Integrand size = 26, antiderivative size = 119 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^4 (a+b x)^5}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^6}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^7}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^8}{2 b^5}+\frac {e^4 (a+b x)^9}{9 b^5} \]
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Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac {6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac {2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac {(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac {e^4 (a+b x)^9}{9 b^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 (d+e x)^4 \, dx \\ & = \int \left (\frac {(b d-a e)^4 (a+b x)^4}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^5}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^7}{b^4}+\frac {e^4 (a+b x)^8}{b^4}\right ) \, dx \\ & = \frac {(b d-a e)^4 (a+b x)^5}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^6}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^7}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^8}{2 b^5}+\frac {e^4 (a+b x)^9}{9 b^5} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(119)=238\).
Time = 0.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.29 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^4 x+2 a^3 d^3 (b d+a e) x^2+\frac {2}{3} a^2 d^2 \left (3 b^2 d^2+8 a b d e+3 a^2 e^2\right ) x^3+a d \left (b^3 d^3+6 a b^2 d^2 e+6 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} \left (b^4 d^4+16 a b^3 d^3 e+36 a^2 b^2 d^2 e^2+16 a^3 b d e^3+a^4 e^4\right ) x^5+\frac {2}{3} b e \left (b^3 d^3+6 a b^2 d^2 e+6 a^2 b d e^2+a^3 e^3\right ) x^6+\frac {2}{7} b^2 e^2 \left (3 b^2 d^2+8 a b d e+3 a^2 e^2\right ) x^7+\frac {1}{2} b^3 e^3 (b d+a e) x^8+\frac {1}{9} b^4 e^4 x^9 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(109)=218\).
Time = 2.26 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.42
method | result | size |
norman | \(\frac {b^{4} e^{4} x^{9}}{9}+\left (\frac {1}{2} a \,b^{3} e^{4}+\frac {1}{2} b^{4} d \,e^{3}\right ) x^{8}+\left (\frac {6}{7} a^{2} b^{2} e^{4}+\frac {16}{7} a \,b^{3} d \,e^{3}+\frac {6}{7} b^{4} d^{2} e^{2}\right ) x^{7}+\left (\frac {2}{3} a^{3} b \,e^{4}+4 a^{2} b^{2} d \,e^{3}+4 a \,b^{3} d^{2} e^{2}+\frac {2}{3} b^{4} d^{3} e \right ) x^{6}+\left (\frac {1}{5} e^{4} a^{4}+\frac {16}{5} b \,e^{3} d \,a^{3}+\frac {36}{5} b^{2} e^{2} d^{2} a^{2}+\frac {16}{5} a \,b^{3} d^{3} e +\frac {1}{5} b^{4} d^{4}\right ) x^{5}+\left (a^{4} d \,e^{3}+6 a^{3} b \,d^{2} e^{2}+6 a^{2} b^{2} d^{3} e +a \,b^{3} d^{4}\right ) x^{4}+\left (2 a^{4} d^{2} e^{2}+\frac {16}{3} a^{3} b \,d^{3} e +2 a^{2} b^{2} d^{4}\right ) x^{3}+\left (2 a^{4} d^{3} e +2 d^{4} a^{3} b \right ) x^{2}+a^{4} d^{4} x\) | \(288\) |
