Optimal. Leaf size=124 \[ \frac{(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac{(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac{c^2 (d+e x)^8}{4 e^4} \]
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Rubi [A] time = 0.189266, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac{(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac{c^2 (d+e x)^8}{4 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^3}+\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^3}-\frac{3 c (2 c d-b e) (d+e x)^6}{e^3}+\frac{2 c^2 (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 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)^6}{6 e^4}-\frac{3 c (2 c d-b e) (d+e x)^7}{7 e^4}+\frac{c^2 (d+e x)^8}{4 e^4}\\ \end{align*}
Mathematica [A] time = 0.0775924, size = 229, normalized size = 1.85 \[ \frac{1}{6} e^2 x^6 \left (2 c e (a e+6 b d)+b^2 e^2+12 c^2 d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a b e^2+8 a c d e+4 b^2 d e+3 b c d^2\right )+\frac{1}{2} d^3 x^2 \left (4 a b e+2 a c d+b^2 d\right )+\frac{1}{5} e x^5 \left (2 c d e (4 a e+9 b d)+b e^2 (a e+4 b d)+8 c^2 d^3\right )+\frac{1}{2} d x^4 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+a b d^4 x+\frac{1}{7} c e^3 x^7 (3 b e+8 c d)+\frac{1}{4} c^2 e^4 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 290, normalized size = 2.3 \begin{align*}{\frac{{c}^{2}{e}^{4}{x}^{8}}{4}}+{\frac{ \left ( \left ( b{e}^{4}+8\,cd{e}^{3} \right ) c+2\,c{e}^{4}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) c+ \left ( b{e}^{4}+8\,cd{e}^{3} \right ) b+2\,c{e}^{4}a \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) c+ \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) b+ \left ( b{e}^{4}+8\,cd{e}^{3} \right ) a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) c+ \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) b+ \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{4}c+ \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) b+ \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ({b}^{2}{d}^{4}+ \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) a \right ){x}^{2}}{2}}+b{d}^{4}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09528, size = 312, normalized size = 2.52 \begin{align*} \frac{1}{4} \, c^{2} e^{4} x^{8} + \frac{1}{7} \,{\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac{1}{6} \,{\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a b d^{3} e +{\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46031, size = 621, normalized size = 5.01 \begin{align*} \frac{1}{4} x^{8} e^{4} c^{2} + \frac{8}{7} x^{7} e^{3} d c^{2} + \frac{3}{7} x^{7} e^{4} c b + 2 x^{6} e^{2} d^{2} c^{2} + 2 x^{6} e^{3} d c b + \frac{1}{6} x^{6} e^{4} b^{2} + \frac{1}{3} x^{6} e^{4} c a + \frac{8}{5} x^{5} e d^{3} c^{2} + \frac{18}{5} x^{5} e^{2} d^{2} c b + \frac{4}{5} x^{5} e^{3} d b^{2} + \frac{8}{5} x^{5} e^{3} d c a + \frac{1}{5} x^{5} e^{4} b a + \frac{1}{2} x^{4} d^{4} c^{2} + 3 x^{4} e d^{3} c b + \frac{3}{2} x^{4} e^{2} d^{2} b^{2} + 3 x^{4} e^{2} d^{2} c a + x^{4} e^{3} d b a + x^{3} d^{4} c b + \frac{4}{3} x^{3} e d^{3} b^{2} + \frac{8}{3} x^{3} e d^{3} c a + 2 x^{3} e^{2} d^{2} b a + \frac{1}{2} x^{2} d^{4} b^{2} + x^{2} d^{4} c a + 2 x^{2} e d^{3} b a + x d^{4} b a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.104212, size = 279, normalized size = 2.25 \begin{align*} a b d^{4} x + \frac{c^{2} e^{4} x^{8}}{4} + x^{7} \left (\frac{3 b c e^{4}}{7} + \frac{8 c^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac{a c e^{4}}{3} + \frac{b^{2} e^{4}}{6} + 2 b c d e^{3} + 2 c^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac{a b e^{4}}{5} + \frac{8 a c d e^{3}}{5} + \frac{4 b^{2} d e^{3}}{5} + \frac{18 b c d^{2} e^{2}}{5} + \frac{8 c^{2} d^{3} e}{5}\right ) + x^{4} \left (a b d e^{3} + 3 a c d^{2} e^{2} + \frac{3 b^{2} d^{2} e^{2}}{2} + 3 b c d^{3} e + \frac{c^{2} d^{4}}{2}\right ) + x^{3} \left (2 a b d^{2} e^{2} + \frac{8 a c d^{3} e}{3} + \frac{4 b^{2} d^{3} e}{3} + b c d^{4}\right ) + x^{2} \left (2 a b d^{3} e + a c d^{4} + \frac{b^{2} d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17103, size = 365, normalized size = 2.94 \begin{align*} \frac{1}{4} \, c^{2} x^{8} e^{4} + \frac{8}{7} \, c^{2} d x^{7} e^{3} + 2 \, c^{2} d^{2} x^{6} e^{2} + \frac{8}{5} \, c^{2} d^{3} x^{5} e + \frac{1}{2} \, c^{2} d^{4} x^{4} + \frac{3}{7} \, b c x^{7} e^{4} + 2 \, b c d x^{6} e^{3} + \frac{18}{5} \, b c d^{2} x^{5} e^{2} + 3 \, b c d^{3} x^{4} e + b c d^{4} x^{3} + \frac{1}{6} \, b^{2} x^{6} e^{4} + \frac{1}{3} \, a c x^{6} e^{4} + \frac{4}{5} \, b^{2} d x^{5} e^{3} + \frac{8}{5} \, a c d x^{5} e^{3} + \frac{3}{2} \, b^{2} d^{2} x^{4} e^{2} + 3 \, a c d^{2} x^{4} e^{2} + \frac{4}{3} \, b^{2} d^{3} x^{3} e + \frac{8}{3} \, a c d^{3} x^{3} e + \frac{1}{2} \, b^{2} d^{4} x^{2} + a c d^{4} x^{2} + \frac{1}{5} \, a b x^{5} e^{4} + a b d x^{4} e^{3} + 2 \, a b d^{2} x^{3} e^{2} + 2 \, a b d^{3} x^{2} e + a b d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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