Integrand size = 24, antiderivative size = 223 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^3 x+\frac {1}{3} a d^2 (2 b d+3 a e) x^3+\frac {1}{5} d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{7} \left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^7+\frac {1}{9} \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{11} e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{11}+\frac {1}{13} c e^2 (3 c d+2 b e) x^{13}+\frac {1}{15} c^2 e^3 x^{15} \]
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Time = 0.14 (sec) , antiderivative size = 223, 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 = {1167} \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac {1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac {1}{3} a d^2 x^3 (3 a e+2 b d)+\frac {1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac {1}{15} c^2 e^3 x^{15} \]
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Rule 1167
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 d^3+a d^2 (2 b d+3 a e) x^2+d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^4+\left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^6+\left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^8+e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{10}+c e^2 (3 c d+2 b e) x^{12}+c^2 e^3 x^{14}\right ) \, dx \\ & = a^2 d^3 x+\frac {1}{3} a d^2 (2 b d+3 a e) x^3+\frac {1}{5} d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{7} \left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^7+\frac {1}{9} \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{11} e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{11}+\frac {1}{13} c e^2 (3 c d+2 b e) x^{13}+\frac {1}{15} c^2 e^3 x^{15} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^3 x+\frac {1}{3} a d^2 (2 b d+3 a e) x^3+\frac {1}{5} d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{7} \left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^7+\frac {1}{9} \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{11} e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{11}+\frac {1}{13} c e^2 (3 c d+2 b e) x^{13}+\frac {1}{15} c^2 e^3 x^{15} \]
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Time = 0.27 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {c^{2} e^{3} x^{15}}{15}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{13}}{13}+\frac {\left (3 d^{2} e \,c^{2}+6 d \,e^{2} b c +e^{3} \left (2 a c +b^{2}\right )\right ) x^{11}}{11}+\frac {\left (c^{2} d^{3}+6 d^{2} e b c +3 d \,e^{2} \left (2 a c +b^{2}\right )+2 e^{3} a b \right ) x^{9}}{9}+\frac {\left (2 b c \,d^{3}+3 d^{2} e \left (2 a c +b^{2}\right )+6 a b d \,e^{2}+e^{3} a^{2}\right ) x^{7}}{7}+\frac {\left (d^{3} \left (2 a c +b^{2}\right )+6 d^{2} e a b +3 d \,e^{2} a^{2}\right ) x^{5}}{5}+\frac {\left (3 d^{2} e \,a^{2}+2 a \,d^{3} b \right ) x^{3}}{3}+a^{2} d^{3} x\) | \(219\) |
