Optimal. Leaf size=240 \[ \frac{2 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^5 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{5 c^2 (d+e x)^7 (2 c d-b e)}{7 e^6}+\frac{c^3 (d+e x)^8}{4 e^6} \]
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Rubi [A] time = 0.231766, 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} \[ \frac{2 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^5 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{5 c^2 (d+e x)^7 (2 c d-b e)}{7 e^6}+\frac{c^3 (d+e x)^8}{4 e^6} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x)^2 \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)^2}{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)^3}{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)^4}{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)^5}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^6}{e^5}+\frac{2 c^3 (d+e x)^7}{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)^3}{3 e^6}+\frac{\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)^4}{2 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)^5}{5 e^6}+\frac{2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{3 e^6}-\frac{5 c^2 (2 c d-b e) (d+e x)^7}{7 e^6}+\frac{c^3 (d+e x)^8}{4 e^6}\\ \end{align*}
Mathematica [A] time = 0.0839736, size = 244, normalized size = 1.02 \[ \frac{1}{3} x^3 \left (4 a^2 c d e+4 a b^2 d e+a b \left (a e^2+6 c d^2\right )+b^3 d^2\right )+a^2 b d^2 x+\frac{1}{3} c x^6 \left (c e (2 a e+5 b d)+2 b^2 e^2+c^2 d^2\right )+\frac{1}{5} x^5 \left (b c \left (6 a e^2+5 c d^2\right )+8 a c^2 d e+8 b^2 c d e+b^3 e^2\right )+\frac{1}{2} x^4 \left (b^2 \left (a e^2+2 c d^2\right )+6 a b c d e+a c \left (a e^2+2 c d^2\right )+b^3 d e\right )+a d x^2 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 e x^7 (5 b e+4 c d)+\frac{1}{4} c^3 e^2 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 302, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{8}}{4}}+{\frac{ \left ( \left ( b{e}^{2}+4\,cde \right ){c}^{2}+4\,{c}^{2}{e}^{2}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,bde+2\,c{d}^{2} \right ){c}^{2}+2\, \left ( b{e}^{2}+4\,cde \right ) bc+2\,c{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{2}{c}^{2}+2\, \left ( 2\,bde+2\,c{d}^{2} \right ) bc+ \left ( b{e}^{2}+4\,cde \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,c{e}^{2}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}{d}^{2}c+ \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( b{e}^{2}+4\,cde \right ) ab+2\,c{e}^{2}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 2\,bde+2\,c{d}^{2} \right ) ab+ \left ( b{e}^{2}+4\,cde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}{d}^{2}a+ \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+b{d}^{2}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03724, size = 325, normalized size = 1.35 \begin{align*} \frac{1}{4} \, c^{3} e^{2} x^{8} + \frac{1}{7} \,{\left (4 \, c^{3} d e + 5 \, b c^{2} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (c^{3} d^{2} + 5 \, b c^{2} d e + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{6} + a^{2} b d^{2} x + \frac{1}{5} \,{\left (5 \, b c^{2} d^{2} + 8 \,{\left (b^{2} c + a c^{2}\right )} d e +{\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (2 \,{\left (b^{2} c + a c^{2}\right )} d^{2} +{\left (b^{3} + 6 \, a b c\right )} d e +{\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (a^{2} b e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e\right )} x^{3} +{\left (a^{2} b d e +{\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64329, size = 666, normalized size = 2.78 \begin{align*} \frac{1}{4} x^{8} e^{2} c^{3} + \frac{4}{7} x^{7} e d c^{3} + \frac{5}{7} x^{7} e^{2} c^{2} b + \frac{1}{3} x^{6} d^{2} c^{3} + \frac{5}{3} x^{6} e d c^{2} b + \frac{2}{3} x^{6} e^{2} c b^{2} + \frac{2}{3} x^{6} e^{2} c^{2} a + x^{5} d^{2} c^{2} b + \frac{8}{5} x^{5} e d c b^{2} + \frac{1}{5} x^{5} e^{2} b^{3} + \frac{8}{5} x^{5} e d c^{2} a + \frac{6}{5} x^{5} e^{2} c b a + x^{4} d^{2} c b^{2} + \frac{1}{2} x^{4} e d b^{3} + x^{4} d^{2} c^{2} a + 3 x^{4} e d c b a + \frac{1}{2} x^{4} e^{2} b^{2} a + \frac{1}{2} x^{4} e^{2} c a^{2} + \frac{1}{3} x^{3} d^{2} b^{3} + 2 x^{3} d^{2} c b a + \frac{4}{3} x^{3} e d b^{2} a + \frac{4}{3} x^{3} e d c a^{2} + \frac{1}{3} x^{3} e^{2} b a^{2} + x^{2} d^{2} b^{2} a + x^{2} d^{2} c a^{2} + x^{2} e d b a^{2} + x d^{2} b a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.105639, size = 294, normalized size = 1.22 \begin{align*} a^{2} b d^{2} x + \frac{c^{3} e^{2} x^{8}}{4} + x^{7} \left (\frac{5 b c^{2} e^{2}}{7} + \frac{4 c^{3} d e}{7}\right ) + x^{6} \left (\frac{2 a c^{2} e^{2}}{3} + \frac{2 b^{2} c e^{2}}{3} + \frac{5 b c^{2} d e}{3} + \frac{c^{3} d^{2}}{3}\right ) + x^{5} \left (\frac{6 a b c e^{2}}{5} + \frac{8 a c^{2} d e}{5} + \frac{b^{3} e^{2}}{5} + \frac{8 b^{2} c d e}{5} + b c^{2} d^{2}\right ) + x^{4} \left (\frac{a^{2} c e^{2}}{2} + \frac{a b^{2} e^{2}}{2} + 3 a b c d e + a c^{2} d^{2} + \frac{b^{3} d e}{2} + b^{2} c d^{2}\right ) + x^{3} \left (\frac{a^{2} b e^{2}}{3} + \frac{4 a^{2} c d e}{3} + \frac{4 a b^{2} d e}{3} + 2 a b c d^{2} + \frac{b^{3} d^{2}}{3}\right ) + x^{2} \left (a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17829, size = 402, normalized size = 1.68 \begin{align*} \frac{1}{4} \, c^{3} x^{8} e^{2} + \frac{4}{7} \, c^{3} d x^{7} e + \frac{1}{3} \, c^{3} d^{2} x^{6} + \frac{5}{7} \, b c^{2} x^{7} e^{2} + \frac{5}{3} \, b c^{2} d x^{6} e + b c^{2} d^{2} x^{5} + \frac{2}{3} \, b^{2} c x^{6} e^{2} + \frac{2}{3} \, a c^{2} x^{6} e^{2} + \frac{8}{5} \, b^{2} c d x^{5} e + \frac{8}{5} \, a c^{2} d x^{5} e + b^{2} c d^{2} x^{4} + a c^{2} d^{2} x^{4} + \frac{1}{5} \, b^{3} x^{5} e^{2} + \frac{6}{5} \, a b c x^{5} e^{2} + \frac{1}{2} \, b^{3} d x^{4} e + 3 \, a b c d x^{4} e + \frac{1}{3} \, b^{3} d^{2} x^{3} + 2 \, a b c d^{2} x^{3} + \frac{1}{2} \, a b^{2} x^{4} e^{2} + \frac{1}{2} \, a^{2} c x^{4} e^{2} + \frac{4}{3} \, a b^{2} d x^{3} e + \frac{4}{3} \, a^{2} c d x^{3} e + a b^{2} d^{2} x^{2} + a^{2} c d^{2} x^{2} + \frac{1}{3} \, a^{2} b x^{3} e^{2} + a^{2} b d x^{2} e + a^{2} b d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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