Optimal. Leaf size=240 \[ \frac{4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac{2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac{2 c^3 (d+e x)^9}{9 e^6} \]
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Rubi [A] time = 0.316337, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac{2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac{2 c^3 (d+e x)^9}{9 e^6} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{e^5}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^4}{e^5}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^5}{e^5}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^7}{e^5}+\frac{2 c^3 (d+e x)^8}{e^5}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^6}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^5}{5 e^6}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^6}{6 e^6}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^6}-\frac{5 c^2 (2 c d-b e) (d+e x)^8}{8 e^6}+\frac{2 c^3 (d+e x)^9}{9 e^6}\\ \end{align*}
Mathematica [A] time = 0.126793, size = 351, normalized size = 1.46 \[ \frac{1}{3} d x^3 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+a^2 b d^3 x+\frac{1}{7} c e x^7 \left (c e (4 a e+15 b d)+4 b^2 e^2+6 c^2 d^2\right )+\frac{1}{6} x^6 \left (3 c^2 d e (4 a e+5 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+2 c^3 d^3\right )+\frac{1}{5} x^5 \left (2 b^2 \left (a e^3+6 c d^2 e\right )+b c d \left (18 a e^2+5 c d^2\right )+2 a c e \left (a e^2+6 c d^2\right )+3 b^3 d e^2\right )+\frac{1}{4} x^4 \left (b^2 \left (6 a d e^2+4 c d^3\right )+a b e \left (a e^2+18 c d^2\right )+2 a c d \left (3 a e^2+2 c d^2\right )+3 b^3 d^2 e\right )+\frac{1}{2} a d^2 x^2 \left (3 a b e+2 a c d+2 b^2 d\right )+\frac{1}{8} c^2 e^2 x^8 (5 b e+6 c d)+\frac{2}{9} c^3 e^3 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 428, normalized size = 1.8 \begin{align*}{\frac{2\,{c}^{3}{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( b{e}^{3}+6\,cd{e}^{2} \right ){c}^{2}+4\,{c}^{2}{e}^{3}b \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,bd{e}^{2}+6\,c{d}^{2}e \right ){c}^{2}+2\, \left ( b{e}^{3}+6\,cd{e}^{2} \right ) bc+2\,c{e}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ){c}^{2}+2\, \left ( 3\,bd{e}^{2}+6\,c{d}^{2}e \right ) bc+ \left ( b{e}^{3}+6\,cd{e}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,c{e}^{3}ab \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{3}{c}^{2}+2\, \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) bc+ \left ( 3\,bd{e}^{2}+6\,c{d}^{2}e \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( b{e}^{3}+6\,cd{e}^{2} \right ) ab+2\,c{e}^{3}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}{d}^{3}c+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,bd{e}^{2}+6\,c{d}^{2}e \right ) ab+ \left ( b{e}^{3}+6\,cd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) ab+ \left ( 3\,bd{e}^{2}+6\,c{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}{d}^{3}a+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ){a}^{2} \right ){x}^{2}}{2}}+b{d}^{3}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01383, size = 463, normalized size = 1.93 \begin{align*} \frac{2}{9} \, c^{3} e^{3} x^{9} + \frac{1}{8} \,{\left (6 \, c^{3} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{3} d^{2} e + 15 \, b c^{2} d e^{2} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{7} + a^{2} b d^{3} x + \frac{1}{6} \,{\left (2 \, c^{3} d^{3} + 15 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} +{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, b c^{2} d^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e + 3 \,{\left (b^{3} + 6 \, a b c\right )} d e^{2} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (a^{2} b e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} + 3 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e + 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a^{2} b