Optimal. Leaf size=38 \[ \frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2} \]
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Rubi [A] time = 0.0178423, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {626, 43} \[ \frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx &=\int (a+b x)^4 (c+d x) \, dx\\ &=\int \left (\frac{(b c-a d) (a+b x)^4}{b}+\frac{d (a+b x)^5}{b}\right ) \, dx\\ &=\frac{(b c-a d) (a+b x)^5}{5 b^2}+\frac{d (a+b x)^6}{6 b^2}\\ \end{align*}
Mathematica [B] time = 0.0180026, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^2 b^2 x^2 (4 c+3 d x)+20 a^3 b x (3 c+2 d x)+15 a^4 (2 c+d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 133, normalized size = 3.5 \begin{align*}{\frac{{b}^{4}d{x}^{6}}{6}}+{\frac{ \left ( 3\,ad{b}^{3}+{b}^{3} \left ( ad+bc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{b}^{2}{a}^{2}d+3\,{b}^{2}a \left ( ad+bc \right ) +a{b}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}bd+3\,b{a}^{2} \left ( ad+bc \right ) +3\,{b}^{2}{a}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3} \left ( ad+bc \right ) +3\,b{a}^{3}c \right ){x}^{2}}{2}}+{a}^{4}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09863, size = 130, normalized size = 3.42 \begin{align*} \frac{1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac{1}{5} \,{\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53655, size = 217, normalized size = 5.71 \begin{align*} \frac{1}{6} x^{6} d b^{4} + \frac{1}{5} x^{5} c b^{4} + \frac{4}{5} x^{5} d b^{3} a + x^{4} c b^{3} a + \frac{3}{2} x^{4} d b^{2} a^{2} + 2 x^{3} c b^{2} a^{2} + \frac{4}{3} x^{3} d b a^{3} + 2 x^{2} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.140041, size = 100, normalized size = 2.63 \begin{align*} a^{4} c x + \frac{b^{4} d x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} d}{5} + \frac{b^{4} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac{4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac{a^{4} d}{2} + 2 a^{3} b c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16292, size = 131, normalized size = 3.45 \begin{align*} \frac{1}{6} \, b^{4} d x^{6} + \frac{1}{5} \, b^{4} c x^{5} + \frac{4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac{3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac{4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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