Optimal. Leaf size=161 \[ \frac{1}{4} b x^4 \left (3 a^2 C+A \left (6 a c+b^2\right )\right )+\frac{1}{3} a x^3 \left (a^2 C+3 A \left (a c+b^2\right )\right )+\frac{3}{2} a^2 A b x^2+a^3 A x+\frac{1}{7} c x^7 \left (3 C \left (a c+b^2\right )+A c^2\right )+\frac{1}{6} b x^6 \left (C \left (6 a c+b^2\right )+3 A c^2\right )+\frac{3}{5} x^5 \left (a c+b^2\right ) (a C+A c)+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9 \]
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Rubi [A] time = 0.193015, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1657} \[ \frac{1}{4} b x^4 \left (3 a^2 C+A \left (6 a c+b^2\right )\right )+\frac{1}{3} a x^3 \left (a^2 C+3 A \left (a c+b^2\right )\right )+\frac{3}{2} a^2 A b x^2+a^3 A x+\frac{1}{7} c x^7 \left (3 C \left (a c+b^2\right )+A c^2\right )+\frac{1}{6} b x^6 \left (C \left (6 a c+b^2\right )+3 A c^2\right )+\frac{3}{5} x^5 \left (a c+b^2\right ) (a C+A c)+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9 \]
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right )^3 \left (A+C x^2\right ) \, dx &=\int \left (a^3 A+3 a^2 A b x+a \left (3 A \left (b^2+a c\right )+a^2 C\right ) x^2+b \left (A \left (b^2+6 a c\right )+3 a^2 C\right ) x^3+3 \left (b^2+a c\right ) (A c+a C) x^4+b \left (3 A c^2+\left (b^2+6 a c\right ) C\right ) x^5+c \left (A c^2+3 \left (b^2+a c\right ) C\right ) x^6+3 b c^2 C x^7+c^3 C x^8\right ) \, dx\\ &=a^3 A x+\frac{3}{2} a^2 A b x^2+\frac{1}{3} a \left (3 A \left (b^2+a c\right )+a^2 C\right ) x^3+\frac{1}{4} b \left (A \left (b^2+6 a c\right )+3 a^2 C\right ) x^4+\frac{3}{5} \left (b^2+a c\right ) (A c+a C) x^5+\frac{1}{6} b \left (3 A c^2+\left (b^2+6 a c\right ) C\right ) x^6+\frac{1}{7} c \left (A c^2+3 \left (b^2+a c\right ) C\right ) x^7+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9\\ \end{align*}
Mathematica [A] time = 0.0444664, size = 163, normalized size = 1.01 \[ \frac{1}{4} b x^4 \left (3 a^2 C+6 a A c+A b^2\right )+\frac{1}{3} a x^3 \left (a^2 C+3 a A c+3 A b^2\right )+\frac{3}{2} a^2 A b x^2+a^3 A x+\frac{1}{7} c x^7 \left (3 a c C+A c^2+3 b^2 C\right )+\frac{1}{6} b x^6 \left (6 a c C+3 A c^2+b^2 C\right )+\frac{3}{5} x^5 \left (a c+b^2\right ) (a C+A c)+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 223, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}C{x}^{9}}{9}}+{\frac{3\,b{c}^{2}C{x}^{8}}{8}}+{\frac{ \left ( \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) C+A{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) C+3\,b{c}^{2}A \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) C+ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,b{a}^{2}C+ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}C+ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) A \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}Ab{x}^{2}}{2}}+{a}^{3}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976297, size = 223, normalized size = 1.39 \begin{align*} \frac{1}{9} \, C c^{3} x^{9} + \frac{3}{8} \, C b c^{2} x^{8} + \frac{1}{7} \,{\left (3 \, C b^{2} c + 3 \, C a c^{2} + A c^{3}\right )} x^{7} + \frac{1}{6} \,{\left (C b^{3} + 6 \, C a b c + 3 \, A b c^{2}\right )} x^{6} + \frac{3}{2} \, A a^{2} b x^{2} + \frac{3}{5} \,{\left (C a b^{2} + A a c^{2} +{\left (C a^{2} + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, C a^{2} b + A b^{3} + 6 \, A a b c\right )} x^{4} + \frac{1}{3} \,{\left (C a^{3} + 3 \, A a b^{2} + 3 \, A a^{2} c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30706, size = 444, normalized size = 2.76 \begin{align*} \frac{1}{9} x^{9} c^{3} C + \frac{3}{8} x^{8} c^{2} b C + \frac{3}{7} x^{7} c b^{2} C + \frac{3}{7} x^{7} c^{2} a C + \frac{1}{7} x^{7} c^{3} A + \frac{1}{6} x^{6} b^{3} C + x^{6} c b a C + \frac{1}{2} x^{6} c^{2} b A + \frac{3}{5} x^{5} b^{2} a C + \frac{3}{5} x^{5} c a^{2} C + \frac{3}{5} x^{5} c b^{2} A + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} b a^{2} C + \frac{1}{4} x^{4} b^{3} A + \frac{3}{2} x^{4} c b a A + \frac{1}{3} x^{3} a^{3} C + x^{3} b^{2} a A + x^{3} c a^{2} A + \frac{3}{2} x^{2} b a^{2} A + x a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09245, size = 197, normalized size = 1.22 \begin{align*} A a^{3} x + \frac{3 A a^{2} b x^{2}}{2} + \frac{3 C b c^{2} x^{8}}{8} + \frac{C c^{3} x^{9}}{9} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 C a c^{2}}{7} + \frac{3 C b^{2} c}{7}\right ) + x^{6} \left (\frac{A b c^{2}}{2} + C a b c + \frac{C b^{3}}{6}\right ) + x^{5} \left (\frac{3 A a c^{2}}{5} + \frac{3 A b^{2} c}{5} + \frac{3 C a^{2} c}{5} + \frac{3 C a b^{2}}{5}\right ) + x^{4} \left (\frac{3 A a b c}{2} + \frac{A b^{3}}{4} + \frac{3 C a^{2} b}{4}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + \frac{C a^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21464, size = 252, normalized size = 1.57 \begin{align*} \frac{1}{9} \, C c^{3} x^{9} + \frac{3}{8} \, C b c^{2} x^{8} + \frac{3}{7} \, C b^{2} c x^{7} + \frac{3}{7} \, C a c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{6} \, C b^{3} x^{6} + C a b c x^{6} + \frac{1}{2} \, A b c^{2} x^{6} + \frac{3}{5} \, C a b^{2} x^{5} + \frac{3}{5} \, C a^{2} c x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, C a^{2} b x^{4} + \frac{1}{4} \, A b^{3} x^{4} + \frac{3}{2} \, A a b c x^{4} + \frac{1}{3} \, C a^{3} x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + \frac{3}{2} \, A a^{2} b x^{2} + A a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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