Integrand size = 24, antiderivative size = 94 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^2 x+\frac {1}{3} e (b c e+a d e+2 a c f) x^3+\frac {1}{5} (a f (2 d e+c f)+b e (d e+2 c f)) x^5+\frac {1}{7} f (2 b d e+b c f+a d f) x^7+\frac {1}{9} b d f^2 x^9 \]
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Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {535} \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{7} f x^7 (a d f+b c f+2 b d e)+\frac {1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac {1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac {1}{9} b d f^2 x^9 \]
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Rule 535
Rubi steps \begin{align*} \text {integral}& = \int \left (a c e^2+e (b c e+a d e+2 a c f) x^2+(a f (2 d e+c f)+b e (d e+2 c f)) x^4+f (2 b d e+b c f+a d f) x^6+b d f^2 x^8\right ) \, dx \\ & = a c e^2 x+\frac {1}{3} e (b c e+a d e+2 a c f) x^3+\frac {1}{5} (a f (2 d e+c f)+b e (d e+2 c f)) x^5+\frac {1}{7} f (2 b d e+b c f+a d f) x^7+\frac {1}{9} b d f^2 x^9 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^2 x+\frac {1}{3} e (b c e+a d e+2 a c f) x^3+\frac {1}{5} \left (b d e^2+2 b c e f+2 a d e f+a c f^2\right ) x^5+\frac {1}{7} f (2 b d e+b c f+a d f) x^7+\frac {1}{9} b d f^2 x^9 \]
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Time = 3.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {b d \,f^{2} x^{9}}{9}+\frac {\left (\left (a d +b c \right ) f^{2}+2 e b d f \right ) x^{7}}{7}+\frac {\left (a c \,f^{2}+2 \left (a d +b c \right ) e f +b d \,e^{2}\right ) x^{5}}{5}+\frac {\left (2 a c e f +\left (a d +b c \right ) e^{2}\right ) x^{3}}{3}+a c \,e^{2} x\) | \(94\) |
norman | \(\frac {b d \,f^{2} x^{9}}{9}+\left (\frac {1}{7} a d \,f^{2}+\frac {1}{7} b c \,f^{2}+\frac {2}{7} e b d f \right ) x^{7}+\left (\frac {1}{5} a c \,f^{2}+\frac {2}{5} a d e f +\frac {2}{5} b c e f +\frac {1}{5} b d \,e^{2}\right ) x^{5}+\left (\frac {2}{3} a c e f +\frac {1}{3} a d \,e^{2}+\frac {1}{3} b c \,e^{2}\right ) x^{3}+a c \,e^{2} x\) | \(100\) |
gosper | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} e b d f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
risch | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} e b d f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
parallelrisch | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} e b d f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {1}{7} \, {\left (2 \, b d e f + {\left (b c + a d\right )} f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b d e^{2} + a c f^{2} + 2 \, {\left (b c + a d\right )} e f\right )} x^{5} + a c e^{2} x + \frac {1}{3} \, {\left (2 \, a c e f + {\left (b c + a d\right )} e^{2}\right )} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^{2} x + \frac {b d f^{2} x^{9}}{9} + x^{7} \left (\frac {a d f^{2}}{7} + \frac {b c f^{2}}{7} + \frac {2 b d e f}{7}\right ) + x^{5} \left (\frac {a c f^{2}}{5} + \frac {2 a d e f}{5} + \frac {2 b c e f}{5} + \frac {b d e^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a c e f}{3} + \frac {a d e^{2}}{3} + \frac {b c e^{2}}{3}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {1}{7} \, {\left (2 \, b d e f + {\left (b c + a d\right )} f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b d e^{2} + a c f^{2} + 2 \, {\left (b c + a d\right )} e f\right )} x^{5} + a c e^{2} x + \frac {1}{3} \, {\left (2 \, a c e f + {\left (b c + a d\right )} e^{2}\right )} x^{3} \]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.21 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {2}{7} \, b d e f x^{7} + \frac {1}{7} \, b c f^{2} x^{7} + \frac {1}{7} \, a d f^{2} x^{7} + \frac {1}{5} \, b d e^{2} x^{5} + \frac {2}{5} \, b c e f x^{5} + \frac {2}{5} \, a d e f x^{5} + \frac {1}{5} \, a c f^{2} x^{5} + \frac {1}{3} \, b c e^{2} x^{3} + \frac {1}{3} \, a d e^{2} x^{3} + \frac {2}{3} \, a c e f x^{3} + a c e^{2} x \]
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Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=x^5\,\left (\frac {a\,c\,f^2}{5}+\frac {b\,d\,e^2}{5}+\frac {2\,a\,d\,e\,f}{5}+\frac {2\,b\,c\,e\,f}{5}\right )+x^3\,\left (\frac {a\,d\,e^2}{3}+\frac {b\,c\,e^2}{3}+\frac {2\,a\,c\,e\,f}{3}\right )+x^7\,\left (\frac {a\,d\,f^2}{7}+\frac {b\,c\,f^2}{7}+\frac {2\,b\,d\,e\,f}{7}\right )+a\,c\,e^2\,x+\frac {b\,d\,f^2\,x^9}{9} \]
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