Optimal. Leaf size=103 \[ \frac{(d+e x)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac{d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac{(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac{c (d+e x)^9}{9 e^4} \]
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Rubi [A] time = 0.163312, 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} \[ \frac{(d+e x)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac{d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac{(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac{c (d+e x)^9}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int x (d+e x)^5 \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)^5}{e^3}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^6}{e^3}+\frac{(-3 c d+b e) (d+e x)^7}{e^3}+\frac{c (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac{d \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{6 e^4}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^7}{7 e^4}-\frac{(3 c d-b e) (d+e x)^8}{8 e^4}+\frac{c (d+e x)^9}{9 e^4}\\ \end{align*}
Mathematica [A] time = 0.0429747, size = 166, normalized size = 1.61 \[ \frac{1}{7} e^3 x^7 \left (a e^2+5 b d e+10 c d^2\right )+\frac{5}{6} d e^2 x^6 \left (a e^2+2 b d e+2 c d^2\right )+d^2 e x^5 \left (2 a e^2+2 b d e+c d^2\right )+\frac{1}{4} d^3 x^4 \left (10 a e^2+5 b d e+c d^2\right )+\frac{1}{3} d^4 x^3 (5 a e+b d)+\frac{1}{2} a d^5 x^2+\frac{1}{8} e^4 x^8 (b e+5 c d)+\frac{1}{9} c e^5 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 172, normalized size = 1.7 \begin{align*}{\frac{{e}^{5}c{x}^{9}}{9}}+{\frac{ \left ({e}^{5}b+5\,d{e}^{4}c \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{5}a+5\,d{e}^{4}b+10\,{d}^{2}{e}^{3}c \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,d{e}^{4}a+10\,{d}^{2}{e}^{3}b+10\,{d}^{3}{e}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}a+10\,{d}^{3}{e}^{2}b+5\,{d}^{4}ec \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}a+5\,{d}^{4}eb+{d}^{5}c \right ){x}^{4}}{4}}+{\frac{ \left ( 5\,{d}^{4}ea+{d}^{5}b \right ){x}^{3}}{3}}+{\frac{{d}^{5}a{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08128, size = 227, normalized size = 2.2 \begin{align*} \frac{1}{9} \, c e^{5} x^{9} + \frac{1}{8} \,{\left (5 \, c d e^{4} + b e^{5}\right )} x^{8} + \frac{1}{2} \, a d^{5} x^{2} + \frac{1}{7} \,{\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{7} + \frac{5}{6} \,{\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{6} +{\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{5} + 5 \, a d^{4} e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16245, size = 427, normalized size = 4.15 \begin{align*} \frac{1}{9} x^{9} e^{5} c + \frac{5}{8} x^{8} e^{4} d c + \frac{1}{8} x^{8} e^{5} b + \frac{10}{7} x^{7} e^{3} d^{2} c + \frac{5}{7} x^{7} e^{4} d b + \frac{1}{7} x^{7} e^{5} a + \frac{5}{3} x^{6} e^{2} d^{3} c + \frac{5}{3} x^{6} e^{3} d^{2} b + \frac{5}{6} x^{6} e^{4} d a + x^{5} e d^{4} c + 2 x^{5} e^{2} d^{3} b + 2 x^{5} e^{3} d^{2} a + \frac{1}{4} x^{4} d^{5} c + \frac{5}{4} x^{4} e d^{4} b + \frac{5}{2} x^{4} e^{2} d^{3} a + \frac{1}{3} x^{3} d^{5} b + \frac{5}{3} x^{3} e d^{4} a + \frac{1}{2} x^{2} d^{5} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.093609, size = 192, normalized size = 1.86 \begin{align*} \frac{a d^{5} x^{2}}{2} + \frac{c e^{5} x^{9}}{9} + x^{8} \left (\frac{b e^{5}}{8} + \frac{5 c d e^{4}}{8}\right ) + x^{7} \left (\frac{a e^{5}}{7} + \frac{5 b d e^{4}}{7} + \frac{10 c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{5 a d e^{4}}{6} + \frac{5 b d^{2} e^{3}}{3} + \frac{5 c d^{3} e^{2}}{3}\right ) + x^{5} \left (2 a d^{2} e^{3} + 2 b d^{3} e^{2} + c d^{4} e\right ) + x^{4} \left (\frac{5 a d^{3} e^{2}}{2} + \frac{5 b d^{4} e}{4} + \frac{c d^{5}}{4}\right ) + x^{3} \left (\frac{5 a d^{4} e}{3} + \frac{b d^{5}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07538, size = 239, normalized size = 2.32 \begin{align*} \frac{1}{9} \, c x^{9} e^{5} + \frac{5}{8} \, c d x^{8} e^{4} + \frac{10}{7} \, c d^{2} x^{7} e^{3} + \frac{5}{3} \, c d^{3} x^{6} e^{2} + c d^{4} x^{5} e + \frac{1}{4} \, c d^{5} x^{4} + \frac{1}{8} \, b x^{8} e^{5} + \frac{5}{7} \, b d x^{7} e^{4} + \frac{5}{3} \, b d^{2} x^{6} e^{3} + 2 \, b d^{3} x^{5} e^{2} + \frac{5}{4} \, b d^{4} x^{4} e + \frac{1}{3} \, b d^{5} x^{3} + \frac{1}{7} \, a x^{7} e^{5} + \frac{5}{6} \, a d x^{6} e^{4} + 2 \, a d^{2} x^{5} e^{3} + \frac{5}{2} \, a d^{3} x^{4} e^{2} + \frac{5}{3} \, a d^{4} x^{3} e + \frac{1}{2} \, a d^{5} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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