default | \(\frac {b^{4} e^{4} x^{9}}{9}+\frac {\left (4 a \,b^{3} e^{4}+4 b^{4} d \,e^{3}\right ) x^{8}}{8}+\frac {\left (6 a^{2} b^{2} e^{4}+16 a \,b^{3} d \,e^{3}+6 b^{4} d^{2} e^{2}\right ) x^{7}}{7}+\frac {\left (4 a^{3} b \,e^{4}+24 a^{2} b^{2} d \,e^{3}+24 a \,b^{3} d^{2} e^{2}+4 b^{4} d^{3} e \right ) x^{6}}{6}+\frac {\left (e^{4} a^{4}+16 b \,e^{3} d \,a^{3}+36 b^{2} e^{2} d^{2} a^{2}+16 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{5}}{5}+\frac {\left (4 a^{4} d \,e^{3}+24 a^{3} b \,d^{2} e^{2}+24 a^{2} b^{2} d^{3} e +4 a \,b^{3} d^{4}\right ) x^{4}}{4}+\frac {\left (6 a^{4} d^{2} e^{2}+16 a^{3} b \,d^{3} e +6 a^{2} b^{2} d^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{4} d^{3} e +4 d^{4} a^{3} b \right ) x^{2}}{2}+a^{4} d^{4} x\) | \(295\) |
risch | \(a^{4} d \,e^{3} x^{4}+\frac {1}{5} d^{4} x^{5} b^{4}+a \,b^{3} d^{4} x^{4}+2 a^{2} b^{2} d^{4} x^{3}+2 a^{3} b \,d^{4} x^{2}+a^{4} d^{4} x +\frac {1}{2} x^{8} a \,b^{3} e^{4}+\frac {1}{2} x^{8} b^{4} d \,e^{3}+\frac {16}{5} x^{5} b \,e^{3} d \,a^{3}+\frac {36}{5} x^{5} b^{2} e^{2} d^{2} a^{2}+\frac {16}{5} x^{5} a \,b^{3} d^{3} e +2 x^{3} a^{4} d^{2} e^{2}+\frac {6}{7} x^{7} a^{2} b^{2} e^{4}+\frac {6}{7} x^{7} b^{4} d^{2} e^{2}+\frac {2}{3} x^{6} a^{3} b \,e^{4}+\frac {2}{3} x^{6} b^{4} d^{3} e +2 d^{3} e \,a^{4} x^{2}+\frac {1}{9} b^{4} e^{4} x^{9}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {16}{3} x^{3} a^{3} b \,d^{3} e +6 a^{2} b^{2} d^{3} e \,x^{4}+6 a^{3} b \,d^{2} e^{2} x^{4}+\frac {16}{7} x^{7} a \,b^{3} d \,e^{3}+4 x^{6} a^{2} b^{2} d \,e^{3}+4 x^{6} a \,b^{3} d^{2} e^{2}\) | \(322\) |
parallelrisch | \(a^{4} d \,e^{3} x^{4}+\frac {1}{5} d^{4} x^{5} b^{4}+a \,b^{3} d^{4} x^{4}+2 a^{2} b^{2} d^{4} x^{3}+2 a^{3} b \,d^{4} x^{2}+a^{4} d^{4} x +\frac {1}{2} x^{8} a \,b^{3} e^{4}+\frac {1}{2} x^{8} b^{4} d \,e^{3}+\frac {16}{5} x^{5} b \,e^{3} d \,a^{3}+\frac {36}{5} x^{5} b^{2} e^{2} d^{2} a^{2}+\frac {16}{5} x^{5} a \,b^{3} d^{3} e +2 x^{3} a^{4} d^{2} e^{2}+\frac {6}{7} x^{7} a^{2} b^{2} e^{4}+\frac {6}{7} x^{7} b^{4} d^{2} e^{2}+\frac {2}{3} x^{6} a^{3} b \,e^{4}+\frac {2}{3} x^{6} b^{4} d^{3} e +2 d^{3} e \,a^{4} x^{2}+\frac {1}{9} b^{4} e^{4} x^{9}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {16}{3} x^{3} a^{3} b \,d^{3} e +6 a^{2} b^{2} d^{3} e \,x^{4}+6 a^{3} b \,d^{2} e^{2} x^{4}+\frac {16}{7} x^{7} a \,b^{3} d \,e^{3}+4 x^{6} a^{2} b^{2} d \,e^{3}+4 x^{6} a \,b^{3} d^{2} e^{2}\) | \(322\) |
gosper | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} a \,b^{3} e^{4}+315 x^{7} b^{4} d \,e^{3}+540 x^{6} a^{2} b^{2} e^{4}+1440 x^{6} a \,b^{3} d \,e^{3}+540 x^{6} b^{4} d^{2} e^{2}+420 x^{5} a^{3} b \,e^{4}+2520 x^{5} a^{2} b^{2} d \,e^{3}+2520 x^{5} a \,b^{3} d^{2} e^{2}+420 x^{5} b^{4} d^{3} e +126 x^{4} e^{4} a^{4}+2016 x^{4} b \,e^{3} d \,a^{3}+4536 x^{4} b^{2} e^{2} d^{2} a^{2}+2016 x^{4} a \,b^{3} d^{3} e +126 x^{4} b^{4} d^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 x^{2} a^{4} d^{2} e^{2}+3360 x^{2} a^{3} b \,d^{3} e +1260 x^{2} a^{2} b^{2} d^{4}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right )}{630}\) | \(323\) |
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (109) = 218\).
