Integrand size = 12, antiderivative size = 133 \[ \int \left (a+b x+c x^2\right )^4 \, dx=a^4 x+2 a^3 b x^2+\frac {2}{3} a^2 \left (3 b^2+2 a c\right ) x^3+a b \left (b^2+3 a c\right ) x^4+\frac {1}{5} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) x^5+\frac {2}{3} b c \left (b^2+3 a c\right ) x^6+\frac {2}{7} c^2 \left (3 b^2+2 a c\right ) x^7+\frac {1}{2} b c^3 x^8+\frac {c^4 x^9}{9} \]
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Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {625} \[ \int \left (a+b x+c x^2\right )^4 \, dx=a^4 x+2 a^3 b x^2+\frac {2}{3} a^2 x^3 \left (2 a c+3 b^2\right )+\frac {1}{5} x^5 \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac {2}{7} c^2 x^7 \left (2 a c+3 b^2\right )+\frac {2}{3} b c x^6 \left (3 a c+b^2\right )+a b x^4 \left (3 a c+b^2\right )+\frac {1}{2} b c^3 x^8+\frac {c^4 x^9}{9} \]
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Rule 625
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4+4 a^3 b x+6 a^2 b^2 \left (1+\frac {2 a c}{3 b^2}\right ) x^2+4 a b^3 \left (1+\frac {3 a c}{b^2}\right ) x^3+b^4 \left (1+\frac {6 a c \left (2 b^2+a c\right )}{b^4}\right ) x^4+4 b^3 c \left (1+\frac {3 a c}{b^2}\right ) x^5+6 b^2 c^2 \left (1+\frac {2 a c}{3 b^2}\right ) x^6+4 b c^3 x^7+c^4 x^8\right ) \, dx \\ & = a^4 x+2 a^3 b x^2+\frac {2}{3} a^2 \left (3 b^2+2 a c\right ) x^3+a b \left (b^2+3 a c\right ) x^4+\frac {1}{5} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) x^5+\frac {2}{3} b c \left (b^2+3 a c\right ) x^6+\frac {2}{7} c^2 \left (3 b^2+2 a c\right ) x^7+\frac {1}{2} b c^3 x^8+\frac {c^4 x^9}{9} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2\right )^4 \, dx=a^4 x+2 a^3 b x^2+\frac {2}{3} a^2 \left (3 b^2+2 a c\right ) x^3+a b \left (b^2+3 a c\right ) x^4+\frac {1}{5} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) x^5+\frac {2}{3} b c \left (b^2+3 a c\right ) x^6+\frac {2}{7} c^2 \left (3 b^2+2 a c\right ) x^7+\frac {1}{2} b c^3 x^8+\frac {c^4 x^9}{9} \]
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Time = 2.91 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {c^{4} x^{9}}{9}+\frac {b \,c^{3} x^{8}}{2}+\left (\frac {4}{7} c^{3} a +\frac {6}{7} b^{2} c^{2}\right ) x^{7}+\left (2 b \,c^{2} a +\frac {2}{3} b^{3} c \right ) x^{6}+\left (\frac {6}{5} a^{2} c^{2}+\frac {12}{5} a \,b^{2} c +\frac {1}{5} b^{4}\right ) x^{5}+\left (3 a^{2} b c +a \,b^{3}\right ) x^{4}+\left (\frac {4}{3} c \,a^{3}+2 a^{2} b^{2}\right ) x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(131\) |
gosper | \(\frac {1}{9} c^{4} x^{9}+\frac {1}{2} b \,c^{3} x^{8}+\frac {4}{7} a \,c^{3} x^{7}+\frac {6}{7} x^{7} b^{2} c^{2}+2 x^{6} b \,c^{2} a +\frac {2}{3} b^{3} c \,x^{6}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {12}{5} a \,b^{2} c \,x^{5}+\frac {1}{5} b^{4} x^{5}+3 a^{2} b c \,x^{4}+a \,b^{3} x^{4}+\frac {4}{3} c \,a^{3} x^{3}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(139\) |
risch | \(\frac {1}{9} c^{4} x^{9}+\frac {1}{2} b \,c^{3} x^{8}+\frac {4}{7} a \,c^{3} x^{7}+\frac {6}{7} x^{7} b^{2} c^{2}+2 x^{6} b \,c^{2} a +\frac {2}{3} b^{3} c \,x^{6}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {12}{5} a \,b^{2} c \,x^{5}+\frac {1}{5} b^{4} x^{5}+3 a^{2} b c \,x^{4}+a \,b^{3} x^{4}+\frac {4}{3} c \,a^{3} x^{3}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(139\) |