d e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} + 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} e + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81814, size = 944, normalized size = 3.93 \begin{align*} \frac{2}{9} x^{9} e^{3} c^{3} + \frac{3}{4} x^{8} e^{2} d c^{3} + \frac{5}{8} x^{8} e^{3} c^{2} b + \frac{6}{7} x^{7} e d^{2} c^{3} + \frac{15}{7} x^{7} e^{2} d c^{2} b + \frac{4}{7} x^{7} e^{3} c b^{2} + \frac{4}{7} x^{7} e^{3} c^{2} a + \frac{1}{3} x^{6} d^{3} c^{3} + \frac{5}{2} x^{6} e d^{2} c^{2} b + 2 x^{6} e^{2} d c b^{2} + \frac{1}{6} x^{6} e^{3} b^{3} + 2 x^{6} e^{2} d c^{2} a + x^{6} e^{3} c b a + x^{5} d^{3} c^{2} b + \frac{12}{5} x^{5} e d^{2} c b^{2} + \frac{3}{5} x^{5} e^{2} d b^{3} + \frac{12}{5} x^{5} e d^{2} c^{2} a + \frac{18}{5} x^{5} e^{2} d c b a + \frac{2}{5} x^{5} e^{3} b^{2} a + \frac{2}{5} x^{5} e^{3} c a^{2} + x^{4} d^{3} c b^{2} + \frac{3}{4} x^{4} e d^{2} b^{3} + x^{4} d^{3} c^{2} a + \frac{9}{2} x^{4} e d^{2} c b a + \frac{3}{2} x^{4} e^{2} d b^{2} a + \frac{3}{2} x^{4} e^{2} d c a^{2} + \frac{1}{4} x^{4} e^{3} b a^{2} + \frac{1}{3} x^{3} d^{3} b^{3} + 2 x^{3} d^{3} c b a + 2 x^{3} e d^{2} b^{2} a + 2 x^{3} e d^{2} c a^{2} + x^{3} e^{2} d b a^{2} + x^{2} d^{3} b^{2} a + x^{2} d^{3} c a^{2} + \frac{3}{2} x^{2} e d^{2} b a^{2} + x d^{3} b a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119929, size = 430, normalized size = 1.79 \begin{align*} a^{2} b d^{3} x + \frac{2 c^{3} e^{3} x^{9}}{9} + x^{8} \left (\frac{5 b c^{2} e^{3}}{8} + \frac{3 c^{3} d e^{2}}{4}\right ) + x^{7} \left (\frac{4 a c^{2} e^{3}}{7} + \frac{4 b^{2} c e^{3}}{7} + \frac{15 b c^{2} d e^{2}}{7} + \frac{6 c^{3} d^{2} e}{7}\right ) + x^{6} \left (a b c e^{3} + 2 a c^{2} d e^{2} + \frac{b^{3} e^{3}}{6} + 2 b^{2} c d e^{2} + \frac{5 b c^{2} d^{2} e}{2} + \frac{c^{3} d^{3}}{3}\right ) + x^{5} \left (\frac{2 a^{2} c e^{3}}{5} + \frac{2 a b^{2} e^{3}}{5} + \frac{18 a b c d e^{2}}{5} + \frac{12 a c^{2} d^{2} e}{5} + \frac{3 b^{3} d e^{2}}{5} + \frac{12 b^{2} c d^{2} e}{5} + b c^{2} d^{3}\right ) + x^{4} \left (\frac{a^{2} b e^{3}}{4} + \frac{3 a^{2} c d e^{2}}{2} + \frac{3 a b^{2} d e^{2}}{2} + \frac{9 a b c d^{2} e}{2} + a c^{2} d^{3} + \frac{3 b^{3} d^{2} e}{4} + b^{2} c d^{3}\right ) + x^{3} \left (a^{2} b d e^{2} + 2 a^{2} c d^{2} e + 2 a b^{2} d^{2} e + 2 a b c d^{3} + \frac{b^{3} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} b d^{2} e}{2} + a^{2} c d^{3} + a b^{2} d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17966, size = 567, normalized size = 2.36 \begin{align*} \frac{2}{9} \, c^{3} x^{9} e^{3} + \frac{3}{4} \, c^{3} d x^{8} e^{2} + \frac{6}{7} \, c^{3} d^{2} x^{7} e + \frac{1}{3} \, c^{3} d^{3} x^{6} + \frac{5}{8} \, b c^{2} x^{8} e^{3} + \frac{15}{7} \, b c^{2} d x^{7} e^{2} + \frac{5}{2} \, b c^{2} d^{2} x^{6} e + b c^{2} d^{3} x^{5} + \frac{4}{7} \, b^{2} c x^{7} e^{3} + \frac{4}{7} \, a c^{2} x^{7} e^{3} + 2 \, b^{2} c d x^{6} e^{2} + 2 \, a c^{2} d x^{6} e^{2} + \frac{12}{5} \, b^{2} c d^{2} x^{5} e + \frac{12}{5} \, a c^{2} d^{2} x^{5} e + b^{2} c d^{3} x^{4} + a c^{2} d^{3} x^{4} + \frac{1}{6} \, b^{3} x^{6} e^{3} + a b c x^{6} e^{3} + \frac{3}{5} \, b^{3} d x^{5} e^{2} + \frac{18}{5} \, a b c d x^{5} e^{2} + \frac{3}{4} \, b^{3} d^{2} x^{4} e + \frac{9}{2} \, a b c d^{2} x^{4} e + \frac{1}{3} \, b^{3} d^{3} x^{3} + 2 \, a b c d^{3} x^{3} + \frac{2}{5} \, a b^{2} x^{5} e^{3} + \frac{2}{5} \, a^{2} c x^{5} e^{3} + \frac{3}{2} \, a b^{2} d x^{4} e^{2} + \frac{3}{2} \, a^{2} c d x^{4} e^{2} + 2 \, a b^{2} d^{2} x^{3} e + 2 \, a^{2} c d^{2} x^{3} e + a b^{2} d^{3} x^{2} + a^{2} c d^{3} x^{2} + \frac{1}{4} \, a^{2} b x^{4} e^{3} + a^{2} b d x^{3} e^{2} + \frac{3}{2} \, a^{2} b d^{2} x^{2} e + a^{2} b d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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