Optimal. Leaf size=156 \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5} \]
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Rubi [A] time = 0.140405, 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} \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^2 \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)^2}{e^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{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)^4}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^5}{e^4}+\frac{c^2 (d+e x)^6}{e^4}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^5}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{2 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)^5}{5 e^5}-\frac{c (2 c d-b e) (d+e x)^6}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.0432366, size = 153, normalized size = 0.98 \[ \frac{1}{3} x^3 \left (a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right )+a^2 d^2 x+\frac{1}{5} x^5 \left (2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{2} x^4 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+a d x^2 (a e+b d)+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 155, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,{c}^{2}de \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}{d}^{2}+4\,bcde+{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2}bc+2\,de \left ( 2\,ac+{b}^{2} \right ) +2\,{e}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,abde+{a}^{2}{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{2}+2\,{d}^{2}ab \right ){x}^{2}}{2}}+{a}^{2}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955823, size = 197, normalized size = 1.26 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \,{\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{2} \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} +{\left (a b d^{2} + a^{2} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72716, size = 412, normalized size = 2.64 \begin{align*} \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{3} x^{6} e^{2} c b + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{4}{5} x^{5} e d c b + \frac{1}{5} x^{5} e^{2} b^{2} + \frac{2}{5} x^{5} e^{2} c a + \frac{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + x^{4} e d c a + \frac{1}{2} x^{4} e^{2} b a + \frac{1}{3} x^{3} d^{2} b^{2} + \frac{2}{3} x^{3} d^{2} c a + \frac{4}{3} x^{3} e d b a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} d^{2} b a + x^{2} e d a^{2} + x d^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089446, size = 173, normalized size = 1.11 \begin{align*} a^{2} d^{2} x + \frac{c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac{b c e^{2}}{3} + \frac{c^{2} d e}{3}\right ) + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{a b e^{2}}{2} + a c d e + \frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{2 a c d^{2}}{3} + \frac{b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08653, size = 240, normalized size = 1.54 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{1}{3} \, b c x^{6} e^{2} + \frac{4}{5} \, b c d x^{5} e + \frac{1}{2} \, b c d^{2} x^{4} + \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{2}{5} \, a c x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + a c d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{2} \, a b x^{4} e^{2} + \frac{4}{3} \, a b d x^{3} e + a b d^{2} x^{2} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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