Integrand size = 20, antiderivative size = 254 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^4+\frac {1}{5} \left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^7+\frac {1}{2} b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
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Time = 0.20 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1671} \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
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Rule 1671
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.01 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A b^2+3 a A c+a^2 C\right ) x^4+\frac {1}{5} \left (A b^4+12 a A b^2 c+6 a^2 A c^2+6 a^2 b^2 C+4 a^3 c C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (6 A b^2 c^2+4 a A c^3+b^4 C+12 a b^2 c C+6 a^2 c^2 C\right ) x^7+\frac {1}{2} b c \left (A c^2+b^2 C+3 a c C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \]
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Time = 0.70 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.05
method | result | size |
norman | \(\frac {c^{4} C \,x^{11}}{11}+\frac {2 b \,c^{3} C \,x^{10}}{5}+\left (\frac {1}{9} c^{4} A +\frac {4}{9} C a \,c^{3}+\frac {2}{3} C \,b^{2} c^{2}\right ) x^{9}+\left (\frac {1}{2} A b \,c^{3}+\frac {3}{2} C a b \,c^{2}+\frac {1}{2} C \,b^{3} c \right ) x^{8}+\left (\frac {4}{7} A a \,c^{3}+\frac {6}{7} A \,b^{2} c^{2}+\frac {6}{7} C \,a^{2} c^{2}+\frac {12}{7} C a \,b^{2} c +\frac {1}{7} C \,b^{4}\right ) x^{7}+\left (2 a A b \,c^{2}+\frac {2}{3} A \,b^{3} c +2 C \,a^{2} b c +\frac {2}{3} C a \,b^{3}\right ) x^{6}+\left (\frac {6}{5} A \,a^{2} c^{2}+\frac {12}{5} A a \,b^{2} c +\frac {1}{5} A \,b^{4}+\frac {4}{5} C \,a^{3} c +\frac {6}{5} C \,a^{2} b^{2}\right ) x^{5}+\left (3 A \,a^{2} b c +A \,b^{3} a +b \,a^{3} C \right ) x^{4}+\left (\frac {4}{3} A \,a^{3} c +2 A \,a^{2} b^{2}+\frac {1}{3} a^{4} C \right ) x^{3}+2 x^{2} A \,a^{3} b +a^{4} A x\) | \(266\) |
gosper | \(\frac {1}{3} x^{3} a^{4} C +\frac {1}{7} x^{7} C \,b^{4}+\frac {4}{3} a^{3} A c \,x^{3}+\frac {6}{5} a^{2} A \,c^{2} x^{5}+\frac {4}{7} a A \,c^{3} x^{7}+x^{4} A \,b^{3} a +2 x^{3} A \,a^{2} b^{2}+2 x^{2} A \,a^{3} b +a^{4} A x +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+C \,a^{3} b \,x^{4}+\frac {2}{3} x^{6} A \,b^{3} c +\frac {2}{3} x^{6} C a \,b^{3}+\frac {1}{2} x^{8} C \,b^{3} c +\frac {6}{7} x^{7} A \,b^{2} c^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {1}{9} A \,c^{4} x^{9}+\frac {12}{7} x^{7} C a \,b^{2} c +2 x^{6} a A b \,c^{2}+2 x^{6} C \,a^{2} b c +\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {1}{5} x^{5} A \,b^{4}+\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
risch | \(\frac {1}{3} x^{3} a^{4} C +\frac {1}{7} x^{7} C \,b^{4}+\frac {4}{3} a^{3} A c \,x^{3}+\frac {6}{5} a^{2} A \,c^{2} x^{5}+\frac {4}{7} a A \,c^{3} x^{7}+x^{4} A \,b^{3} a +2 x^{3} A \,a^{2} b^{2}+2 x^{2} A \,a^{3} b +a^{4} A x +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+C \,a^{3} b \,x^{4}+\frac {2}{3} x^{6} A \,b^{3} c +\frac {2}{3} x^{6} C a \,b^{3}+\frac {1}{2} x^{8} C \,b^{3} c +\frac {6}{7} x^{7} A \,b^{2} c^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {1}{9} A \,c^{4} x^{9}+\frac {12}{7} x^{7} C a \,b^{2} c +2 x^{6} a A b \,c^{2}+2 x^{6} C \,a^{2} b c +\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {1}{5} x^{5} A \,b^{4}+\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
parallelrisch | \(\frac {1}{3} x^{3} a^{4} C +\frac {1}{7} x^{7} C \,b^{4}+\frac {4}{3} a^{3} A c \,x^{3}+\frac {6}{5} a^{2} A \,c^{2} x^{5}+\frac {4}{7} a A \,c^{3} x^{7}+x^{4} A \,b^{3} a +2 x^{3} A \,a^{2} b^{2}+2 x^{2} A \,a^{3} b +a^{4} A x +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+C \,a^{3} b \,x^{4}+\frac {2}{3} x^{6} A \,b^{3} c +\frac {2}{3} x^{6} C a \,b^{3}+\frac {1}{2} x^{8} C \,b^{3} c +\frac {6}{7} x^{7} A \,b^{2} c^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {1}{9} A \,c^{4} x^{9}+\frac {12}{7} x^{7} C a \,b^{2} c +2 x^{6} a A b \,c^{2}+2 x^{6} C \,a^{2} b c +\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {1}{5} x^{5} A \,b^{4}+\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
