Optimal. Leaf size=254 \[ a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
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Rubi [A] time = 0.33, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1657} \[ \frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx &=\int \left (a^4 A+4 a^3 A b x+a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^2+4 a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^3+\left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^4+4 b \left (b^2+3 a c\right ) (A c+a C) x^5+\left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^6+4 b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^7+c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^8+4 b c^3 C x^9+c^4 C x^{10}\right ) \, dx\\ &=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^4+\frac {1}{5} \left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^7+\frac {1}{2} b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 256, normalized size = 1.01 \[ a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+3 a A c+A b^2\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (6 a^2 c^2 C+4 a A c^3+12 a b^2 c C+6 A b^2 c^2+b^4 C\right )+\frac {1}{5} x^5 \left (4 a^3 c C+6 a^2 A c^2+6 a^2 b^2 C+12 a A b^2 c+A b^4\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (3 a c C+A c^2+b^2 C\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 308, normalized size = 1.21 \[ \frac {1}{11} x^{11} c^{4} C + \frac {2}{5} x^{10} c^{3} b C + \frac {2}{3} x^{9} c^{2} b^{2} C + \frac {4}{9} x^{9} c^{3} a C + \frac {1}{9} x^{9} c^{4} A + \frac {1}{2} x^{8} c b^{3} C + \frac {3}{2} x^{8} c^{2} b a C + \frac {1}{2} x^{8} c^{3} b A + \frac {1}{7} x^{7} b^{4} C + \frac {12}{7} x^{7} c b^{2} a C + \frac {6}{7} x^{7} c^{2} a^{2} C + \frac {6}{7} x^{7} c^{2} b^{2} A + \frac {4}{7} x^{7} c^{3} a A + \frac {2}{3} x^{6} b^{3} a C + 2 x^{6} c b a^{2} C + \frac {2}{3} x^{6} c b^{3} A + 2 x^{6} c^{2} b a A + \frac {6}{5} x^{5} b^{2} a^{2} C + \frac {4}{5} x^{5} c a^{3} C + \frac {1}{5} x^{5} b^{4} A + \frac {12}{5} x^{5} c b^{2} a A + \frac {6}{5} x^{5} c^{2} a^{2} A + x^{4} b a^{3} C + x^{4} b^{3} a A + 3 x^{4} c b a^{2} A + \frac {1}{3} x^{3} a^{4} C + 2 x^{3} b^{2} a^{2} A + \frac {4}{3} x^{3} c a^{3} A + 2 x^{2} b a^{3} A + x a^{4} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 308, normalized size = 1.21 \[ \frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {2}{3} \, C b^{2} c^{2} x^{9} + \frac {4}{9} \, C a c^{3} x^{9} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, C b^{3} c x^{8} + \frac {3}{2} \, C a b c^{2} x^{8} + \frac {1}{2} \, A b c^{3} x^{8} + \frac {1}{7} \, C b^{4} x^{7} + \frac {12}{7} \, C a b^{2} c x^{7} + \frac {6}{7} \, C a^{2} c^{2} x^{7} + \frac {6}{7} \, A b^{2} c^{2} x^{7} + \frac {4}{7} \, A a c^{3} x^{7} + \frac {2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac {2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac {6}{5} \, C a^{2} b^{2} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {4}{5} \, C a^{3} c x^{5} + \frac {12}{5} \, A a b^{2} c x^{5} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac {1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 343, normalized size = 1.35 \[ \frac {C \,c^{4} x^{11}}{11}+\frac {2 C b \,c^{3} x^{10}}{5}+\frac {\left (A \,c^{4}+\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) C \right ) x^{9}}{9}+\frac {\left (4 A b \,c^{3}+\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) C \right ) x^{8}}{8}+2 A \,a^{3} b \,x^{2}+\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) A +\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) C \right ) x^{7}}{7}+A \,a^{4} x +\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) A +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) C \right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) A +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) C \right ) x^{5}}{5}+\frac {\left (4 C \,a^{3} b +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) A \right ) x^{4}}{4}+\frac {\left (C \,a^{4}+\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) A \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 263, normalized size = 1.04 \[ \frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 244, normalized size = 0.96 \[ x^5\,\left (\frac {4\,C\,a^3\,c}{5}+\frac {6\,C\,a^2\,b^2}{5}+\frac {6\,A\,a^2\,c^2}{5}+\frac {12\,A\,a\,b^2\,c}{5}+\frac {A\,b^4}{5}\right )+x^7\,\left (\frac {6\,C\,a^2\,c^2}{7}+\frac {12\,C\,a\,b^2\,c}{7}+\frac {4\,A\,a\,c^3}{7}+\frac {C\,b^4}{7}+\frac {6\,A\,b^2\,c^2}{7}\right )+x^3\,\left (\frac {C\,a^4}{3}+\frac {4\,A\,c\,a^3}{3}+2\,A\,a^2\,b^2\right )+x^9\,\left (\frac {2\,C\,b^2\,c^2}{3}+\frac {A\,c^4}{9}+\frac {4\,C\,a\,c^3}{9}\right )+\frac {C\,c^4\,x^{11}}{11}+A\,a^4\,x+\frac {2\,b\,x^6\,\left (b^2+3\,a\,c\right )\,\left (A\,c+C\,a\right )}{3}+a\,b\,x^4\,\left (C\,a^2+3\,A\,c\,a+A\,b^2\right )+\frac {b\,c\,x^8\,\left (C\,b^2+A\,c^2+3\,C\,a\,c\right )}{2}+2\,A\,a^3\,b\,x^2+\frac {2\,C\,b\,c^3\,x^{10}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 320, normalized size = 1.26 \[ A a^{4} x + 2 A a^{3} b x^{2} + \frac {2 C b c^{3} x^{10}}{5} + \frac {C c^{4} x^{11}}{11} + x^{9} \left (\frac {A c^{4}}{9} + \frac {4 C a c^{3}}{9} + \frac {2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac {A b c^{3}}{2} + \frac {3 C a b c^{2}}{2} + \frac {C b^{3} c}{2}\right ) + x^{7} \left (\frac {4 A a c^{3}}{7} + \frac {6 A b^{2} c^{2}}{7} + \frac {6 C a^{2} c^{2}}{7} + \frac {12 C a b^{2} c}{7} + \frac {C b^{4}}{7}\right ) + x^{6} \left (2 A a b c^{2} + \frac {2 A b^{3} c}{3} + 2 C a^{2} b c + \frac {2 C a b^{3}}{3}\right ) + x^{5} \left (\frac {6 A a^{2} c^{2}}{5} + \frac {12 A a b^{2} c}{5} + \frac {A b^{4}}{5} + \frac {4 C a^{3} c}{5} + \frac {6 C a^{2} b^{2}}{5}\right ) + x^{4} \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \left (\frac {4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac {C a^{4}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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