Optimal. Leaf size=55 \[ \frac{1}{3} b^2 d x^3+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{4} b x^4 (b e+2 c d)+\frac{1}{6} c^2 e x^6 \]
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Rubi [A] time = 0.0400556, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{1}{3} b^2 d x^3+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{4} b x^4 (b e+2 c d)+\frac{1}{6} c^2 e x^6 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (b x+c x^2\right )^2 \, dx &=\int \left (b^2 d x^2+b (2 c d+b e) x^3+c (c d+2 b e) x^4+c^2 e x^5\right ) \, dx\\ &=\frac{1}{3} b^2 d x^3+\frac{1}{4} b (2 c d+b e) x^4+\frac{1}{5} c (c d+2 b e) x^5+\frac{1}{6} c^2 e x^6\\ \end{align*}
Mathematica [A] time = 0.0101526, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^3 \left (5 b^2 (4 d+3 e x)+6 b c x (5 d+4 e x)+2 c^2 x^2 (6 d+5 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 52, normalized size = 1. \begin{align*}{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{ \left ( 2\,bce+d{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{2}e+2\,bcd \right ){x}^{4}}{4}}+{\frac{{b}^{2}d{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15484, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{3} \, b^{2} d x^{3} + \frac{1}{5} \,{\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d + b^{2} e\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45162, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{2}{5} x^{5} e c b + \frac{1}{2} x^{4} d c b + \frac{1}{4} x^{4} e b^{2} + \frac{1}{3} x^{3} d b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.278477, size = 54, normalized size = 0.98 \begin{align*} \frac{b^{2} d x^{3}}{3} + \frac{c^{2} e x^{6}}{6} + x^{5} \left (\frac{2 b c e}{5} + \frac{c^{2} d}{5}\right ) + x^{4} \left (\frac{b^{2} e}{4} + \frac{b c d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4691, size = 76, normalized size = 1.38 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{2}{5} \, b c x^{5} e + \frac{1}{2} \, b c d x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{3} \, b^{2} d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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