Optimal. Leaf size=391 \[ \frac {\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac {\left (5600 d^3+945 d^2 e+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac {\left (7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}-\frac {\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac {\left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}+\frac {25 (d+e x)^{12}}{3 e^9}-\frac {5 (160 d+9 e) (d+e x)^{11}}{11 e^9} \]
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Rubi [A] time = 0.39, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1628} \[ \frac {\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac {\left (945 d^2 e+5600 d^3+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac {\left (1665 d^2 e^2+1575 d^3 e+7000 d^4+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}-\frac {\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}+\frac {\left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}+\frac {25 (d+e x)^{12}}{3 e^9}-\frac {5 (160 d+9 e) (d+e x)^{11}}{11 e^9} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int (d+e x)^3 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^3}{e^8}+\frac {\left (-800 d^7-315 d^6 e-666 d^5 e^2-185 d^4 e^3-592 d^3 e^4+195 d^2 e^5-214 d e^6+33 e^7\right ) (d+e x)^4}{e^8}+\frac {\left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right ) (d+e x)^5}{e^8}+\frac {\left (-5600 d^5-1575 d^4 e-2220 d^3 e^2-370 d^2 e^3-592 d e^4+65 e^5\right ) (d+e x)^6}{e^8}+\frac {\left (7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right ) (d+e x)^7}{e^8}+\frac {\left (-5600 d^3-945 d^2 e-666 d e^2-37 e^3\right ) (d+e x)^8}{e^8}+\frac {\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^9}{e^8}-\frac {5 (160 d+9 e) (d+e x)^{10}}{e^8}+\frac {100 (d+e x)^{11}}{e^8}\right ) \, dx\\ &=\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac {\left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}-\frac {\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}+\frac {\left (7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}-\frac {\left (5600 d^3+945 d^2 e+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac {\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac {5 (160 d+9 e) (d+e x)^{11}}{11 e^9}+\frac {25 (d+e x)^{12}}{3 e^9}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 277, normalized size = 0.71 \[ 18 d^3 x+\frac {3}{10} e x^{10} \left (100 d^2-45 d e+37 e^2\right )+\frac {1}{3} d x^3 \left (107 d^2+99 d e+54 e^2\right )+\frac {3}{2} d^2 x^2 (11 d+18 e)+\frac {1}{9} x^9 \left (100 d^3-135 d^2 e+333 d e^2-37 e^3\right )+\frac {1}{8} x^8 \left (-45 d^3+333 d^2 e-111 d e^2+148 e^3\right )+\frac {1}{7} x^7 \left (111 d^3-111 d^2 e+444 d e^2+65 e^3\right )+\frac {1}{6} x^6 \left (-37 d^3+444 d^2 e+195 d e^2+107 e^3\right )+\frac {1}{5} x^5 \left (148 d^3+195 d^2 e+321 d e^2+33 e^3\right )+\frac {1}{4} x^4 \left (65 d^3+321 d^2 e+99 d e^2+18 e^3\right )+\frac {15}{11} e^2 x^{11} (20 d-3 e)+\frac {25 e^3 x^{12}}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 305, normalized