Integrand size = 22, antiderivative size = 135 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=a d^4 x+\frac {1}{3} d^3 (b d+4 a e) x^3+\frac {1}{5} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^5+\frac {2}{7} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^9+\frac {1}{11} e^3 (4 c d+b e) x^{11}+\frac {1}{13} c e^4 x^{13} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1167} \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{9} e^2 x^9 \left (e (a e+4 b d)+6 c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+4 b d e+c d^2\right )+\frac {2}{7} d e x^7 \left (e (2 a e+3 b d)+2 c d^2\right )+\frac {1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac {1}{11} e^3 x^{11} (b e+4 c d)+\frac {1}{13} c e^4 x^{13} \]
[In]
[Out]
Rule 1167
Rubi steps \begin{align*} \text {integral}& = \int \left (a d^4+d^3 (b d+4 a e) x^2+d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^4+2 d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^6+e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^8+e^3 (4 c d+b e) x^{10}+c e^4 x^{12}\right ) \, dx \\ & = a d^4 x+\frac {1}{3} d^3 (b d+4 a e) x^3+\frac {1}{5} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^5+\frac {2}{7} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^9+\frac {1}{11} e^3 (4 c d+b e) x^{11}+\frac {1}{13} c e^4 x^{13} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=a d^4 x+\frac {1}{3} d^3 (b d+4 a e) x^3+\frac {1}{5} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^5+\frac {2}{7} d e \left (2 c d^2+3 b d e+2 a e^2\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+4 b d e+a e^2\right ) x^9+\frac {1}{11} e^3 (4 c d+b e) x^{11}+\frac {1}{13} c e^4 x^{13} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {c \,e^{4} x^{13}}{13}+\left (\frac {1}{11} e^{4} b +\frac {4}{11} d \,e^{3} c \right ) x^{11}+\left (\frac {1}{9} e^{4} a +\frac {4}{9} d \,e^{3} b +\frac {2}{3} e^{2} d^{2} c \right ) x^{9}+\left (\frac {4}{7} d \,e^{3} a +\frac {6}{7} e^{2} d^{2} b +\frac {4}{7} d^{3} e c \right ) x^{7}+\left (\frac {6}{5} e^{2} d^{2} a +\frac {4}{5} d^{3} e b +\frac {1}{5} d^{4} c \right ) x^{5}+\left (\frac {4}{3} d^{3} e a +\frac {1}{3} d^{4} b \right ) x^{3}+a \,d^{4} x\) | \(135\) |
default | \(\frac {c \,e^{4} x^{13}}{13}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{11}}{11}+\frac {\left (e^{4} a +4 d \,e^{3} b +6 e^{2} d^{2} c \right ) x^{9}}{9}+\frac {\left (4 d \,e^{3} a +6 e^{2} d^{2} b +4 d^{3} e c \right ) x^{7}}{7}+\frac {\left (6 e^{2} d^{2} a +4 d^{3} e b +d^{4} c \right ) x^{5}}{5}+\frac {\left (4 d^{3} e a +d^{4} b \right ) x^{3}}{3}+a \,d^{4} x\) | \(136\) |
gosper | \(\frac {1}{13} c \,e^{4} x^{13}+\frac {1}{11} x^{11} e^{4} b +\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {4}{9} x^{9} d \,e^{3} b +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {6}{7} x^{7} e^{2} d^{2} b +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {4}{5} x^{5} d^{3} e b +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+\frac {1}{3} x^{3} d^{4} b +a \,d^{4} x\) | \(149\) |
risch | \(\frac {1}{13} c \,e^{4} x^{13}+\frac {1}{11} x^{11} e^{4} b +\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {4}{9} x^{9} d \,e^{3} b +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {6}{7} x^{7} e^{2} d^{2} b +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {4}{5} x^{5} d^{3} e b +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+\frac {1}{3} x^{3} d^{4} b +a \,d^{4} x\) | \(149\) |
parallelrisch | \(\frac {1}{13} c \,e^{4} x^{13}+\frac {1}{11} x^{11} e^{4} b +\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {4}{9} x^{9} d \,e^{3} b +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {6}{7} x^{7} e^{2} d^{2} b +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {4}{5} x^{5} d^{3} e b +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+\frac {1}{3} x^{3} d^{4} b +a \,d^{4} x\) | \(149\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {1}{11} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.16 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=a d^{4} x + \frac {c e^{4} x^{13}}{13} + x^{11} \left (\frac {b e^{4}}{11} + \frac {4 c d e^{3}}{11}\right ) + x^{9} \left (\frac {a e^{4}}{9} + \frac {4 b d e^{3}}{9} + \frac {2 c d^{2} e^{2}}{3}\right ) + x^{7} \cdot \left (\frac {4 a d e^{3}}{7} + \frac {6 b d^{2} e^{2}}{7} + \frac {4 c d^{3} e}{7}\right ) + x^{5} \cdot \left (\frac {6 a d^{2} e^{2}}{5} + \frac {4 b d^{3} e}{5} + \frac {c d^{4}}{5}\right ) + x^{3} \cdot \left (\frac {4 a d^{3} e}{3} + \frac {b d^{4}}{3}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {1}{11} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {4}{11} \, c d e^{3} x^{11} + \frac {1}{11} \, b e^{4} x^{11} + \frac {2}{3} \, c d^{2} e^{2} x^{9} + \frac {4}{9} \, b d e^{3} x^{9} + \frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, c d^{3} e x^{7} + \frac {6}{7} \, b d^{2} e^{2} x^{7} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {1}{5} \, c d^{4} x^{5} + \frac {4}{5} \, b d^{3} e x^{5} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {1}{3} \, b d^{4} x^{3} + \frac {4}{3} \, a d^{3} e x^{3} + a d^{4} x \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right )^4 \left (a+b x^2+c x^4\right ) \, dx=x^3\,\left (\frac {b\,d^4}{3}+\frac {4\,a\,e\,d^3}{3}\right )+x^{11}\,\left (\frac {b\,e^4}{11}+\frac {4\,c\,d\,e^3}{11}\right )+x^5\,\left (\frac {c\,d^4}{5}+\frac {4\,b\,d^3\,e}{5}+\frac {6\,a\,d^2\,e^2}{5}\right )+x^9\,\left (\frac {2\,c\,d^2\,e^2}{3}+\frac {4\,b\,d\,e^3}{9}+\frac {a\,e^4}{9}\right )+\frac {c\,e^4\,x^{13}}{13}+a\,d^4\,x+\frac {2\,d\,e\,x^7\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{7} \]
[In]
[Out]