Integrand size = 22, antiderivative size = 54 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^7}{7}+\frac {1}{4} a b x^8+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {1}{5} b c x^{10}+\frac {c^2 x^{11}}{11} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1599, 712} \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^7}{7}+\frac {1}{9} x^9 \left (2 a c+b^2\right )+\frac {1}{4} a b x^8+\frac {1}{5} b c x^{10}+\frac {c^2 x^{11}}{11} \]
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Rule 712
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int x^6 \left (a+b x+c x^2\right )^2 \, dx \\ & = \int \left (a^2 x^6+2 a b x^7+\left (b^2+2 a c\right ) x^8+2 b c x^9+c^2 x^{10}\right ) \, dx \\ & = \frac {a^2 x^7}{7}+\frac {1}{4} a b x^8+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {1}{5} b c x^{10}+\frac {c^2 x^{11}}{11} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^7}{7}+\frac {1}{4} a b x^8+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {1}{5} b c x^{10}+\frac {c^2 x^{11}}{11} \]
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Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {a^{2} x^{7}}{7}+\frac {a b \,x^{8}}{4}+\frac {\left (2 a c +b^{2}\right ) x^{9}}{9}+\frac {b c \,x^{10}}{5}+\frac {c^{2} x^{11}}{11}\) | \(45\) |
norman | \(\frac {c^{2} x^{11}}{11}+\frac {b c \,x^{10}}{5}+\left (\frac {2 a c}{9}+\frac {b^{2}}{9}\right ) x^{9}+\frac {a b \,x^{8}}{4}+\frac {a^{2} x^{7}}{7}\) | \(46\) |
gosper | \(\frac {x^{7} \left (1260 c^{2} x^{4}+2772 b c \,x^{3}+3080 a c \,x^{2}+1540 b^{2} x^{2}+3465 a b x +1980 a^{2}\right )}{13860}\) | \(47\) |
risch | \(\frac {1}{7} a^{2} x^{7}+\frac {1}{4} a b \,x^{8}+\frac {2}{9} x^{9} a c +\frac {1}{9} b^{2} x^{9}+\frac {1}{5} b c \,x^{10}+\frac {1}{11} c^{2} x^{11}\) | \(47\) |
parallelrisch | \(\frac {1}{7} a^{2} x^{7}+\frac {1}{4} a b \,x^{8}+\frac {2}{9} x^{9} a c +\frac {1}{9} b^{2} x^{9}+\frac {1}{5} b c \,x^{10}+\frac {1}{11} c^{2} x^{11}\) | \(47\) |
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Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {1}{5} \, b c x^{10} + \frac {1}{4} \, a b x^{8} + \frac {1}{9} \, {\left (b^{2} + 2 \, a c\right )} x^{9} + \frac {1}{7} \, a^{2} x^{7} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^{2} x^{7}}{7} + \frac {a b x^{8}}{4} + \frac {b c x^{10}}{5} + \frac {c^{2} x^{11}}{11} + x^{9} \cdot \left (\frac {2 a c}{9} + \frac {b^{2}}{9}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {1}{5} \, b c x^{10} + \frac {1}{4} \, a b x^{8} + \frac {1}{9} \, {\left (b^{2} + 2 \, a c\right )} x^{9} + \frac {1}{7} \, a^{2} x^{7} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {1}{5} \, b c x^{10} + \frac {1}{9} \, b^{2} x^{9} + \frac {2}{9} \, a c x^{9} + \frac {1}{4} \, a b x^{8} + \frac {1}{7} \, a^{2} x^{7} \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx=x^9\,\left (\frac {b^2}{9}+\frac {2\,a\,c}{9}\right )+\frac {a^2\,x^7}{7}+\frac {c^2\,x^{11}}{11}+\frac {a\,b\,x^8}{4}+\frac {b\,c\,x^{10}}{5} \]
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