size = 0.78 \[ \frac {25}{3} x^{12} e^{3} - \frac {45}{11} x^{11} e^{3} + \frac {300}{11} x^{11} e^{2} d + \frac {111}{10} x^{10} e^{3} - \frac {27}{2} x^{10} e^{2} d + 30 x^{10} e d^{2} - \frac {37}{9} x^{9} e^{3} + 37 x^{9} e^{2} d - 15 x^{9} e d^{2} + \frac {100}{9} x^{9} d^{3} + \frac {37}{2} x^{8} e^{3} - \frac {111}{8} x^{8} e^{2} d + \frac {333}{8} x^{8} e d^{2} - \frac {45}{8} x^{8} d^{3} + \frac {65}{7} x^{7} e^{3} + \frac {444}{7} x^{7} e^{2} d - \frac {111}{7} x^{7} e d^{2} + \frac {111}{7} x^{7} d^{3} + \frac {107}{6} x^{6} e^{3} + \frac {65}{2} x^{6} e^{2} d + 74 x^{6} e d^{2} - \frac {37}{6} x^{6} d^{3} + \frac {33}{5} x^{5} e^{3} + \frac {321}{5} x^{5} e^{2} d + 39 x^{5} e d^{2} + \frac {148}{5} x^{5} d^{3} + \frac {9}{2} x^{4} e^{3} + \frac {99}{4} x^{4} e^{2} d + \frac {321}{4} x^{4} e d^{2} + \frac {65}{4} x^{4} d^{3} + 18 x^{3} e^{2} d + 33 x^{3} e d^{2} + \frac {107}{3} x^{3} d^{3} + 27 x^{2} e d^{2} + \frac {33}{2} x^{2} d^{3} + 18 x d^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 296, normalized size = 0.76 \[ \frac {25}{3} \, x^{12} e^{3} + \frac {300}{11} \, d x^{11} e^{2} + 30 \, d^{2} x^{10} e + \frac {100}{9} \, d^{3} x^{9} - \frac {45}{11} \, x^{11} e^{3} - \frac {27}{2} \, d x^{10} e^{2} - 15 \, d^{2} x^{9} e - \frac {45}{8} \, d^{3} x^{8} + \frac {111}{10} \, x^{10} e^{3} + 37 \, d x^{9} e^{2} + \frac {333}{8} \, d^{2} x^{8} e + \frac {111}{7} \, d^{3} x^{7} - \frac {37}{9} \, x^{9} e^{3} - \frac {111}{8} \, d x^{8} e^{2} - \frac {111}{7} \, d^{2} x^{7} e - \frac {37}{6} \, d^{3} x^{6} + \frac {37}{2} \, x^{8} e^{3} + \frac {444}{7} \, d x^{7} e^{2} + 74 \, d^{2} x^{6} e + \frac {148}{5} \, d^{3} x^{5} + \frac {65}{7} \, x^{7} e^{3} + \frac {65}{2} \, d x^{6} e^{2} + 39 \, d^{2} x^{5} e + \frac {65}{4} \, d^{3} x^{4} + \frac {107}{6} \, x^{6} e^{3} + \frac {321}{5} \, d x^{5} e^{2} + \frac {321}{4} \, d^{2} x^{4} e + \frac {107}{3} \, d^{3} x^{3} + \frac {33}{5} \, x^{5} e^{3} + \frac {99}{4} \, d x^{4} e^{2} + 33 \, d^{2} x^{3} e + \frac {33}{2} \, d^{3} x^{2} + \frac {9}{2} \, x^{4} e^{3} + 18 \, d x^{3} e^{2} + 27 \, d^{2} x^{2} e + 18 \, d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 264, normalized size = 0.68 \[ \frac {25 e^{3} x^{12}}{3}+\frac {\left (300 d \,e^{2}-45 e^{3}\right ) x^{11}}{11}+\frac {\left (300 d^{2} e -135 d \,e^{2}+111 e^{3}\right ) x^{10}}{10}+\frac {\left (100 d^{3}-135 d^{2} e +333 d \,e^{2}-37 e^{3}\right ) x^{9}}{9}+\frac {\left (-45 d^{3}+333 d^{2} e -111 d \,e^{2}+148 e^{3}\right ) x^{8}}{8}+\frac {\left (111 d^{3}-111 d^{2} e +444 d \,e^{2}+65 e^{3}\right ) x^{7}}{7}+\frac {\left (-37 d^{3}+444 d^{2} e +195 d \,e^{2}+107 e^{3}\right ) x^{6}}{6}+\frac {\left (148 d^{3}+195 d^{2} e +321 d \,e^{2}+33 e^{3}\right ) x^{5}}{5}+18 d^{3} x +\frac {\left (65 d^{3}+321 d^{2} e +99 d \,e^{2}+18 e^{3}\right ) x^{4}}{4}+\frac {\left (107 d^{3}+99 d^{2} e +54 d \,e^{2}\right ) x^{3}}{3}+\frac {\left (33 d^{3}+54 d^{2} e \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 263, normalized size = 0.67 \[ \frac {25}{3} \, e^{3} x^{12} + \frac {15}{11} \, {\left (20 \, d e^{2} - 3 \, e^{3}\right )} x^{11} + \frac {3}{10} \, {\left (100 \, d^{2} e - 45 \, d e^{2} + 37 \, e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (100 \, d^{3} - 135 \, d^{2} e + 333 \, d e^{2} - 37 \, e^{3}\right )} x^{9} - \frac {1}{8} \, {\left (45 \, d^{3} - 333 \, d^{2} e + 111 \, d e^{2} - 148 \, e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (111 \, d^{3} - 111 \, d^{2} e + 444 \, d e^{2} + 65 \, e^{3}\right )} x^{7} - \frac {1}{6} \, {\left (37 \, d^{3} - 444 \, d^{2} e - 195 \, d e^{2} - 107 \, e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d^{3} + 195 \, d^{2} e + 321 \, d e^{2} + 33 \, e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d^{3} + 321 \, d^{2} e + 99 \, d e^{2} + 18 \, e^{3}\right )} x^{4} + 18 \, d^{3} x + \frac {1}{3} \, {\left (107 \, d^{3} + 99 \, d^{2} e + 54 \, d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (11 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 251, normalized size = 0.64 \[ 18\,d^3\,x+x^3\,\left (\frac {107\,d^3}{3}+33\,d^2\,e+18\,d\,e^2\right )+x^9\,\left (\frac {100\,d^3}{9}-15\,d^2\,e+37\,d\,e^2-\frac {37\,e^3}{9}\right )+x^6\,\left (-\frac {37\,d^3}{6}+74\,d^2\,e+\frac {65\,d\,e^2}{2}+\frac {107\,e^3}{6}\right )+x^4\,\left (\frac {65\,d^3}{4}+\frac {321\,d^2\,e}{4}+\frac {99\,d\,e^2}{4}+\frac {9\,e^3}{2}\right )-x^8\,\left (\frac {45\,d^3}{8}-\frac {333\,d^2\,e}{8}+\frac {111\,d\,e^2}{8}-\frac {37\,e^3}{2}\right )+x^5\,\left (\frac {148\,d^3}{5}+39\,d^2\,e+\frac {321\,d\,e^2}{5}+\frac {33\,e^3}{5}\right )+x^7\,\left (\frac {111\,d^3}{7}-\frac {111\,d^2\,e}{7}+\frac {444\,d\,e^2}{7}+\frac {65\,e^3}{7}\right )+\frac {25\,e^3\,x^{12}}{3}+\frac {3\,e\,x^{10}\,\left (100\,d^2-45\,d\,e+37\,e^2\right )}{10}+\frac {3\,d^2\,x^2\,\left (11\,d+18\,e\right )}{2}+\frac {15\,e^2\,x^{11}\,\left (20\,d-3\,e\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 298, normalized size = 0.76 \[ 18 d^{3} x + \frac {25 e^{3} x^{12}}{3} + x^{11} \left (\frac {300 d e^{2}}{11} - \frac {45 e^{3}}{11}\right ) + x^{10} \left (30 d^{2} e - \frac {27 d e^{2}}{2} + \frac {111 e^{3}}{10}\right ) + x^{9} \left (\frac {100 d^{3}}{9} - 15 d^{2} e + 37 d e^{2} - \frac {37 e^{3}}{9}\right ) + x^{8} \left (- \frac {45 d^{3}}{8} + \frac {333 d^{2} e}{8} - \frac {111 d e^{2}}{8} + \frac {37 e^{3}}{2}\right ) + x^{7} \left (\frac {111 d^{3}}{7} - \frac {111 d^{2} e}{7} + \frac {444 d e^{2}}{7} + \frac {65 e^{3}}{7}\right ) + x^{6} \left (- \frac {37 d^{3}}{6} + 74 d^{2} e + \frac {65 d e^{2}}{2} + \frac {107 e^{3}}{6}\right ) + x^{5} \left (\frac {148 d^{3}}{5} + 39 d^{2} e + \frac {321 d e^{2}}{5} + \frac {33 e^{3}}{5}\right ) + x^{4} \left (\frac {65 d^{3}}{4} + \frac {321 d^{2} e}{4} + \frac {99 d e^{2}}{4} + \frac {9 e^{3}}{2}\right ) + x^{3} \left (\frac {107 d^{3}}{3} + 33 d^{2} e + 18 d e^{2}\right ) + x^{2} \left (\frac {33 d^{3}}{2} + 27 d^{2} e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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