Integrand size = 14, antiderivative size = 33 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 276} \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]
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Rule 12
Rule 276
Rubi steps \begin{align*} \text {integral}& = c \int x^2 \left (e+f x^4\right )^2 \, dx \\ & = c \int \left (e^2 x^2+2 e f x^6+f^2 x^{10}\right ) \, dx \\ & = \frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]
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Time = 1.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
method | result | size |
default | \(\left (\frac {1}{11} f^{2} x^{11}+\frac {2}{7} e f \,x^{7}+\frac {1}{3} e^{2} x^{3}\right ) c\) | \(27\) |
parallelrisch | \(\left (\frac {1}{11} f^{2} x^{11}+\frac {2}{7} e f \,x^{7}+\frac {1}{3} e^{2} x^{3}\right ) c\) | \(27\) |
gosper | \(\frac {x^{3} \left (21 f^{2} x^{8}+66 e f \,x^{4}+77 e^{2}\right ) c}{231}\) | \(28\) |
norman | \(\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{7} c e f \,x^{7}+\frac {1}{11} c \,f^{2} x^{11}\) | \(28\) |
risch | \(\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{7} c e f \,x^{7}+\frac {1}{11} c \,f^{2} x^{11}\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{11} \, c f^{2} x^{11} + \frac {2}{7} \, c e f x^{7} + \frac {1}{3} \, c e^{2} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \]
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {c\,x^3\,\left (77\,e^2+66\,e\,f\,x^4+21\,f^2\,x^8\right )}{231} \]
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