Integrand size = 14, antiderivative size = 49 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=a^2 x+\frac {2}{3} a b x^3+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1104} \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=a^2 x+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {2}{3} a b x^3+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
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Rule 1104
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2+2 a b x^2+b^2 \left (1+\frac {2 a c}{b^2}\right ) x^4+2 b c x^6+c^2 x^8\right ) \, dx \\ & = a^2 x+\frac {2}{3} a b x^3+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=a^2 x+\frac {2}{3} a b x^3+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
default | \(a^{2} x +\frac {2 a b \,x^{3}}{3}+\frac {\left (2 a c +b^{2}\right ) x^{5}}{5}+\frac {2 b c \,x^{7}}{7}+\frac {c^{2} x^{9}}{9}\) | \(42\) |
norman | \(\frac {c^{2} x^{9}}{9}+\frac {2 b c \,x^{7}}{7}+\left (\frac {2 a c}{5}+\frac {b^{2}}{5}\right ) x^{5}+\frac {2 a b \,x^{3}}{3}+a^{2} x\) | \(43\) |
gosper | \(\frac {1}{9} c^{2} x^{9}+\frac {2}{7} b c \,x^{7}+\frac {2}{5} x^{5} a c +\frac {1}{5} b^{2} x^{5}+\frac {2}{3} a b \,x^{3}+a^{2} x\) | \(44\) |
risch | \(\frac {1}{9} c^{2} x^{9}+\frac {2}{7} b c \,x^{7}+\frac {2}{5} x^{5} a c +\frac {1}{5} b^{2} x^{5}+\frac {2}{3} a b \,x^{3}+a^{2} x\) | \(44\) |
parallelrisch | \(\frac {1}{9} c^{2} x^{9}+\frac {2}{7} b c \,x^{7}+\frac {2}{5} x^{5} a c +\frac {1}{5} b^{2} x^{5}+\frac {2}{3} a b \,x^{3}+a^{2} x\) | \(44\) |
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Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} x + \frac {2 a b x^{3}}{3} + \frac {2 b c x^{7}}{7} + \frac {c^{2} x^{9}}{9} + x^{5} \cdot \left (\frac {2 a c}{5} + \frac {b^{2}}{5}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, b^{2} x^{5} + a^{2} x + \frac {2}{15} \, {\left (3 \, c x^{5} + 5 \, b x^{3}\right )} a \]
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, b^{2} x^{5} + \frac {2}{5} \, a c x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^2+c x^4\right )^2 \, dx=a^2\,x+x^5\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+\frac {c^2\,x^9}{9}+\frac {2\,a\,b\,x^3}{3}+\frac {2\,b\,c\,x^7}{7} \]
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