Integrand size = 25, antiderivative size = 154 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \]
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Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1671} \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{2} a b e x^4+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{4} b c e x^8+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \]
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Rule 1671
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 d+a^2 e x+a (2 b d+a f) x^2+2 a b e x^3+\left (b^2 d+2 a c d+2 a b f\right ) x^4+\left (b^2+2 a c\right ) e x^5+\left (2 b c d+b^2 f+2 a c f\right ) x^6+2 b c e x^7+c (c d+2 b f) x^8+c^2 e x^9+c^2 f x^{10}\right ) \, dx \\ & = a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \]
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Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {c^{2} f \,x^{11}}{11}+\frac {c^{2} e \,x^{10}}{10}+\frac {\left (2 f b c +c^{2} d \right ) x^{9}}{9}+\frac {b c e \,x^{8}}{4}+\frac {\left (2 b c d +f \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 a c +b^{2}\right ) e \,x^{6}}{6}+\frac {\left (d \left (2 a c +b^{2}\right )+2 a b f \right ) x^{5}}{5}+\frac {a b e \,x^{4}}{2}+\frac {\left (f \,a^{2}+2 d a b \right ) x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(139\) |
norman | \(\frac {c^{2} f \,x^{11}}{11}+\frac {c^{2} e \,x^{10}}{10}+\left (\frac {2}{9} f b c +\frac {1}{9} c^{2} d \right ) x^{9}+\frac {b c e \,x^{8}}{4}+\left (\frac {2}{7} a c f +\frac {1}{7} b^{2} f +\frac {2}{7} b c d \right ) x^{7}+\left (\frac {1}{3} a c e +\frac {1}{6} b^{2} e \right ) x^{6}+\left (\frac {2}{5} a b f +\frac {2}{5} a c d +\frac {1}{5} b^{2} d \right ) x^{5}+\frac {a b e \,x^{4}}{2}+\left (\frac {1}{3} f \,a^{2}+\frac {2}{3} d a b \right ) x^{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(141\) |
gosper | \(\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(152\) |
risch | \(\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(152\) |
parallelrisch | \(\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(152\) |
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Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{4} \, b c e x^{8} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {a b e x^{4}}{2} + \frac {b c e x^{8}}{4} + \frac {c^{2} e x^{10}}{10} + \frac {c^{2} f x^{11}}{11} + x^{9} \cdot \left (\frac {2 b c f}{9} + \frac {c^{2} d}{9}\right ) + x^{7} \cdot \left (\frac {2 a c f}{7} + \frac {b^{2} f}{7} + \frac {2 b c d}{7}\right ) + x^{6} \left (\frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \cdot \left (\frac {2 a b f}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b d}{3}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{4} \, b c e x^{8} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \]
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Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.98 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c f x^{9} + \frac {1}{4} \, b c e x^{8} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} f x^{7} + \frac {2}{7} \, a c f x^{7} + \frac {1}{6} \, b^{2} e x^{6} + \frac {1}{3} \, a c e x^{6} + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b f x^{5} + \frac {1}{2} \, a b e x^{4} + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \]
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Time = 7.85 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,e\,x^{10}}{10}+\frac {c^2\,f\,x^{11}}{11}+\frac {e\,x^6\,\left (b^2+2\,a\,c\right )}{6}+a^2\,d\,x+\frac {a\,b\,e\,x^4}{2}+\frac {b\,c\,e\,x^8}{4} \]
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