Integrand size = 17, antiderivative size = 132 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=a^4 d^2 x+\frac {1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac {2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac {2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac {1}{11} c^4 e^2 x^{11}+\frac {d e \left (a+c x^2\right )^5}{5 c} \]
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Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824} \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=a^4 d^2 x+\frac {1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+\frac {2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac {1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac {2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^5}{5 c}+\frac {1}{11} c^4 e^2 x^{11} \]
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Rule 710
Rule 1824
Rubi steps \begin{align*} \text {integral}& = \frac {d e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-2 d e x+(d+e x)^2\right ) \, dx \\ & = \frac {d e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^2+a^3 \left (4 c d^2+a e^2\right ) x^2+2 a^2 c \left (3 c d^2+2 a e^2\right ) x^4+2 a c^2 \left (2 c d^2+3 a e^2\right ) x^6+c^3 \left (c d^2+4 a e^2\right ) x^8+c^4 e^2 x^{10}\right ) \, dx \\ & = a^4 d^2 x+\frac {1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac {2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac {2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac {1}{11} c^4 e^2 x^{11}+\frac {d e \left (a+c x^2\right )^5}{5 c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.12 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=\frac {2}{15} a^3 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+\frac {2}{35} a^2 c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac {1}{63} a c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+\frac {1}{495} c^4 x^9 \left (55 d^2+99 d e x+45 e^2 x^2\right )+a^4 \left (d^2 x+d e x^2+\frac {e^2 x^3}{3}\right ) \]
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Time = 2.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.27
method | result | size |
norman | \(\frac {c^{4} e^{2} x^{11}}{11}+\frac {d e \,c^{4} x^{10}}{5}+\left (\frac {4}{9} e^{2} c^{3} a +\frac {1}{9} c^{4} d^{2}\right ) x^{9}+d e \,c^{3} a \,x^{8}+\left (\frac {6}{7} a^{2} c^{2} e^{2}+\frac {4}{7} a \,c^{3} d^{2}\right ) x^{7}+2 a^{2} c^{2} d e \,x^{6}+\left (\frac {4}{5} e^{2} c \,a^{3}+\frac {6}{5} a^{2} c^{2} d^{2}\right ) x^{5}+2 d e c \,a^{3} x^{4}+\left (\frac {1}{3} a^{4} e^{2}+\frac {4}{3} a^{3} c \,d^{2}\right ) x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\) | \(168\) |
default | \(\frac {c^{4} e^{2} x^{11}}{11}+\frac {d e \,c^{4} x^{10}}{5}+\frac {\left (4 e^{2} c^{3} a +c^{4} d^{2}\right ) x^{9}}{9}+d e \,c^{3} a \,x^{8}+\frac {\left (6 a^{2} c^{2} e^{2}+4 a \,c^{3} d^{2}\right ) x^{7}}{7}+2 a^{2} c^{2} d e \,x^{6}+\frac {\left (4 e^{2} c \,a^{3}+6 a^{2} c^{2} d^{2}\right ) x^{5}}{5}+2 d e c \,a^{3} x^{4}+\frac {\left (a^{4} e^{2}+4 a^{3} c \,d^{2}\right ) x^{3}}{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\) | \(170\) |
gosper | \(\frac {1}{11} c^{4} e^{2} x^{11}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {4}{9} x^{9} e^{2} c^{3} a +\frac {1}{9} x^{9} c^{4} d^{2}+d e \,c^{3} a \,x^{8}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+2 a^{2} c^{2} d e \,x^{6}+\frac {4}{5} x^{5} e^{2} c \,a^{3}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+2 d e c \,a^{3} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\) | \(172\) |
risch | \(\frac {1}{11} c^{4} e^{2} x^{11}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {4}{9} x^{9} e^{2} c^{3} a +\frac {1}{9} x^{9} c^{4} d^{2}+d e \,c^{3} a \,x^{8}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+2 a^{2} c^{2} d e \,x^{6}+\frac {4}{5} x^{5} e^{2} c \,a^{3}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+2 d e c \,a^{3} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\) | \(172\) |
parallelrisch | \(\frac {1}{11} c^{4} e^{2} x^{11}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {4}{9} x^{9} e^{2} c^{3} a +\frac {1}{9} x^{9} c^{4} d^{2}+d e \,c^{3} a \,x^{8}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+2 a^{2} c^{2} d e \,x^{6}+\frac {4}{5} x^{5} e^{2} c \,a^{3}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+2 d e c \,a^{3} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\) | \(172\) |
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Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.28 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, c^{4} d e x^{10} + a c^{3} d e x^{8} + 2 \, a^{2} c^{2} d e x^{6} + 2 \, a^{3} c d e x^{4} + \frac {1}{9} \, {\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + a^{4} d e x^{2} + \frac {2}{7} \, {\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {2}{5} \, {\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{2} + a^{4} e^{2}\right )} x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.42 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=a^{4} d^{2} x + a^{4} d e x^{2} + 2 a^{3} c d e x^{4} + 2 a^{2} c^{2} d e x^{6} + a c^{3} d e x^{8} + \frac {c^{4} d e x^{10}}{5} + \frac {c^{4} e^{2} x^{11}}{11} + x^{9} \cdot \left (\frac {4 a c^{3} e^{2}}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{7} \cdot \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {4 a c^{3} d^{2}}{7}\right ) + x^{5} \cdot \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {4 a^{3} c d^{2}}{3}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.28 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, c^{4} d e x^{10} + a c^{3} d e x^{8} + 2 \, a^{2} c^{2} d e x^{6} + 2 \, a^{3} c d e x^{4} + \frac {1}{9} \, {\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + a^{4} d e x^{2} + \frac {2}{7} \, {\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {2}{5} \, {\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{2} + a^{4} e^{2}\right )} x^{3} \]
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Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.30 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, c^{4} d e x^{10} + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {4}{9} \, a c^{3} e^{2} x^{9} + a c^{3} d e x^{8} + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {6}{7} \, a^{2} c^{2} e^{2} x^{7} + 2 \, a^{2} c^{2} d e x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {4}{5} \, a^{3} c e^{2} x^{5} + 2 \, a^{3} c d e x^{4} + \frac {4}{3} \, a^{3} c d^{2} x^{3} + \frac {1}{3} \, a^{4} e^{2} x^{3} + a^{4} d e x^{2} + a^{4} d^{2} x \]
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Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (a+c x^2\right )^4 \, dx=x^3\,\left (\frac {a^4\,e^2}{3}+\frac {4\,c\,a^3\,d^2}{3}\right )+x^9\,\left (\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^4\,d\,e\,x^2+\frac {c^4\,d\,e\,x^{10}}{5}+\frac {2\,a^2\,c\,x^5\,\left (3\,c\,d^2+2\,a\,e^2\right )}{5}+\frac {2\,a\,c^2\,x^7\,\left (2\,c\,d^2+3\,a\,e^2\right )}{7}+2\,a^3\,c\,d\,e\,x^4+a\,c^3\,d\,e\,x^8+2\,a^2\,c^2\,d\,e\,x^6 \]
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