norman | \(a^{2} d^{3} x +\left (d^{2} e \,a^{2}+\frac {2}{3} a \,d^{3} b \right ) x^{3}+\left (\frac {3}{5} d \,e^{2} a^{2}+\frac {6}{5} d^{2} e a b +\frac {2}{5} d^{3} a c +\frac {1}{5} b^{2} d^{3}\right ) x^{5}+\left (\frac {1}{7} e^{3} a^{2}+\frac {6}{7} a b d \,e^{2}+\frac {6}{7} d^{2} e a c +\frac {3}{7} b^{2} d^{2} e +\frac {2}{7} b c \,d^{3}\right ) x^{7}+\left (\frac {2}{9} e^{3} a b +\frac {2}{3} a c d \,e^{2}+\frac {1}{3} b^{2} d \,e^{2}+\frac {2}{3} d^{2} e b c +\frac {1}{9} c^{2} d^{3}\right ) x^{9}+\left (\frac {2}{11} e^{3} a c +\frac {1}{11} b^{2} e^{3}+\frac {6}{11} d \,e^{2} b c +\frac {3}{11} d^{2} e \,c^{2}\right ) x^{11}+\left (\frac {2}{13} e^{3} b c +\frac {3}{13} d \,e^{2} c^{2}\right ) x^{13}+\frac {c^{2} e^{3} x^{15}}{15}\) | \(226\) |
gosper | \(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\frac {2}{3} x^{3} a \,d^{3} b +\frac {3}{5} x^{5} d \,e^{2} a^{2}+\frac {6}{5} x^{5} d^{2} e a b +\frac {2}{5} x^{5} d^{3} a c +\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{7} x^{7} e^{3} a^{2}+\frac {6}{7} x^{7} a b d \,e^{2}+\frac {6}{7} x^{7} d^{2} e a c +\frac {3}{7} x^{7} b^{2} d^{2} e +\frac {2}{7} x^{7} b c \,d^{3}+\frac {2}{9} x^{9} e^{3} a b +\frac {2}{3} x^{9} a c d \,e^{2}+\frac {1}{3} x^{9} b^{2} d \,e^{2}+\frac {2}{3} x^{9} d^{2} e b c +\frac {1}{9} x^{9} c^{2} d^{3}+\frac {2}{11} x^{11} e^{3} a c +\frac {1}{11} x^{11} b^{2} e^{3}+\frac {6}{11} x^{11} d \,e^{2} b c +\frac {3}{11} x^{11} d^{2} e \,c^{2}+\frac {2}{13} x^{13} e^{3} b c +\frac {3}{13} c^{2} d \,e^{2} x^{13}+\frac {1}{15} c^{2} e^{3} x^{15}\) | \(262\) |
risch | \(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\frac {2}{3} x^{3} a \,d^{3} b +\frac {3}{5} x^{5} d \,e^{2} a^{2}+\frac {6}{5} x^{5} d^{2} e a b +\frac {2}{5} x^{5} d^{3} a c +\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{7} x^{7} e^{3} a^{2}+\frac {6}{7} x^{7} a b d \,e^{2}+\frac {6}{7} x^{7} d^{2} e a c +\frac {3}{7} x^{7} b^{2} d^{2} e +\frac {2}{7} x^{7} b c \,d^{3}+\frac {2}{9} x^{9} e^{3} a b +\frac {2}{3} x^{9} a c d \,e^{2}+\frac {1}{3} x^{9} b^{2} d \,e^{2}+\frac {2}{3} x^{9} d^{2} e b c +\frac {1}{9} x^{9} c^{2} d^{3}+\frac {2}{11} x^{11} e^{3} a c +\frac {1}{11} x^{11} b^{2} e^{3}+\frac {6}{11} x^{11} d \,e^{2} b c +\frac {3}{11} x^{11} d^{2} e \,c^{2}+\frac {2}{13} x^{13} e^{3} b c +\frac {3}{13} c^{2} d \,e^{2} x^{13}+\frac {1}{15} c^{2} e^{3} x^{15}\) | \(262\) |
parallelrisch | \(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\frac {2}{3} x^{3} a \,d^{3} b +\frac {3}{5} x^{5} d \,e^{2} a^{2}+\frac {6}{5} x^{5} d^{2} e a b +\frac {2}{5} x^{5} d^{3} a c +\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{7} x^{7} e^{3} a^{2}+\frac {6}{7} x^{7} a b d \,e^{2}+\frac {6}{7} x^{7} d^{2} e a c +\frac {3}{7} x^{7} b^{2} d^{2} e +\frac {2}{7} x^{7} b c \,d^{3}+\frac {2}{9} x^{9} e^{3} a b +\frac {2}{3} x^{9} a c d \,e^{2}+\frac {1}{3} x^{9} b^{2} d \,e^{2}+\frac {2}{3} x^{9} d^{2} e b c +\frac {1}{9} x^{9} c^{2} d^{3}+\frac {2}{11} x^{11} e^{3} a c +\frac {1}{11} x^{11} b^{2} e^{3}+\frac {6}{11} x^{11} d \,e^{2} b c +\frac {3}{11} x^{11} d^{2} e \,c^{2}+\frac {2}{13} x^{13} e^{3} b c +\frac {3}{13} c^{2} d \,e^{2} x^{13}+\frac {1}{15} c^{2} e^{3} x^{15}\) | \(262\) |
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Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{15} \, c^{2} e^{3} x^{15} + \frac {1}{13} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{7} + a^{2} d^{3} x + \frac {1}{5} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{3} \]