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.39 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} + {\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (105) = 210\).
Time = 0.04 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.67 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{4} x + \frac {b^{4} e^{4} x^{9}}{9} + x^{8} \left (\frac {a b^{3} e^{4}}{2} + \frac {b^{4} d e^{3}}{2}\right ) + x^{7} \cdot \left (\frac {6 a^{2} b^{2} e^{4}}{7} + \frac {16 a b^{3} d e^{3}}{7} + \frac {6 b^{4} d^{2} e^{2}}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} b e^{4}}{3} + 4 a^{2} b^{2} d e^{3} + 4 a b^{3} d^{2} e^{2} + \frac {2 b^{4} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {16 a^{3} b d e^{3}}{5} + \frac {36 a^{2} b^{2} d^{2} e^{2}}{5} + \frac {16 a b^{3} d^{3} e}{5} + \frac {b^{4} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 6 a^{3} b d^{2} e^{2} + 6 a^{2} b^{2} d^{3} e + a b^{3} d^{4}\right ) + x^{3} \cdot \left (2 a^{4} d^{2} e^{2} + \frac {16 a^{3} b d^{3} e}{3} + 2 a^{2} b^{2} d^{4}\right ) + x^{2} \cdot \left (2 a^{4} d^{3} e + 2 a^{3} b d^{4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (109) = 218\).
Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.39 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} + {\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (109) = 218\).
Time = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.70 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{9} \, b^{4} e^{4} x^{9} + \frac {1}{2} \, b^{4} d e^{3} x^{8} + \frac {1}{2} \, a b^{3} e^{4} x^{8} + \frac {6}{7} \, b^{4} d^{2} e^{2} x^{7} + \frac {16}{7} \, a b^{3} d e^{3} x^{7} + \frac {6}{7} \, a^{2} b^{2} e^{4} x^{7} + \frac {2}{3} \, b^{4} d^{3} e x^{6} + 4 \, a b^{3} d^{2} e^{2} x^{6} + 4 \, a^{2} b^{2} d e^{3} x^{6} + \frac {2}{3} \, a^{3} b e^{4} x^{6} + \frac {1}{5} \, b^{4} d^{4} x^{5} + \frac {16}{5} \, a b^{3} d^{3} e x^{5} + \frac {36}{5} \, a^{2} b^{2} d^{2} e^{2} x^{5} + \frac {16}{5} \, a^{3} b d e^{3} x^{5} + \frac {1}{5} \, a^{4} e^{4} x^{5} + a b^{3} d^{4} x^{4} + 6 \, a^{2} b^{2} d^{3} e x^{4} + 6 \, a^{3} b d^{2} e^{2} x^{4} + a^{4} d e^{3} x^{4} + 2 \, a^{2} b^{2} d^{4} x^{3} + \frac {16}{3} \, a^{3} b d^{3} e x^{3} + 2 \, a^{4} d^{2} e^{2} x^{3} + 2 \, a^{3} b d^{4} x^{2} + 2 \, a^{4} d^{3} e x^{2} + a^{4} d^{4} x \]
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Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.28 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^5\,\left (\frac {a^4\,e^4}{5}+\frac {16\,a^3\,b\,d\,e^3}{5}+\frac {36\,a^2\,b^2\,d^2\,e^2}{5}+\frac {16\,a\,b^3\,d^3\,e}{5}+\frac {b^4\,d^4}{5}\right )+x^4\,\left (a^4\,d\,e^3+6\,a^3\,b\,d^2\,e^2+6\,a^2\,b^2\,d^3\,e+a\,b^3\,d^4\right )+x^6\,\left (\frac {2\,a^3\,b\,e^4}{3}+4\,a^2\,b^2\,d\,e^3+4\,a\,b^3\,d^2\,e^2+\frac {2\,b^4\,d^3\,e}{3}\right )+a^4\,d^4\,x+\frac {b^4\,e^4\,x^9}{9}+2\,a^3\,d^3\,x^2\,\left (a\,e+b\,d\right )+\frac {b^3\,e^3\,x^8\,\left (a\,e+b\,d\right )}{2}+\frac {2\,a^2\,d^2\,x^3\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {2\,b^2\,e^2\,x^7\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,b^2\,d^2\right )}{7} \]
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