Optimal. Leaf size=157 \[ \frac{1}{7} x^7 \left (20 d^2-34 d e+17 e^2\right )-\frac{1}{6} x^6 \left (17 d^2-34 d e+4 e^2\right )+\frac{1}{5} x^5 \left (17 d^2-8 d e+21 e^2\right )-\frac{1}{4} x^4 \left (4 d^2-42 d e-7 e^2\right )+\frac{1}{3} x^3 \left (21 d^2+14 d e+6 e^2\right )+6 d^2 x+\frac{1}{8} e x^8 (40 d-17 e)+\frac{1}{2} d x^2 (7 d+12 e)+\frac{20 e^2 x^9}{9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.166052, antiderivative size = 157, normalized size of antiderivative = 1., 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.
[In]
[Out]
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*}
Mathematica [A] time = 0.0332222, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 146, normalized size = 0.9 \begin{align*}{\frac{20\,{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( 40\,de-17\,{e}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,{d}^{2}-34\,de+17\,{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( -17\,{d}^{2}+34\,de-4\,{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,{d}^{2}-8\,de+21\,{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( -4\,{d}^{2}+42\,de+7\,{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{d}^{2}+14\,de+6\,{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{d}^{2}+12\,de \right ){x}^{2}}{2}}+6\,{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.95401, size = 196, normalized size = 1.25 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.840011, size = 389, normalized size = 2.48 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.094552, size = 158, normalized size = 1.01 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12306, size = 216, normalized size = 1.38 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]