Optimal. Leaf size=201 \[ 18 d^2 x+\frac {3}{2} d (11 d+12 e) x^2+\frac {1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac {1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac {1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5-\frac {1}{6} \left (37 d^2-296 d e-65 e^2\right ) x^6+\frac {37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7-\frac {1}{8} \left (45 d^2-222 d e+37 e^2\right ) x^8+\frac {1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac {1}{2} (40 d-9 e) e x^{10}+\frac {100 e^2 x^{11}}{11} \]
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Rubi [A]
time = 0.16, antiderivative size = 201, 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 = {1642}
\begin {gather*} \frac {1}{9} x^9 \left (100 d^2-90 d e+111 e^2\right )-\frac {1}{8} x^8 \left (45 d^2-222 d e+37 e^2\right )+\frac {37}{7} x^7 \left (3 d^2-2 d e+4 e^2\right )-\frac {1}{6} x^6 \left (37 d^2-296 d e-65 e^2\right )+\frac {1}{5} x^5 \left (148 d^2+130 d e+107 e^2\right )+\frac {1}{4} x^4 \left (65 d^2+214 d e+33 e^2\right )+\frac {1}{3} x^3 \left (107 d^2+66 d e+18 e^2\right )+18 d^2 x+\frac {1}{2} e x^{10} (40 d-9 e)+\frac {3}{2} d x^2 (11 d+12 e)+\frac {100 e^2 x^{11}}{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int (d+e x)^2 \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 (18 d^2+3 d (11 d+12 e) x+\left (107 d^2+66 d e+18 e^2\right ) x^2+\left (65 d^2+214 d e+33 e^2\right ) x^3+\left (148 d^2+130 d e+107 e^2\right ) x^4-\left (37 d^2-296 d e-65 e^2\right ) x^5+37 \left (3 d^2-2 d e+4 e^2\right ) x^6-\left (45 d^2-222 d e+37 e^2\right ) x^7+\left (100 d^2-90 d e+111 e^2\right ) x^8+5 (40 d-9 e) e x^9+100 e^2 x^{10}\right ) \, dx\\ &=18 d^2 x+\frac {3}{2} d (11 d+12 e) x^2+\frac {1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac {1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac {1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5-\frac {1}{6} \left (37 d^2-296 d e-65 e^2\right ) x^6+\frac {37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7-\frac {1}{8} \left (45 d^2-222 d e+37 e^2\right ) x^8+\frac {1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac {1}{2} (40 d-9 e) e x^{10}+\frac {100 e^2 x^{11}}{11}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 201, normalized size = 1.00 \begin {gather*} 18 d^2 x+\frac {3}{2} d (11 d+12 e) x^2+\frac {1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac {1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac {1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5+\frac {1}{6} \left (-37 d^2+296 d e+65 e^2\right ) x^6+\frac {37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7+\frac {1}{8} \left (-45 d^2+222 d e-37 e^2\right ) x^8+\frac {1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac {1}{2} (40 d-9 e) e x^{10}+\frac {100 e^2 x^{11}}{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 186, normalized size = 0.93
method | result | size |
