Optimal. Leaf size=156 \[ \frac{(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^5}-\frac{2 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{4 e^5}-\frac{2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac{c^2 (d+e x)^8}{8 e^5} \]
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Rubi [A] time = 0.174232, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^5}-\frac{2 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{4 e^5}-\frac{2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac{c^2 (d+e x)^8}{8 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{e^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^4}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^6}{e^4}+\frac{c^2 (d+e x)^7}{e^4}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^5}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{6 e^5}-\frac{2 c (2 c d-b e) (d+e x)^7}{7 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}\\ \end{align*}
Mathematica [A] time = 0.0646776, size = 223, normalized size = 1.43 \[ \frac{1}{4} x^4 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac{1}{6} e x^6 \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac{1}{3} d x^3 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{1}{5} x^5 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 219, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{e}^{3}{x}^{8}}{8}}+{\frac{ \left ( 2\,{e}^{3}bc+3\,d{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{2}e{c}^{2}+6\,d{e}^{2}bc+{e}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}{d}^{3}+6\,{d}^{2}ebc+3\,d{e}^{2} \left ( 2\,ac+{b}^{2} \right ) +2\,ab{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{3}bc+3\,{d}^{2}e \left ( 2\,ac+{b}^{2} \right ) +6\,d{e}^{2}ab+{a}^{2}{e}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3} \left ( 2\,ac+{b}^{2} \right ) +6\,{d}^{2}eab+3\,d{e}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}+2\,{d}^{3}ab \right ){x}^{2}}{2}}+{d}^{3}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977418, size = 294, normalized size = 1.88 \begin{align*} \frac{1}{8} \, c^{2} e^{3} x^{8} + \frac{1}{7} \,{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80195, size = 587, normalized size = 3.76 \begin{align*} \frac{1}{8} x^{8} e^{3} c^{2} + \frac{3}{7} x^{7} e^{2} d c^{2} + \frac{2}{7} x^{7} e^{3} c b + \frac{1}{2} x^{6} e d^{2} c^{2} + x^{6} e^{2} d c b + \frac{1}{6} x^{6} e^{3} b^{2} + \frac{1}{3} x^{6} e^{3} c a + \frac{1}{5} x^{5} d^{3} c^{2} + \frac{6}{5} x^{5} e d^{2} c b + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{6}{5} x^{5} e^{2} d c a + \frac{2}{5} x^{5} e^{3} b a + \frac{1}{2} x^{4} d^{3} c b + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{3}{2} x^{4} e d^{2} c a + \frac{3}{2} x^{4} e^{2} d b a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{2} + \frac{2}{3} x^{3} d^{3} c a + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098688, size = 260, normalized size = 1.67 \begin{align*} a^{2} d^{3} x + \frac{c^{2} e^{3} x^{8}}{8} + x^{7} \left (\frac{2 b c e^{3}}{7} + \frac{3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac{a c e^{3}}{3} + \frac{b^{2} e^{3}}{6} + b c d e^{2} + \frac{c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{2 a b e^{3}}{5} + \frac{6 a c d e^{2}}{5} + \frac{3 b^{2} d e^{2}}{5} + \frac{6 b c d^{2} e}{5} + \frac{c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a b d e^{2}}{2} + \frac{3 a c d^{2} e}{2} + \frac{3 b^{2} d^{2} e}{4} + \frac{b c d^{3}}{2}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac{2 a c d^{3}}{3} + \frac{b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09897, size = 342, normalized size = 2.19 \begin{align*} \frac{1}{8} \, c^{2} x^{8} e^{3} + \frac{3}{7} \, c^{2} d x^{7} e^{2} + \frac{1}{2} \, c^{2} d^{2} x^{6} e + \frac{1}{5} \, c^{2} d^{3} x^{5} + \frac{2}{7} \, b c x^{7} e^{3} + b c d x^{6} e^{2} + \frac{6}{5} \, b c d^{2} x^{5} e + \frac{1}{2} \, b c d^{3} x^{4} + \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{1}{3} \, a c x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{6}{5} \, a c d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{3}{2} \, a c d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{2}{3} \, a c d^{3} x^{3} + \frac{2}{5} \, a b x^{5} e^{3} + \frac{3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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