parallelrisch | \(\frac {1}{9} c^{4} x^{9}+\frac {1}{2} b \,c^{3} x^{8}+\frac {4}{7} a \,c^{3} x^{7}+\frac {6}{7} x^{7} b^{2} c^{2}+2 x^{6} b \,c^{2} a +\frac {2}{3} b^{3} c \,x^{6}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {12}{5} a \,b^{2} c \,x^{5}+\frac {1}{5} b^{4} x^{5}+3 a^{2} b c \,x^{4}+a \,b^{3} x^{4}+\frac {4}{3} c \,a^{3} x^{3}+2 a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(139\) |
default | \(\frac {c^{4} x^{9}}{9}+\frac {b \,c^{3} x^{8}}{2}+\frac {\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right ) x^{6}}{6}+\frac {\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right ) x^{3}}{3}+2 a^{3} b \,x^{2}+a^{4} x\) | \(168\) |
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Time = 0.35 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{9} \, c^{4} x^{9} + \frac {1}{2} \, b c^{3} x^{8} + \frac {2}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{7} + \frac {2}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{6} + 2 \, a^{3} b x^{2} + \frac {1}{5} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{5} + a^{4} x + {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06 \[ \int \left (a+b x+c x^2\right )^4 \, dx=a^{4} x + 2 a^{3} b x^{2} + \frac {b c^{3} x^{8}}{2} + \frac {c^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {4 a c^{3}}{7} + \frac {6 b^{2} c^{2}}{7}\right ) + x^{6} \cdot \left (2 a b c^{2} + \frac {2 b^{3} c}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} c^{2}}{5} + \frac {12 a b^{2} c}{5} + \frac {b^{4}}{5}\right ) + x^{4} \cdot \left (3 a^{2} b c + a b^{3}\right ) + x^{3} \cdot \left (\frac {4 a^{3} c}{3} + 2 a^{2} b^{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.02 \[ \int \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{9} \, c^{4} x^{9} + \frac {1}{2} \, b c^{3} x^{8} + \frac {6}{7} \, b^{2} c^{2} x^{7} + \frac {2}{3} \, b^{3} c x^{6} + \frac {1}{5} \, b^{4} x^{5} + a^{4} x + \frac {2}{3} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{3} + \frac {1}{5} \, {\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a^{2} + \frac {1}{35} \, {\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} a \]
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.04 \[ \int \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{9} \, c^{4} x^{9} + \frac {1}{2} \, b c^{3} x^{8} + \frac {6}{7} \, b^{2} c^{2} x^{7} + \frac {4}{7} \, a c^{3} x^{7} + \frac {2}{3} \, b^{3} c x^{6} + 2 \, a b c^{2} x^{6} + \frac {1}{5} \, b^{4} x^{5} + \frac {12}{5} \, a b^{2} c x^{5} + \frac {6}{5} \, a^{2} c^{2} x^{5} + a b^{3} x^{4} + 3 \, a^{2} b c x^{4} + 2 \, a^{2} b^{2} x^{3} + \frac {4}{3} \, a^{3} c x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \]
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Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93 \[ \int \left (a+b x+c x^2\right )^4 \, dx=x^5\,\left (\frac {6\,a^2\,c^2}{5}+\frac {12\,a\,b^2\,c}{5}+\frac {b^4}{5}\right )+a^4\,x+\frac {c^4\,x^9}{9}+x^3\,\left (\frac {4\,c\,a^3}{3}+2\,a^2\,b^2\right )+x^7\,\left (\frac {6\,b^2\,c^2}{7}+\frac {4\,a\,c^3}{7}\right )+2\,a^3\,b\,x^2+\frac {b\,c^3\,x^8}{2}+a\,b\,x^4\,\left (b^2+3\,a\,c\right )+\frac {2\,b\,c\,x^6\,\left (b^2+3\,a\,c\right )}{3} \]
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