Optimal. Leaf size=38 \[ \frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2} \]
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Rubi [A] time = 0.0268233, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 (c+d x) \, dx &=\int \left (\frac{(b c-a d) (a+b x)^2}{b}+\frac{d (a+b x)^3}{b}\right ) \, dx\\ &=\frac{(b c-a d) (a+b x)^3}{3 b^2}+\frac{d (a+b x)^4}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0078615, size = 46, normalized size = 1.21 \[ \frac{1}{12} x \left (6 a^2 (2 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 49, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}d{x}^{4}}{4}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{2}}{2}}+{a}^{2}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95809, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75886, size = 115, normalized size = 3.03 \begin{align*} \frac{1}{4} x^{4} d b^{2} + \frac{1}{3} x^{3} c b^{2} + \frac{2}{3} x^{3} d b a + x^{2} c b a + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078587, size = 49, normalized size = 1.29 \begin{align*} a^{2} c x + \frac{b^{2} d x^{4}}{4} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06686, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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