Optimal. Leaf size=87 \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]
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Rubi [A] time = 0.0764796, antiderivative size = 87, 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} \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx &=\int \left (b^2 d^2 x^2+2 b d (c d+b e) x^3+\left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^4+2 c e (c d+b e) x^5+c^2 e^2 x^6\right ) \, dx\\ &=\frac{1}{3} b^2 d^2 x^3+\frac{1}{2} b d (c d+b e) x^4+\frac{1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac{1}{3} c e (c d+b e) x^6+\frac{1}{7} c^2 e^2 x^7\\ \end{align*}
Mathematica [A] time = 0.0141169, size = 87, normalized size = 1. \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 90, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,de{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}{e}^{2}+4\,bcde+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}de+2\,{d}^{2}bc \right ){x}^{4}}{4}}+{\frac{{b}^{2}{d}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11869, size = 115, normalized size = 1.32 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, b^{2} d^{2} x^{3} + \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 + b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{2} + b^{2} d e\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40058, size = 217, normalized size = 2.49 \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{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + \frac{1}{3} x^{3} d^{2} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.266324, size = 94, normalized size = 1.08 \begin{align*} \frac{b^{2} d^{2} x^{3}}{3} + \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{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16886, size = 127, normalized size = 1.46 \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{1}{2} \, b^{2} d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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