Optimal. Leaf size=157 \[ \frac {1}{7} x^7 \left (20 d^2-34 d e+17 e^2\right )-\frac {1}{6} x^6 \left (17 d^2-34 d e+4 e^2\right )+\frac {1}{5} x^5 \left (17 d^2-8 d e+21 e^2\right )-\frac {1}{4} x^4 \left (4 d^2-42 d e-7 e^2\right )+\frac {1}{3} x^3 \left (21 d^2+14 d e+6 e^2\right )+6 d^2 x+\frac {1}{8} e x^8 (40 d-17 e)+\frac {1}{2} d x^2 (7 d+12 e)+\frac {20 e^2 x^9}{9} \]
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Rubi [A] time = 0.17, antiderivative size = 157, 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 = {1628} \[ \frac {1}{7} x^7 \left (20 d^2-34 d e+17 e^2\right )-\frac {1}{6} x^6 \left (17 d^2-34 d e+4 e^2\right )+\frac {1}{5} x^5 \left (17 d^2-8 d e+21 e^2\right )-\frac {1}{4} x^4 \left (4 d^2-42 d e-7 e^2\right )+\frac {1}{3} x^3 \left (21 d^2+14 d e+6 e^2\right )+6 d^2 x+\frac {1}{8} e x^8 (40 d-17 e)+\frac {1}{2} d x^2 (7 d+12 e)+\frac {20 e^2 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int (d+e x)^2 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (6 d^2+d (7 d+12 e) x+\left (21 d^2+14 d e+6 e^2\right ) x^2-\left (4 d^2-42 d e-7 e^2\right ) x^3+\left (17 d^2-8 d e+21 e^2\right ) x^4-\left (17 d^2-34 d e+4 e^2\right ) x^5+\left (20 d^2-34 d e+17 e^2\right ) x^6+(40 d-17 e) e x^7+20 e^2 x^8\right ) \, dx\\ &=6 d^2 x+\frac {1}{2} d (7 d+12 e) x^2+\frac {1}{3} \left (21 d^2+14 d e+6 e^2\right ) x^3-\frac {1}{4} \left (4 d^2-42 d e-7 e^2\right ) x^4+\frac {1}{5} \left (17 d^2-8 d e+21 e^2\right ) x^5-\frac {1}{6} \left (17 d^2-34 d e+4 e^2\right ) x^6+\frac {1}{7} \left (20 d^2-34 d e+17 e^2\right ) x^7+\frac {1}{8} (40 d-17 e) e x^8+\frac {20 e^2 x^9}{9}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 136, normalized size = 0.87 \[ d^2 \left (\frac {20 x^7}{7}-\frac {17 x^6}{6}+\frac {17 x^5}{5}-x^4+7 x^3+\frac {7 x^2}{2}+6 x\right )+d e \left (5 x^8-\frac {34 x^7}{7}+\frac {17 x^6}{3}-\frac {8 x^5}{5}+\frac {21 x^4}{2}+\frac {14 x^3}{3}+6 x^2\right )+\frac {e^2 \left (5600 x^6-5355 x^5+6120 x^4-1680 x^3+10584 x^2+4410 x+5040\right ) x^3}{2520} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 160, normalized size = 1.02 \[ \frac {20}{9} x^{9} e^{2} - \frac {17}{8} x^{8} e^{2} + 5 x^{8} e d + \frac {17}{7} x^{7} e^{2} - \frac {34}{7} x^{7} e d + \frac {20}{7} x^{7} d^{2} - \frac {2}{3} x^{6} e^{2} + \frac {17}{3} x^{6} e d - \frac {17}{6} x^{6} d^{2} + \frac {21}{5} x^{5} e^{2} - \frac {8}{5} x^{5} e d + \frac {17}{5} x^{5} d^{2} + \frac {7}{4} x^{4} e^{2} + \frac {21}{2} x^{4} e d - x^{4} d^{2} + 2 x^{3} e^{2} + \frac {14}{3} x^{3} e d + 7 x^{3} d^{2} + 6 x^{2} e d + \frac {7}{2} x^{2} d^{2} + 6 x d^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 160, normalized size = 1.02 \[ \frac {20}{9} \, x^{9} e^{2} + 5 \, d x^{8} e + \frac {20}{7} \, d^{2} x^{7} - \frac {17}{8} \, x^{8} e^{2} - \frac {34}{7} \, d x^{7} e - \frac {17}{6} \, d^{2} x^{6} + \frac {17}{7} \, x^{7} e^{2} + \frac {17}{3} \, d x^{6} e + \frac {17}{5} \, d^{2} x^{5} - \frac {2}{3} \, x^{6} e^{2} - \frac {8}{5} \, d x^{5} e - d^{2} x^{4} + \frac {21}{5} \, x^{5} e^{2} + \frac {21}{2} \, d x^{4} e + 7 \, d^{2} x^{3} + \frac {7}{4} \, x^{4} e^{2} + \frac {14}{3} \, d x^{3} e + \frac {7}{2} \, d^{2} x^{2} + 2 \, x^{3} e^{2} + 6 \, d x^{2} e + 6 \, d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 146, normalized size = 0.93 \[ \frac {20 e^{2} x^{9}}{9}+\frac {\left (40 d e -17 e^{2}\right ) x^{8}}{8}+\frac {\left (20 d^{2}-34 d e +17 e^{2}\right ) x^{7}}{7}+\frac {\left (-17 d^{2}+34 d e -4 e^{2}\right ) x^{6}}{6}+\frac {\left (17 d^{2}-8 d e +21 e^{2}\right ) x^{5}}{5}+\frac {\left (-4 d^{2}+42 d e +7 e^{2}\right ) x^{4}}{4}+6 d^{2} x +\frac {\left (21 d^{2}+14 d e +6 e^{2}\right ) x^{3}}{3}+\frac {\left (7 d^{2}+12 d e \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 0.92 \[ \frac {20}{9} \, e^{2} x^{9} + \frac {1}{8} \, {\left (40 \, d e - 17 \, e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{2} - 34 \, d e + 17 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{2} - 34 \, d e + 4 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{2} - 8 \, d e + 21 \, e^{2}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{2} - 42 \, d e - 7 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, d^{2} + 14 \, d e + 6 \, e^{2}\right )} x^{3} + 6 \, d^{2} x + \frac {1}{2} \, {\left (7 \, d^{2} + 12 \, d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.11, size = 137, normalized size = 0.87 \[ x^3\,\left (7\,d^2+\frac {14\,d\,e}{3}+2\,e^2\right )+x^4\,\left (-d^2+\frac {21\,d\,e}{2}+\frac {7\,e^2}{4}\right )-x^6\,\left (\frac {17\,d^2}{6}-\frac {17\,d\,e}{3}+\frac {2\,e^2}{3}\right )+x^5\,\left (\frac {17\,d^2}{5}-\frac {8\,d\,e}{5}+\frac {21\,e^2}{5}\right )+x^7\,\left (\frac {20\,d^2}{7}-\frac {34\,d\,e}{7}+\frac {17\,e^2}{7}\right )+6\,d^2\,x+\frac {20\,e^2\,x^9}{9}+\frac {d\,x^2\,\left (7\,d+12\,e\right )}{2}+\frac {e\,x^8\,\left (40\,d-17\,e\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 158, normalized size = 1.01 \[ 6 d^{2} x + \frac {20 e^{2} x^{9}}{9} + x^{8} \left (5 d e - \frac {17 e^{2}}{8}\right ) + x^{7} \left (\frac {20 d^{2}}{7} - \frac {34 d e}{7} + \frac {17 e^{2}}{7}\right ) + x^{6} \left (- \frac {17 d^{2}}{6} + \frac {17 d e}{3} - \frac {2 e^{2}}{3}\right ) + x^{5} \left (\frac {17 d^{2}}{5} - \frac {8 d e}{5} + \frac {21 e^{2}}{5}\right ) + x^{4} \left (- d^{2} + \frac {21 d e}{2} + \frac {7 e^{2}}{4}\right ) + x^{3} \left (7 d^{2} + \frac {14 d e}{3} + 2 e^{2}\right ) + x^{2} \left (\frac {7 d^{2}}{2} + 6 d e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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