norman | \(\frac {100 e^{2} x^{11}}{11}+\left (20 d e -\frac {9}{2} e^{2}\right ) x^{10}+\left (\frac {100}{9} d^{2}-10 d e +\frac {37}{3} e^{2}\right ) x^{9}+\left (-\frac {45}{8} d^{2}+\frac {111}{4} d e -\frac {37}{8} e^{2}\right ) x^{8}+\left (\frac {111}{7} d^{2}-\frac {74}{7} d e +\frac {148}{7} e^{2}\right ) x^{7}+\left (-\frac {37}{6} d^{2}+\frac {148}{3} d e +\frac {65}{6} e^{2}\right ) x^{6}+\left (\frac {148}{5} d^{2}+26 d e +\frac {107}{5} e^{2}\right ) x^{5}+\left (\frac {65}{4} d^{2}+\frac {107}{2} d e +\frac {33}{4} e^{2}\right ) x^{4}+\left (\frac {107}{3} d^{2}+22 d e +6 e^{2}\right ) x^{3}+\left (\frac {33}{2} d^{2}+18 d e \right ) x^{2}+18 d^{2} x\) | \(177\) |
default | \(\frac {100 e^{2} x^{11}}{11}+\frac {\left (200 d e -45 e^{2}\right ) x^{10}}{10}+\frac {\left (100 d^{2}-90 d e +111 e^{2}\right ) x^{9}}{9}+\frac {\left (-45 d^{2}+222 d e -37 e^{2}\right ) x^{8}}{8}+\frac {\left (111 d^{2}-74 d e +148 e^{2}\right ) x^{7}}{7}+\frac {\left (-37 d^{2}+296 d e +65 e^{2}\right ) x^{6}}{6}+\frac {\left (148 d^{2}+130 d e +107 e^{2}\right ) x^{5}}{5}+\frac {\left (65 d^{2}+214 d e +33 e^{2}\right ) x^{4}}{4}+\frac {\left (107 d^{2}+66 d e +18 e^{2}\right ) x^{3}}{3}+\frac {\left (33 d^{2}+36 d e \right ) x^{2}}{2}+18 d^{2} x\) | \(186\) |
gosper | \(\frac {65}{4} x^{4} d^{2}+\frac {33}{4} x^{4} e^{2}+\frac {107}{3} x^{3} d^{2}-\frac {37}{6} x^{6} d^{2}+\frac {65}{6} x^{6} e^{2}+\frac {148}{5} x^{5} d^{2}+\frac {111}{7} x^{7} d^{2}+\frac {148}{7} x^{7} e^{2}-\frac {37}{8} x^{8} e^{2}+\frac {33}{2} d^{2} x^{2}+22 x^{3} d e +\frac {107}{2} x^{4} d e +\frac {148}{3} x^{6} d e +26 x^{5} d e +\frac {111}{4} x^{8} d e -\frac {74}{7} x^{7} d e +6 e^{2} x^{3}-\frac {9}{2} x^{10} e^{2}+\frac {100}{9} x^{9} d^{2}-\frac {45}{8} x^{8} d^{2}+18 d^{2} x -10 x^{9} d e +20 x^{10} d e +\frac {100}{11} e^{2} x^{11}+18 d e \,x^{2}+\frac {107}{5} e^{2} x^{5}+\frac {37}{3} e^{2} x^{9}\) | \(207\) |
risch | \(\frac {65}{4} x^{4} d^{2}+\frac {33}{4} x^{4} e^{2}+\frac {107}{3} x^{3} d^{2}-\frac {37}{6} x^{6} d^{2}+\frac {65}{6} x^{6} e^{2}+\frac {148}{5} x^{5} d^{2}+\frac {111}{7} x^{7} d^{2}+\frac {148}{7} x^{7} e^{2}-\frac {37}{8} x^{8} e^{2}+\frac {33}{2} d^{2} x^{2}+22 x^{3} d e +\frac {107}{2} x^{4} d e +\frac {148}{3} x^{6} d e +26 x^{5} d e +\frac {111}{4} x^{8} d e -\frac {74}{7} x^{7} d e +6 e^{2} x^{3}-\frac {9}{2} x^{10} e^{2}+\frac {100}{9} x^{9} d^{2}-\frac {45}{8} x^{8} d^{2}+18 d^{2} x -10 x^{9} d e +20 x^{10} d e +\frac {100}{11} e^{2} x^{11}+18 d e \,x^{2}+\frac {107}{5} e^{2} x^{5}+\frac {37}{3} e^{2} x^{9}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 185, normalized size = 0.92 \begin {gather*} \frac {100}{11} \, x^{11} e^{2} + \frac {1}{2} \, {\left (40 \, d e - 9 \, e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (100 \, d^{2} - 90 \, d e + 111 \, e^{2}\right )} x^{9} - \frac {1}{8} \, {\left (45 \, d^{2} - 222 \, d e + 37 \, e^{2}\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d^{2} - 2 \, d e + 4 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (37 \, d^{2} - 296 \, d e - 65 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d^{2} + 130 \, d e + 107 \, e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d^{2} + 214 \, d e + 33 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d^{2} + 66 \, d e + 18 \, e^{2}\right )} x^{3} + 18 \, d^{2} x + \frac {3}{2} \, {\left (11 \, d^{2} + 12 \, d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 180, normalized size = 0.90 \begin {gather*} \frac {100}{9} \, d^{2} x^{9} - \frac {45}{8} \, d^{2} x^{8} + \frac {111}{7} \, d^{2} x^{7} - \frac {37}{6} \, d^{2} x^{6} + \frac {148}{5} \, d^{2} x^{5} + \frac {65}{4} \, d^{2} x^{4} + \frac {107}{3} \, d^{2} x^{3} + \frac {33}{2} \, d^{2} x^{2} + 18 \, d^{2} x + \frac {1}{9240} \, {\left (84000 \, x^{11} - 41580 \, x^{10} + 113960 \, x^{9} - 42735 \, x^{8} + 195360 \, x^{7} + 100100 \, x^{6} + 197736 \, x^{5} + 76230 \, x^{4} + 55440 \, x^{3}\right )} e^{2} + \frac {1}{84} \, {\left (1680 \, d x^{10} - 840 \, d x^{9} + 2331 \, d x^{8} - 888 \, d x^{7} + 4144 \, d x^{6} + 2184 \, d x^{5} + 4494 \, d x^{4} + 1848 \, d x^{3} + 1512 \, d x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 206, normalized size = 1.02 \begin {gather*} 18 d^{2} x + \frac {100 e^{2} x^{11}}{11} + x^{10} \cdot \left (20 d e - \frac {9 e^{2}}{2}\right ) + x^{9} \cdot \left (\frac {100 d^{2}}{9} - 10 d e + \frac {37 e^{2}}{3}\right ) + x^{8} \left (- \frac {45 d^{2}}{8} + \frac {111 d e}{4} - \frac {37 e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {111 d^{2}}{7} - \frac {74 d e}{7} + \frac {148 e^{2}}{7}\right ) + x^{6} \left (- \frac {37 d^{2}}{6} + \frac {148 d e}{3} + \frac {65 e^{2}}{6}\right ) + x^{5} \cdot \left (\frac {148 d^{2}}{5} + 26 d e + \frac {107 e^{2}}{5}\right ) + x^{4} \cdot \left (\frac {65 d^{2}}{4} + \frac {107 d e}{2} + \frac {33 e^{2}}{4}\right ) + x^{3} \cdot \left (\frac {107 d^{2}}{3} + 22 d e + 6 e^{2}\right ) + x^{2} \cdot \left (\frac {33 d^{2}}{2} + 18 d e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.19, size = 206, normalized size = 1.02 \begin {gather*} \frac {100}{11} \, x^{11} e^{2} + 20 \, d x^{10} e + \frac {100}{9} \, d^{2} x^{9} - \frac {9}{2} \, x^{10} e^{2} - 10 \, d x^{9} e - \frac {45}{8} \, d^{2} x^{8} + \frac {37}{3} \, x^{9} e^{2} + \frac {111}{4} \, d x^{8} e + \frac {111}{7} \, d^{2} x^{7} - \frac {37}{8} \, x^{8} e^{2} - \frac {74}{7} \, d x^{7} e - \frac {37}{6} \, d^{2} x^{6} + \frac {148}{7} \, x^{7} e^{2} + \frac {148}{3} \, d x^{6} e + \frac {148}{5} \, d^{2} x^{5} + \frac {65}{6} \, x^{6} e^{2} + 26 \, d x^{5} e + \frac {65}{4} \, d^{2} x^{4} + \frac {107}{5} \, x^{5} e^{2} + \frac {107}{2} \, d x^{4} e + \frac {107}{3} \, d^{2} x^{3} + \frac {33}{4} \, x^{4} e^{2} + 22 \, d x^{3} e + \frac {33}{2} \, d^{2} x^{2} + 6 \, x^{3} e^{2} + 18 \, d x^{2} e + 18 \, d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 175, normalized size = 0.87 \begin {gather*} x^3\,\left (\frac {107\,d^2}{3}+22\,d\,e+6\,e^2\right )+x^9\,\left (\frac {100\,d^2}{9}-10\,d\,e+\frac {37\,e^2}{3}\right )+x^4\,\left (\frac {65\,d^2}{4}+\frac {107\,d\,e}{2}+\frac {33\,e^2}{4}\right )-x^8\,\left (\frac {45\,d^2}{8}-\frac {111\,d\,e}{4}+\frac {37\,e^2}{8}\right )+x^6\,\left (-\frac {37\,d^2}{6}+\frac {148\,d\,e}{3}+\frac {65\,e^2}{6}\right )+x^5\,\left (\frac {148\,d^2}{5}+26\,d\,e+\frac {107\,e^2}{5}\right )+x^7\,\left (\frac {111\,d^2}{7}-\frac {74\,d\,e}{7}+\frac {148\,e^2}{7}\right )+18\,d^2\,x+\frac {100\,e^2\,x^{11}}{11}+\frac {3\,d\,x^2\,\left (11\,d+12\,e\right )}{2}+\frac {e\,x^{10}\,\left (40\,d-9\,e\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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