Optimal. Leaf size=103 \[ \frac{(d+e x)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac{d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac{c (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.125834, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ \frac{(d+e x)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac{d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac{c (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int x (d+e x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac{d \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^3}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^5}{e^3}+\frac{(-3 c d+b e) (d+e x)^6}{e^3}+\frac{c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{d \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^4}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^6}{6 e^4}-\frac{(3 c d-b e) (d+e x)^7}{7 e^4}+\frac{c (d+e x)^8}{8 e^4}\\ \end{align*}
Mathematica [A] time = 0.0358542, size = 140, normalized size = 1.36 \[ \frac{1}{6} e^2 x^6 \left (a e^2+4 b d e+6 c d^2\right )+\frac{2}{5} d e x^5 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{4} d^2 x^4 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{3} d^3 x^3 (4 a e+b d)+\frac{1}{2} a d^4 x^2+\frac{1}{7} e^3 x^7 (b e+4 c d)+\frac{1}{8} c e^4 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 139, normalized size = 1.4 \begin{align*}{\frac{{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{d}^{2}{e}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,d{e}^{3}a+6\,{d}^{2}{e}^{2}b+4\,{d}^{3}ec \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+4\,{d}^{3}eb+{d}^{4}c \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,{d}^{3}ea+{d}^{4}b \right ){x}^{3}}{3}}+{\frac{{d}^{4}a{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08931, size = 186, normalized size = 1.81 \begin{align*} \frac{1}{8} \, c e^{4} x^{8} + \frac{1}{7} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{7} + \frac{1}{2} \, a d^{4} x^{2} + \frac{1}{6} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15082, size = 344, normalized size = 3.34 \begin{align*} \frac{1}{8} x^{8} e^{4} c + \frac{4}{7} x^{7} e^{3} d c + \frac{1}{7} x^{7} e^{4} b + x^{6} e^{2} d^{2} c + \frac{2}{3} x^{6} e^{3} d b + \frac{1}{6} x^{6} e^{4} a + \frac{4}{5} x^{5} e d^{3} c + \frac{6}{5} x^{5} e^{2} d^{2} b + \frac{4}{5} x^{5} e^{3} d a + \frac{1}{4} x^{4} d^{4} c + x^{4} e d^{3} b + \frac{3}{2} x^{4} e^{2} d^{2} a + \frac{1}{3} x^{3} d^{4} b + \frac{4}{3} x^{3} e d^{3} a + \frac{1}{2} x^{2} d^{4} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.085898, size = 153, normalized size = 1.49 \begin{align*} \frac{a d^{4} x^{2}}{2} + \frac{c e^{4} x^{8}}{8} + x^{7} \left (\frac{b e^{4}}{7} + \frac{4 c d e^{3}}{7}\right ) + x^{6} \left (\frac{a e^{4}}{6} + \frac{2 b d e^{3}}{3} + c d^{2} e^{2}\right ) + x^{5} \left (\frac{4 a d e^{3}}{5} + \frac{6 b d^{2} e^{2}}{5} + \frac{4 c d^{3} e}{5}\right ) + x^{4} \left (\frac{3 a d^{2} e^{2}}{2} + b d^{3} e + \frac{c d^{4}}{4}\right ) + x^{3} \left (\frac{4 a d^{3} e}{3} + \frac{b d^{4}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09192, size = 193, normalized size = 1.87 \begin{align*} \frac{1}{8} \, c x^{8} e^{4} + \frac{4}{7} \, c d x^{7} e^{3} + c d^{2} x^{6} e^{2} + \frac{4}{5} \, c d^{3} x^{5} e + \frac{1}{4} \, c d^{4} x^{4} + \frac{1}{7} \, b x^{7} e^{4} + \frac{2}{3} \, b d x^{6} e^{3} + \frac{6}{5} \, b d^{2} x^{5} e^{2} + b d^{3} x^{4} e + \frac{1}{3} \, b d^{4} x^{3} + \frac{1}{6} \, a x^{6} e^{4} + \frac{4}{5} \, a d x^{5} e^{3} + \frac{3}{2} \, a d^{2} x^{4} e^{2} + \frac{4}{3} \, a d^{3} x^{3} e + \frac{1}{2} \, a d^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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