default | \(\frac {c^{4} C \,x^{11}}{11}+\frac {2 b \,c^{3} C \,x^{10}}{5}+\frac {\left (\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) C +c^{4} A \right ) x^{9}}{9}+\frac {\left (\left (4 b a \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) C +4 A b \,c^{3}\right ) x^{8}}{8}+\frac {\left (\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) C +\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) A \right ) x^{7}}{7}+\frac {\left (\left (4 a^{2} b c +4 b a \left (2 a c +b^{2}\right )\right ) C +\left (4 b a \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) A \right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} \left (2 a c +b^{2}\right )+4 b^{2} a^{2}\right ) C +\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (4 b \,a^{3} C +\left (4 a^{2} b c +4 b a \left (2 a c +b^{2}\right )\right ) A \right ) x^{4}}{4}+\frac {\left (a^{4} C +\left (2 a^{2} \left (2 a c +b^{2}\right )+4 b^{2} a^{2}\right ) A \right ) x^{3}}{3}+2 x^{2} A \,a^{3} b +a^{4} A x\) | \(343\) |
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Time = 0.26 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \]
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Time = 0.04 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.26 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=A a^{4} x + 2 A a^{3} b x^{2} + \frac {2 C b c^{3} x^{10}}{5} + \frac {C c^{4} x^{11}}{11} + x^{9} \left (\frac {A c^{4}}{9} + \frac {4 C a c^{3}}{9} + \frac {2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac {A b c^{3}}{2} + \frac {3 C a b c^{2}}{2} + \frac {C b^{3} c}{2}\right ) + x^{7} \cdot \left (\frac {4 A a c^{3}}{7} + \frac {6 A b^{2} c^{2}}{7} + \frac {6 C a^{2} c^{2}}{7} + \frac {12 C a b^{2} c}{7} + \frac {C b^{4}}{7}\right ) + x^{6} \cdot \left (2 A a b c^{2} + \frac {2 A b^{3} c}{3} + 2 C a^{2} b c + \frac {2 C a b^{3}}{3}\right ) + x^{5} \cdot \left (\frac {6 A a^{2} c^{2}}{5} + \frac {12 A a b^{2} c}{5} + \frac {A b^{4}}{5} + \frac {4 C a^{3} c}{5} + \frac {6 C a^{2} b^{2}}{5}\right ) + x^{4} \cdot \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \cdot \left (\frac {4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac {C a^{4}}{3}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \]
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Time = 0.26 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.21 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {2}{3} \, C b^{2} c^{2} x^{9} + \frac {4}{9} \, C a c^{3} x^{9} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, C b^{3} c x^{8} + \frac {3}{2} \, C a b c^{2} x^{8} + \frac {1}{2} \, A b c^{3} x^{8} + \frac {1}{7} \, C b^{4} x^{7} + \frac {12}{7} \, C a b^{2} c x^{7} + \frac {6}{7} \, C a^{2} c^{2} x^{7} + \frac {6}{7} \, A b^{2} c^{2} x^{7} + \frac {4}{7} \, A a c^{3} x^{7} + \frac {2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac {2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac {6}{5} \, C a^{2} b^{2} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {4}{5} \, C a^{3} c x^{5} + \frac {12}{5} \, A a b^{2} c x^{5} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac {1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \]
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Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.96 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=x^5\,\left (\frac {4\,C\,a^3\,c}{5}+\frac {6\,C\,a^2\,b^2}{5}+\frac {6\,A\,a^2\,c^2}{5}+\frac {12\,A\,a\,b^2\,c}{5}+\frac {A\,b^4}{5}\right )+x^7\,\left (\frac {6\,C\,a^2\,c^2}{7}+\frac {12\,C\,a\,b^2\,c}{7}+\frac {4\,A\,a\,c^3}{7}+\frac {C\,b^4}{7}+\frac {6\,A\,b^2\,c^2}{7}\right )+x^3\,\left (\frac {C\,a^4}{3}+\frac {4\,A\,c\,a^3}{3}+2\,A\,a^2\,b^2\right )+x^9\,\left (\frac {2\,C\,b^2\,c^2}{3}+\frac {A\,c^4}{9}+\frac {4\,C\,a\,c^3}{9}\right )+\frac {C\,c^4\,x^{11}}{11}+A\,a^4\,x+\frac {2\,b\,x^6\,\left (b^2+3\,a\,c\right )\,\left (A\,c+C\,a\right )}{3}+a\,b\,x^4\,\left (C\,a^2+3\,A\,c\,a+A\,b^2\right )+\frac {b\,c\,x^8\,\left (C\,b^2+A\,c^2+3\,C\,a\,c\right )}{2}+2\,A\,a^3\,b\,x^2+\frac {2\,C\,b\,c^3\,x^{10}}{5} \]
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