \,c^{2}+\frac {1}{7} x^{7} d^{4} c^{3}+x^{6} d^{2} e^{2} b^{3}+2 x^{6} d^{3} e \,b^{2} c +\frac {1}{2} x^{6} d^{4} b \,c^{2}+\frac {4}{5} x^{5} d^{3} e \,b^{3}+\frac {3}{5} x^{5} d^{4} b^{2} c +\frac {1}{4} b^{3} d^{4} x^{4}\) | \(251\) |
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Time = 0.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.02 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{4} \, b^{3} d^{4} x^{4} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, b^{2} c d^{2} e^{2} + 4 \, b^{3} d e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + 4 \, b^{2} c d^{3} e + 2 \, b^{3} d^{2} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{4} + 4 \, b^{3} d^{3} e\right )} x^{5} \]
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Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {b^{3} d^{4} x^{4}}{4} + \frac {c^{3} e^{4} x^{11}}{11} + x^{10} \cdot \left (\frac {3 b c^{2} e^{4}}{10} + \frac {2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac {b^{2} c e^{4}}{3} + \frac {4 b c^{2} d e^{3}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac {b^{3} e^{4}}{8} + \frac {3 b^{2} c d e^{3}}{2} + \frac {9 b c^{2} d^{2} e^{2}}{4} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {4 b^{3} d e^{3}}{7} + \frac {18 b^{2} c d^{2} e^{2}}{7} + \frac {12 b c^{2} d^{3} e}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \left (b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac {b c^{2} d^{4}}{2}\right ) + x^{5} \cdot \left (\frac {4 b^{3} d^{3} e}{5} + \frac {3 b^{2} c d^{4}}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.02 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{4} \, b^{3} d^{4} x^{4} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, b^{2} c d^{2} e^{2} + 4 \, b^{3} d e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + 4 \, b^{2} c d^{3} e + 2 \, b^{3} d^{2} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{4} + 4 \, b^{3} d^{3} e\right )} x^{5} \]
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Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.11 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {3}{10} \, b c^{2} e^{4} x^{10} + \frac {2}{3} \, c^{3} d^{2} e^{2} x^{9} + \frac {4}{3} \, b c^{2} d e^{3} x^{9} + \frac {1}{3} \, b^{2} c e^{4} x^{9} + \frac {1}{2} \, c^{3} d^{3} e x^{8} + \frac {9}{4} \, b c^{2} d^{2} e^{2} x^{8} + \frac {3}{2} \, b^{2} c d e^{3} x^{8} + \frac {1}{8} \, b^{3} e^{4} x^{8} + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {12}{7} \, b c^{2} d^{3} e x^{7} + \frac {18}{7} \, b^{2} c d^{2} e^{2} x^{7} + \frac {4}{7} \, b^{3} d e^{3} x^{7} + \frac {1}{2} \, b c^{2} d^{4} x^{6} + 2 \, b^{2} c d^{3} e x^{6} + b^{3} d^{2} e^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{4} x^{5} + \frac {4}{5} \, b^{3} d^{3} e x^{5} + \frac {1}{4} \, b^{3} d^{4} x^{4} \]
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Time = 0.09 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.95 \[ \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {4\,b^3\,d\,e^3}{7}+\frac {18\,b^2\,c\,d^2\,e^2}{7}+\frac {12\,b\,c^2\,d^3\,e}{7}+\frac {c^3\,d^4}{7}\right )+x^8\,\left (\frac {b^3\,e^4}{8}+\frac {3\,b^2\,c\,d\,e^3}{2}+\frac {9\,b\,c^2\,d^2\,e^2}{4}+\frac {c^3\,d^3\,e}{2}\right )+\frac {b^3\,d^4\,x^4}{4}+\frac {c^3\,e^4\,x^{11}}{11}+\frac {b^2\,d^3\,x^5\,\left (4\,b\,e+3\,c\,d\right )}{5}+\frac {c^2\,e^3\,x^{10}\,\left (3\,b\,e+4\,c\,d\right )}{10}+\frac {b\,d^2\,x^6\,\left (2\,b^2\,e^2+4\,b\,c\,d\,e+c^2\,d^2\right )}{2}+\frac {c\,e^2\,x^9\,\left (b^2\,e^2+4\,b\,c\,d\,e+2\,c^2\,d^2\right )}{3} \]
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