Optimal. Leaf size=137 \[ \frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5} \]
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Rubi [A] time = 0.140859, antiderivative size = 137, 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 = {698} \[ \frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^4}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^5}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^6}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^7}{e^4}+\frac{c^2 (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac{d^2 (c d-b e)^2 (d+e x)^5}{5 e^5}-\frac{d (c d-b e) (2 c d-b e) (d+e x)^6}{3 e^5}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^7}{7 e^5}-\frac{c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}\\ \end{align*}
Mathematica [A] time = 0.0255949, size = 159, normalized size = 1.16 \[ \frac{1}{7} e^2 x^7 \left (b^2 e^2+8 b c d e+6 c^2 d^2\right )+\frac{2}{3} d e x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{5} d^2 x^5 \left (6 b^2 e^2+8 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^4 x^3+\frac{1}{2} b d^3 x^4 (2 b e+c d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 166, normalized size = 1.2 \begin{align*}{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,{e}^{4}bc+4\,d{e}^{3}{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{4}{b}^{2}+8\,d{e}^{3}bc+6\,{d}^{2}{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,d{e}^{3}{b}^{2}+12\,{d}^{2}{e}^{2}bc+4\,{d}^{3}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{b}^{2}+8\,{d}^{3}ebc+{c}^{2}{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{3}e{b}^{2}+2\,{d}^{4}bc \right ){x}^{4}}{4}}+{\frac{{d}^{4}{b}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16143, size = 217, normalized size = 1.58 \begin{align*} \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{3} \, b^{2} d^{4} x^{3} + \frac{1}{4} \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac{2}{3} \,{\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38888, size = 387, normalized size = 2.82 \begin{align*} \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{1}{4} x^{8} e^{4} c b + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{8}{7} x^{7} e^{3} d c b + \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{8}{5} x^{5} e d^{3} c b + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + \frac{1}{3} x^{3} d^{4} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.374781, size = 178, normalized size = 1.3 \begin{align*} \frac{b^{2} d^{4} x^{3}}{3} + \frac{c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac{b c e^{4}}{4} + \frac{c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac{b^{2} e^{4}}{7} + \frac{8 b c d e^{3}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{6 b^{2} d^{2} e^{2}}{5} + \frac{8 b c d^{3} e}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac{b c d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25349, size = 228, normalized size = 1.66 \begin{align*} \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{1}{4} \, b c x^{8} e^{4} + \frac{8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac{8}{5} \, b c d^{3} x^{5} e + \frac{1}{2} \, b c d^{4} x^{4} + \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac{1}{3} \, b^{2} d^{4} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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