Optimal. Leaf size=101 \[ -\frac{3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4} \]
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Rubi [A] time = 0.12897, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3 (b d+2 c d x)^2}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^6}{64 c^3 d^4}+\frac{(b d+2 c d x)^8}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac{3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4}\\ \end{align*}
Mathematica [A] time = 0.0302483, size = 179, normalized size = 1.77 \[ d^2 \left (\frac{1}{5} c x^5 \left (12 a^2 c^2+39 a b^2 c+7 b^4\right )+\frac{1}{4} b x^4 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+\frac{1}{3} a x^3 \left (4 a^2 c^2+15 a b^2 c+3 b^4\right )+\frac{1}{2} a^2 b x^2 \left (4 a c+3 b^2\right )+a^3 b^2 x+\frac{1}{7} c^3 x^7 \left (12 a c+25 b^2\right )+\frac{1}{6} b c^2 x^6 \left (36 a c+19 b^2\right )+2 b c^4 x^8+\frac{4 c^5 x^9}{9}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 396, normalized size = 3.9 \begin{align*}{\frac{4\,{c}^{5}{d}^{2}{x}^{9}}{9}}+2\,b{d}^{2}{c}^{4}{x}^{8}+{\frac{ \left ( 13\,{b}^{2}{d}^{2}{c}^{3}+4\,{c}^{2}{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{3}{d}^{2}{c}^{2}+4\,b{d}^{2}c \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{c}^{2}{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,b{d}^{2}c \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{c}^{2}{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,b{d}^{2}c \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,{a}^{2}b{c}^{2}{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,{b}^{2}{d}^{2}c{a}^{2}+4\,{c}^{2}{d}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,b{d}^{2}c{a}^{3}+3\,{b}^{3}{d}^{2}{a}^{2} \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}{a}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14134, size = 267, normalized size = 2.64 \begin{align*} \frac{4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac{1}{7} \,{\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{7} + \frac{1}{6} \,{\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{6} + a^{3} b^{2} d^{2} x + \frac{1}{5} \,{\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{5} + \frac{1}{4} \,{\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70073, size = 500, normalized size = 4.95 \begin{align*} \frac{4}{9} x^{9} d^{2} c^{5} + 2 x^{8} d^{2} c^{4} b + \frac{25}{7} x^{7} d^{2} c^{3} b^{2} + \frac{12}{7} x^{7} d^{2} c^{4} a + \frac{19}{6} x^{6} d^{2} c^{2} b^{3} + 6 x^{6} d^{2} c^{3} b a + \frac{7}{5} x^{5} d^{2} c b^{4} + \frac{39}{5} x^{5} d^{2} c^{2} b^{2} a + \frac{12}{5} x^{5} d^{2} c^{3} a^{2} + \frac{1}{4} x^{4} d^{2} b^{5} + \frac{9}{2} x^{4} d^{2} c b^{3} a + 6 x^{4} d^{2} c^{2} b a^{2} + x^{3} d^{2} b^{4} a + 5 x^{3} d^{2} c b^{2} a^{2} + \frac{4}{3} x^{3} d^{2} c^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b^{3} a^{2} + 2 x^{2} d^{2} c b a^{3} + x d^{2} b^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.108754, size = 246, normalized size = 2.44 \begin{align*} a^{3} b^{2} d^{2} x + 2 b c^{4} d^{2} x^{8} + \frac{4 c^{5} d^{2} x^{9}}{9} + x^{7} \left (\frac{12 a c^{4} d^{2}}{7} + \frac{25 b^{2} c^{3} d^{2}}{7}\right ) + x^{6} \left (6 a b c^{3} d^{2} + \frac{19 b^{3} c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{12 a^{2} c^{3} d^{2}}{5} + \frac{39 a b^{2} c^{2} d^{2}}{5} + \frac{7 b^{4} c d^{2}}{5}\right ) + x^{4} \left (6 a^{2} b c^{2} d^{2} + \frac{9 a b^{3} c d^{2}}{2} + \frac{b^{5} d^{2}}{4}\right ) + x^{3} \left (\frac{4 a^{3} c^{2} d^{2}}{3} + 5 a^{2} b^{2} c d^{2} + a b^{4} d^{2}\right ) + x^{2} \left (2 a^{3} b c d^{2} + \frac{3 a^{2} b^{3} d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17085, size = 317, normalized size = 3.14 \begin{align*} \frac{4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac{25}{7} \, b^{2} c^{3} d^{2} x^{7} + \frac{12}{7} \, a c^{4} d^{2} x^{7} + \frac{19}{6} \, b^{3} c^{2} d^{2} x^{6} + 6 \, a b c^{3} d^{2} x^{6} + \frac{7}{5} \, b^{4} c d^{2} x^{5} + \frac{39}{5} \, a b^{2} c^{2} d^{2} x^{5} + \frac{12}{5} \, a^{2} c^{3} d^{2} x^{5} + \frac{1}{4} \, b^{5} d^{2} x^{4} + \frac{9}{2} \, a b^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{2} d^{2} x^{4} + a b^{4} d^{2} x^{3} + 5 \, a^{2} b^{2} c d^{2} x^{3} + \frac{4}{3} \, a^{3} c^{2} d^{2} x^{3} + \frac{3}{2} \, a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b c d^{2} x^{2} + a^{3} b^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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