Optimal. Leaf size=93 \[ \frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)-\frac{1}{4} x^4 (4 d-21 e)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2} \]
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Rubi [A] time = 0.107599, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1628} \[ \frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)-\frac{1}{4} x^4 (4 d-21 e)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int (d+e x) \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+(7 d+6 e) x+7 (3 d+e) x^2-(4 d-21 e) x^3+(17 d-4 e) x^4-17 (d-e) x^5+(20 d-17 e) x^6+20 e x^7\right ) \, dx\\ &=6 d x+\frac{1}{2} (7 d+6 e) x^2+\frac{7}{3} (3 d+e) x^3-\frac{1}{4} (4 d-21 e) x^4+\frac{1}{5} (17 d-4 e) x^5-\frac{17}{6} (d-e) x^6+\frac{1}{7} (20 d-17 e) x^7+\frac{5 e x^8}{2}\\ \end{align*}
Mathematica [A] time = 0.0144416, size = 93, normalized size = 1. \[ \frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)+\frac{1}{4} x^4 (21 e-4 d)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 84, normalized size = 0.9 \begin{align*}{\frac{5\,e{x}^{8}}{2}}+{\frac{ \left ( 20\,d-17\,e \right ){x}^{7}}{7}}+{\frac{ \left ( -17\,d+17\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,d-4\,e \right ){x}^{5}}{5}}+{\frac{ \left ( -4\,d+21\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,d+7\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,d+6\,e \right ){x}^{2}}{2}}+6\,dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991202, size = 107, normalized size = 1.15 \begin{align*} \frac{5}{2} \, e x^{8} + \frac{1}{7} \,{\left (20 \, d - 17 \, e\right )} x^{7} - \frac{17}{6} \,{\left (d - e\right )} x^{6} + \frac{1}{5} \,{\left (17 \, d - 4 \, e\right )} x^{5} - \frac{1}{4} \,{\left (4 \, d - 21 \, e\right )} x^{4} + \frac{7}{3} \,{\left (3 \, d + e\right )} x^{3} + \frac{1}{2} \,{\left (7 \, d + 6 \, e\right )} x^{2} + 6 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.839921, size = 217, normalized size = 2.33 \begin{align*} \frac{5}{2} x^{8} e - \frac{17}{7} x^{7} e + \frac{20}{7} x^{7} d + \frac{17}{6} x^{6} e - \frac{17}{6} x^{6} d - \frac{4}{5} x^{5} e + \frac{17}{5} x^{5} d + \frac{21}{4} x^{4} e - x^{4} d + \frac{7}{3} x^{3} e + 7 x^{3} d + 3 x^{2} e + \frac{7}{2} x^{2} d + 6 x d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083282, size = 87, normalized size = 0.94 \begin{align*} 6 d x + \frac{5 e x^{8}}{2} + x^{7} \left (\frac{20 d}{7} - \frac{17 e}{7}\right ) + x^{6} \left (- \frac{17 d}{6} + \frac{17 e}{6}\right ) + x^{5} \left (\frac{17 d}{5} - \frac{4 e}{5}\right ) + x^{4} \left (- d + \frac{21 e}{4}\right ) + x^{3} \left (7 d + \frac{7 e}{3}\right ) + x^{2} \left (\frac{7 d}{2} + 3 e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14264, size = 122, normalized size = 1.31 \begin{align*} \frac{5}{2} \, x^{8} e + \frac{20}{7} \, d x^{7} - \frac{17}{7} \, x^{7} e - \frac{17}{6} \, d x^{6} + \frac{17}{6} \, x^{6} e + \frac{17}{5} \, d x^{5} - \frac{4}{5} \, x^{5} e - d x^{4} + \frac{21}{4} \, x^{4} e + 7 \, d x^{3} + \frac{7}{3} \, x^{3} e + \frac{7}{2} \, d x^{2} + 3 \, x^{2} e + 6 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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