Optimal. Leaf size=45 \[ \frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2} \]
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Rubi [A] time = 0.0535636, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^3}{4 c}+\frac{(b d+2 c d x)^5}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) d^3 (b+2 c x)^4}{32 c^2}+\frac{d^3 (b+2 c x)^6}{48 c^2}\\ \end{align*}
Mathematica [A] time = 0.0145711, size = 66, normalized size = 1.47 \[ \frac{1}{6} d^3 x (b+c x) \left (6 a \left (b^2+2 b c x+2 c^2 x^2\right )+x \left (11 b^2 c x+3 b^3+16 b c^2 x^2+8 c^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 108, normalized size = 2.4 \begin{align*}{\frac{4\,{c}^{4}{d}^{3}{x}^{6}}{3}}+4\,b{d}^{3}{c}^{3}{x}^{5}+{\frac{ \left ( 8\,{c}^{3}{d}^{3}a+18\,{b}^{2}{d}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 12\,b{d}^{3}{c}^{2}a+7\,{b}^{3}{d}^{3}c \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}ca+{b}^{4}{d}^{3} \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23227, size = 131, normalized size = 2.91 \begin{align*} \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + a b^{3} d^{3} x + \frac{1}{2} \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{3} + \frac{1}{2} \,{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02252, size = 225, normalized size = 5. \begin{align*} \frac{4}{3} x^{6} d^{3} c^{4} + 4 x^{5} d^{3} c^{3} b + \frac{9}{2} x^{4} d^{3} c^{2} b^{2} + 2 x^{4} d^{3} c^{3} a + \frac{7}{3} x^{3} d^{3} c b^{3} + 4 x^{3} d^{3} c^{2} b a + \frac{1}{2} x^{2} d^{3} b^{4} + 3 x^{2} d^{3} c b^{2} a + x d^{3} b^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.120087, size = 114, normalized size = 2.53 \begin{align*} a b^{3} d^{3} x + 4 b c^{3} d^{3} x^{5} + \frac{4 c^{4} d^{3} x^{6}}{3} + x^{4} \left (2 a c^{3} d^{3} + \frac{9 b^{2} c^{2} d^{3}}{2}\right ) + x^{3} \left (4 a b c^{2} d^{3} + \frac{7 b^{3} c d^{3}}{3}\right ) + x^{2} \left (3 a b^{2} c d^{3} + \frac{b^{4} d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22207, size = 146, normalized size = 3.24 \begin{align*} \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + \frac{9}{2} \, b^{2} c^{2} d^{3} x^{4} + 2 \, a c^{3} d^{3} x^{4} + \frac{7}{3} \, b^{3} c d^{3} x^{3} + 4 \, a b c^{2} d^{3} x^{3} + \frac{1}{2} \, b^{4} d^{3} x^{2} + 3 \, a b^{2} c d^{3} x^{2} + a b^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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