Optimal. Leaf size=55 \[ \frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
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Rubi [A] time = 0.0244556, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {692, 629} \[ \frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3+\frac{1}{4} \left (\left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx\\ &=\frac{1}{12} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3\\ \end{align*}
Mathematica [A] time = 0.0198803, size = 97, normalized size = 1.76 \[ \frac{1}{3} d^3 x (b+c x) \left (3 a^2 \left (b^2+2 b c x+2 c^2 x^2\right )+a x \left (11 b^2 c x+3 b^3+16 b c^2 x^2+8 c^3 x^3\right )+x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 237, normalized size = 4.3 \begin{align*}{c}^{5}{d}^{3}{x}^{8}+4\,b{d}^{3}{c}^{4}{x}^{7}+{\frac{ \left ( 30\,{b}^{2}{d}^{3}{c}^{3}+8\,{c}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{b}^{3}{d}^{3}{c}^{2}+12\,b{d}^{3}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +16\,{c}^{3}{d}^{3}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{4}{d}^{3}c+6\,{b}^{2}{d}^{3}c \left ( 2\,ac+{b}^{2} \right ) +24\,{b}^{2}{d}^{3}{c}^{2}a+8\,{c}^{3}{d}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +12\,{b}^{3}{d}^{3}ca+12\,b{d}^{3}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}c{a}^{2}+2\,{b}^{4}{d}^{3}a \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20458, size = 217, normalized size = 3.95 \begin{align*} c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{1}{3} \,{\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x +{\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} +{\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (a b^{4} + 3 \, a^{2} 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 = 1.72548, size = 393, normalized size = 7.15 \begin{align*} x^{8} d^{3} c^{5} + 4 x^{7} d^{3} c^{4} b + \frac{19}{3} x^{6} d^{3} c^{3} b^{2} + \frac{8}{3} x^{6} d^{3} c^{4} a + 5 x^{5} d^{3} c^{2} b^{3} + 8 x^{5} d^{3} c^{3} b a + 2 x^{4} d^{3} c b^{4} + 9 x^{4} d^{3} c^{2} b^{2} a + 2 x^{4} d^{3} c^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{5} + \frac{14}{3} x^{3} d^{3} c b^{3} a + 4 x^{3} d^{3} c^{2} b a^{2} + x^{2} d^{3} b^{4} a + 3 x^{2} d^{3} c b^{2} a^{2} + x d^{3} b^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.100154, size = 194, normalized size = 3.53 \begin{align*} a^{2} b^{3} d^{3} x + 4 b c^{4} d^{3} x^{7} + c^{5} d^{3} x^{8} + x^{6} \left (\frac{8 a c^{4} d^{3}}{3} + \frac{19 b^{2} c^{3} d^{3}}{3}\right ) + x^{5} \left (8 a b c^{3} d^{3} + 5 b^{3} c^{2} d^{3}\right ) + x^{4} \left (2 a^{2} c^{3} d^{3} + 9 a b^{2} c^{2} d^{3} + 2 b^{4} c d^{3}\right ) + x^{3} \left (4 a^{2} b c^{2} d^{3} + \frac{14 a b^{3} c d^{3}}{3} + \frac{b^{5} d^{3}}{3}\right ) + x^{2} \left (3 a^{2} b^{2} c d^{3} + a b^{4} d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21721, size = 261, normalized size = 4.75 \begin{align*} c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{19}{3} \, b^{2} c^{3} d^{3} x^{6} + \frac{8}{3} \, a c^{4} d^{3} x^{6} + 5 \, b^{3} c^{2} d^{3} x^{5} + 8 \, a b c^{3} d^{3} x^{5} + 2 \, b^{4} c d^{3} x^{4} + 9 \, a b^{2} c^{2} d^{3} x^{4} + 2 \, a^{2} c^{3} d^{3} x^{4} + \frac{1}{3} \, b^{5} d^{3} x^{3} + \frac{14}{3} \, a b^{3} c d^{3} x^{3} + 4 \, a^{2} b c^{2} d^{3} x^{3} + a b^{4} d^{3} x^{2} + 3 \, a^{2} b^{2} c d^{3} x^{2} + a^{2} b^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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