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Time = 0.04 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.22 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d^{3} x + \frac {c^{2} e^{3} x^{15}}{15} + x^{13} \cdot \left (\frac {2 b c e^{3}}{13} + \frac {3 c^{2} d e^{2}}{13}\right ) + x^{11} \cdot \left (\frac {2 a c e^{3}}{11} + \frac {b^{2} e^{3}}{11} + \frac {6 b c d e^{2}}{11} + \frac {3 c^{2} d^{2} e}{11}\right ) + x^{9} \cdot \left (\frac {2 a b e^{3}}{9} + \frac {2 a c d e^{2}}{3} + \frac {b^{2} d e^{2}}{3} + \frac {2 b c d^{2} e}{3} + \frac {c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac {a^{2} e^{3}}{7} + \frac {6 a b d e^{2}}{7} + \frac {6 a c d^{2} e}{7} + \frac {3 b^{2} d^{2} e}{7} + \frac {2 b c d^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} d e^{2}}{5} + \frac {6 a b d^{2} e}{5} + \frac {2 a c d^{3}}{5} + \frac {b^{2} d^{3}}{5}\right ) + x^{3} \left (a^{2} d^{2} e + \frac {2 a b d^{3}}{3}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{15} \, c^{2} e^{3} x^{15} + \frac {1}{13} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{7} + a^{2} d^{3} x + \frac {1}{5} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{3} \]
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Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.17 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{15} \, c^{2} e^{3} x^{15} + \frac {3}{13} \, c^{2} d e^{2} x^{13} + \frac {2}{13} \, b c e^{3} x^{13} + \frac {3}{11} \, c^{2} d^{2} e x^{11} + \frac {6}{11} \, b c d e^{2} x^{11} + \frac {1}{11} \, b^{2} e^{3} x^{11} + \frac {2}{11} \, a c e^{3} x^{11} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {2}{3} \, b c d^{2} e x^{9} + \frac {1}{3} \, b^{2} d e^{2} x^{9} + \frac {2}{3} \, a c d e^{2} x^{9} + \frac {2}{9} \, a b e^{3} x^{9} + \frac {2}{7} \, b c d^{3} x^{7} + \frac {3}{7} \, b^{2} d^{2} e x^{7} + \frac {6}{7} \, a c d^{2} e x^{7} + \frac {6}{7} \, a b d e^{2} x^{7} + \frac {1}{7} \, a^{2} e^{3} x^{7} + \frac {1}{5} \, b^{2} d^{3} x^{5} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {6}{5} \, a b d^{2} e x^{5} + \frac {3}{5} \, a^{2} d e^{2} x^{5} + \frac {2}{3} \, a b d^{3} x^{3} + a^{2} d^{2} e x^{3} + a^{2} d^{3} x \]
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Time = 0.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx=x^7\,\left (\frac {a^2\,e^3}{7}+\frac {6\,a\,b\,d\,e^2}{7}+\frac {6\,c\,a\,d^2\,e}{7}+\frac {3\,b^2\,d^2\,e}{7}+\frac {2\,c\,b\,d^3}{7}\right )+x^9\,\left (\frac {b^2\,d\,e^2}{3}+\frac {2\,b\,c\,d^2\,e}{3}+\frac {2\,a\,b\,e^3}{9}+\frac {c^2\,d^3}{9}+\frac {2\,a\,c\,d\,e^2}{3}\right )+x^5\,\left (\frac {3\,a^2\,d\,e^2}{5}+\frac {6\,a\,b\,d^2\,e}{5}+\frac {2\,c\,a\,d^3}{5}+\frac {b^2\,d^3}{5}\right )+x^{11}\,\left (\frac {b^2\,e^3}{11}+\frac {6\,b\,c\,d\,e^2}{11}+\frac {3\,c^2\,d^2\,e}{11}+\frac {2\,a\,c\,e^3}{11}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^{15}}{15}+\frac {a\,d^2\,x^3\,\left (3\,a\,e+2\,b\,d\right )}{3}+\frac {c\,e^2\,x^{13}\,\left (2\,b\,e+3\,c\,d\right )}{13} \]
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