Optimal. Leaf size=65 \[ -\frac{2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac{(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac{b^2 (c+d x)^6}{6 d^3} \]
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Rubi [A] time = 0.0644285, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac{(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac{b^2 (c+d x)^6}{6 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 (c+d x)^3 \, dx &=\int \left (\frac{(-b c+a d)^2 (c+d x)^3}{d^2}-\frac{2 b (b c-a d) (c+d x)^4}{d^2}+\frac{b^2 (c+d x)^5}{d^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (c+d x)^4}{4 d^3}-\frac{2 b (b c-a d) (c+d x)^5}{5 d^3}+\frac{b^2 (c+d x)^6}{6 d^3}\\ \end{align*}
Mathematica [A] time = 0.0140066, size = 122, normalized size = 1.88 \[ \frac{1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{6} b^2 d^3 x^6 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 125, normalized size = 1.9 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{6}}{6}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{2}}{2}}+{a}^{2}{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0054, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7463, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} d^{3} b^{2} + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} d^{3} b a + \frac{3}{4} x^{4} d c^{2} b^{2} + \frac{3}{2} x^{4} d^{2} c b a + \frac{1}{4} x^{4} d^{3} a^{2} + \frac{1}{3} x^{3} c^{3} b^{2} + 2 x^{3} d c^{2} b a + x^{3} d^{2} c a^{2} + x^{2} c^{3} b a + \frac{3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.082265, size = 133, normalized size = 2.05 \begin{align*} a^{2} c^{3} x + \frac{b^{2} d^{3} x^{6}}{6} + x^{5} \left (\frac{2 a b d^{3}}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + \frac{3 a b c d^{2}}{2} + \frac{3 b^{2} c^{2} d}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac{b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05247, size = 176, normalized size = 2.71 \begin{align*} \frac{1}{6} \, b^{2} d^{3} x^{6} + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{2}{5} \, a b d^{3} x^{5} + \frac{3}{4} \, b^{2} c^{2} d x^{4} + \frac{3}{2} \, a b c d^{2} x^{4} + \frac{1}{4} \, a^{2} d^{3} x^{4} + \frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + a b c^{3} x^{2} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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