Integrand size = 15, antiderivative size = 73 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {1}{9} c^4 d x^9+\frac {e \left (a+c x^2\right )^5}{10 c} \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200} \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {e \left (a+c x^2\right )^5}{10 c}+\frac {1}{9} c^4 d x^9 \]
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Rule 200
Rule 655
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+c x^2\right )^5}{10 c}+d \int \left (a+c x^2\right )^4 \, dx \\ & = \frac {e \left (a+c x^2\right )^5}{10 c}+d \int \left (a^4+4 a^3 c x^2+6 a^2 c^2 x^4+4 a c^3 x^6+c^4 x^8\right ) \, dx \\ & = a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {1}{9} c^4 d x^9+\frac {e \left (a+c x^2\right )^5}{10 c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {4}{3} a^3 c d x^3+a^3 c e x^4+\frac {6}{5} a^2 c^2 d x^5+a^2 c^2 e x^6+\frac {4}{7} a c^3 d x^7+\frac {1}{2} a c^3 e x^8+\frac {1}{9} c^4 d x^9+\frac {1}{10} c^4 e x^{10} \]
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Time = 2.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33
method | result | size |
gosper | \(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\) | \(97\) |
default | \(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\) | \(97\) |
norman | \(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\) | \(97\) |
risch | \(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\) | \(97\) |
parallelrisch | \(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\) | \(97\) |
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Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]
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Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^{4} d x + \frac {a^{4} e x^{2}}{2} + \frac {4 a^{3} c d x^{3}}{3} + a^{3} c e x^{4} + \frac {6 a^{2} c^{2} d x^{5}}{5} + a^{2} c^{2} e x^{6} + \frac {4 a c^{3} d x^{7}}{7} + \frac {a c^{3} e x^{8}}{2} + \frac {c^{4} d x^{9}}{9} + \frac {c^{4} e x^{10}}{10} \]
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Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]
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Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {e\,a^4\,x^2}{2}+d\,a^4\,x+e\,a^3\,c\,x^4+\frac {4\,d\,a^3\,c\,x^3}{3}+e\,a^2\,c^2\,x^6+\frac {6\,d\,a^2\,c^2\,x^5}{5}+\frac {e\,a\,c^3\,x^8}{2}+\frac {4\,d\,a\,c^3\,x^7}{7}+\frac {e\,c^4\,x^{10}}{10}+\frac {d\,c^4\,x^9}{9} \]
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