Optimal. Leaf size=121 \[ 18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \]
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Rubi [A]
time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1642}
\begin {gather*} \frac {5}{9} x^9 (20 d-9 e)-\frac {3}{8} x^8 (15 d-37 e)+\frac {37}{7} x^7 (3 d-e)-\frac {37}{6} x^6 (d-4 e)+\frac {1}{5} x^5 (148 d+65 e)+\frac {1}{4} x^4 (65 d+107 e)+\frac {1}{3} x^3 (107 d+33 e)+\frac {3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int (d+e x) \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+3 (11 d+6 e) x+(107 d+33 e) x^2+(65 d+107 e) x^3+(148 d+65 e) x^4-37 (d-4 e) x^5+37 (3 d-e) x^6-3 (15 d-37 e) x^7+5 (20 d-9 e) x^8+100 e x^9\right ) \, dx\\ &=18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 121, normalized size = 1.00 \begin {gather*} 18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 108, normalized size = 0.89
method | result | size |
norman | \(10 e \,x^{10}+\left (\frac {100 d}{9}-5 e \right ) x^{9}+\left (-\frac {45 d}{8}+\frac {111 e}{8}\right ) x^{8}+\left (\frac {111 d}{7}-\frac {37 e}{7}\right ) x^{7}+\left (-\frac {37 d}{6}+\frac {74 e}{3}\right ) x^{6}+\left (\frac {148 d}{5}+13 e \right ) x^{5}+\left (\frac {65 d}{4}+\frac {107 e}{4}\right ) x^{4}+\left (\frac {107 d}{3}+11 e \right ) x^{3}+\left (\frac {33 d}{2}+9 e \right ) x^{2}+18 d x\) | \(100\) |
gosper | \(10 e \,x^{10}+\frac {100}{9} x^{9} d -5 x^{9} e -\frac {45}{8} x^{8} d +\frac {111}{8} e \,x^{8}+\frac {111}{7} x^{7} d -\frac {37}{7} x^{7} e -\frac {37}{6} x^{6} d +\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\) | \(108\) |
default | \(10 e \,x^{10}+\frac {\left (100 d -45 e \right ) x^{9}}{9}+\frac {\left (-45 d +111 e \right ) x^{8}}{8}+\frac {\left (111 d -37 e \right ) x^{7}}{7}+\frac {\left (-37 d +148 e \right ) x^{6}}{6}+\frac {\left (148 d +65 e \right ) x^{5}}{5}+\frac {\left (65 d +107 e \right ) x^{4}}{4}+\frac {\left (107 d +33 e \right ) x^{3}}{3}+\frac {\left (33 d +18 e \right ) x^{2}}{2}+18 d x\) | \(108\) |
risch | \(10 e \,x^{10}+\frac {100}{9} x^{9} d -5 x^{9} e -\frac {45}{8} x^{8} d +\frac {111}{8} e \,x^{8}+\frac {111}{7} x^{7} d -\frac {37}{7} x^{7} e -\frac {37}{6} x^{6} d +\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 114, normalized size = 0.94 \begin {gather*} 10 \, x^{10} e + \frac {5}{9} \, {\left (20 \, d - 9 \, e\right )} x^{9} - \frac {3}{8} \, {\left (15 \, d - 37 \, e\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d - e\right )} x^{7} - \frac {37}{6} \, {\left (d - 4 \, e\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d + 65 \, e\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d + 107 \, e\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d + 33 \, e\right )} x^{3} + \frac {3}{2} \, {\left (11 \, d + 6 \, e\right )} x^{2} + 18 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 103, normalized size = 0.85 \begin {gather*} \frac {100}{9} \, d x^{9} - \frac {45}{8} \, d x^{8} + \frac {111}{7} \, d x^{7} - \frac {37}{6} \, d x^{6} + \frac {148}{5} \, d x^{5} + \frac {65}{4} \, d x^{4} + \frac {107}{3} \, d x^{3} + \frac {33}{2} \, d x^{2} + 18 \, d x + \frac {1}{168} \, {\left (1680 \, x^{10} - 840 \, x^{9} + 2331 \, x^{8} - 888 \, x^{7} + 4144 \, x^{6} + 2184 \, x^{5} + 4494 \, x^{4} + 1848 \, x^{3} + 1512 \, 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 = 112, normalized size = 0.93 \begin {gather*} 18 d x + 10 e x^{10} + x^{9} \cdot \left (\frac {100 d}{9} - 5 e\right ) + x^{8} \left (- \frac {45 d}{8} + \frac {111 e}{8}\right ) + x^{7} \cdot \left (\frac {111 d}{7} - \frac {37 e}{7}\right ) + x^{6} \left (- \frac {37 d}{6} + \frac {74 e}{3}\right ) + x^{5} \cdot \left (\frac {148 d}{5} + 13 e\right ) + x^{4} \cdot \left (\frac {65 d}{4} + \frac {107 e}{4}\right ) + x^{3} \cdot \left (\frac {107 d}{3} + 11 e\right ) + x^{2} \cdot \left (\frac {33 d}{2} + 9 e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.96, size = 116, normalized size = 0.96 \begin {gather*} 10 \, x^{10} e + \frac {100}{9} \, d x^{9} - 5 \, x^{9} e - \frac {45}{8} \, d x^{8} + \frac {111}{8} \, x^{8} e + \frac {111}{7} \, d x^{7} - \frac {37}{7} \, x^{7} e - \frac {37}{6} \, d x^{6} + \frac {74}{3} \, x^{6} e + \frac {148}{5} \, d x^{5} + 13 \, x^{5} e + \frac {65}{4} \, d x^{4} + \frac {107}{4} \, x^{4} e + \frac {107}{3} \, d x^{3} + 11 \, x^{3} e + \frac {33}{2} \, d x^{2} + 9 \, x^{2} e + 18 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.17, size = 101, normalized size = 0.83 \begin {gather*} 10\,e\,x^{10}+\left (\frac {100\,d}{9}-5\,e\right )\,x^9+\left (\frac {111\,e}{8}-\frac {45\,d}{8}\right )\,x^8+\left (\frac {111\,d}{7}-\frac {37\,e}{7}\right )\,x^7+\left (\frac {74\,e}{3}-\frac {37\,d}{6}\right )\,x^6+\left (\frac {148\,d}{5}+13\,e\right )\,x^5+\left (\frac {65\,d}{4}+\frac {107\,e}{4}\right )\,x^4+\left (\frac {107\,d}{3}+11\,e\right )\,x^3+\left (\frac {33\,d}{2}+9\,e\right )\,x^2+18\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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