Integrand size = 38, antiderivative size = 23 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=1-4 \left (-2+x+\frac {4 \left (e^4+x^2\right )}{3 x^2}\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 14} \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=-\frac {64 e^8}{9 x^4}-4 x^2+\frac {64 e^4}{9 x^2}+\frac {16 x}{3}-\frac {32 e^4}{3 x} \]
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Rule 12
Rule 14
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{x^5} \, dx \\ & = \frac {1}{9} \int \left (48+\frac {256 e^8}{x^5}-\frac {128 e^4}{x^3}+\frac {96 e^4}{x^2}-72 x\right ) \, dx \\ & = -\frac {64 e^8}{9 x^4}+\frac {64 e^4}{9 x^2}-\frac {32 e^4}{3 x}+\frac {16 x}{3}-4 x^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=\frac {8}{9} \left (-\frac {8 e^8}{x^4}+\frac {8 e^4}{x^2}-\frac {12 e^4}{x}+6 x-\frac {9 x^2}{2}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35
method | result | size |
default | \(-4 x^{2}+\frac {16 x}{3}-\frac {64 \,{\mathrm e}^{8}}{9 x^{4}}-\frac {32 \,{\mathrm e}^{4}}{3 x}+\frac {64 \,{\mathrm e}^{4}}{9 x^{2}}\) | \(31\) |
risch | \(-4 x^{2}+\frac {16 x}{3}+\frac {-96 x^{3} {\mathrm e}^{4}+64 x^{2} {\mathrm e}^{4}-64 \,{\mathrm e}^{8}}{9 x^{4}}\) | \(34\) |
norman | \(\frac {\frac {16 x^{5}}{3}-4 x^{6}-\frac {64 \,{\mathrm e}^{8}}{9}+\frac {64 x^{2} {\mathrm e}^{4}}{9}-\frac {32 x^{3} {\mathrm e}^{4}}{3}}{x^{4}}\) | \(36\) |
gosper | \(-\frac {4 \left (9 x^{6}-12 x^{5}+24 x^{3} {\mathrm e}^{4}-16 x^{2} {\mathrm e}^{4}+16 \,{\mathrm e}^{8}\right )}{9 x^{4}}\) | \(37\) |
parallelrisch | \(-\frac {36 x^{6}-48 x^{5}+96 x^{3} {\mathrm e}^{4}-64 x^{2} {\mathrm e}^{4}+64 \,{\mathrm e}^{8}}{9 x^{4}}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=-\frac {4 \, {\left (9 \, x^{6} - 12 \, x^{5} + 8 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e^{4} + 16 \, e^{8}\right )}}{9 \, x^{4}} \]
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Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=- 4 x^{2} + \frac {16 x}{3} - \frac {96 x^{3} e^{4} - 64 x^{2} e^{4} + 64 e^{8}}{9 x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=-4 \, x^{2} + \frac {16}{3} \, x - \frac {32 \, {\left (3 \, x^{3} e^{4} - 2 \, x^{2} e^{4} + 2 \, e^{8}\right )}}{9 \, x^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=-4 \, x^{2} + \frac {16}{3} \, x - \frac {32 \, {\left (3 \, x^{3} e^{4} - 2 \, x^{2} e^{4} + 2 \, e^{8}\right )}}{9 \, x^{4}} \]
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Time = 8.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {256 e^8+48 x^5-72 x^6+e^4 \left (-128 x^2+96 x^3\right )}{9 x^5} \, dx=-\frac {4\,\left (9\,x^6-12\,x^5+24\,{\mathrm {e}}^4\,x^3-16\,{\mathrm {e}}^4\,x^2+16\,{\mathrm {e}}^8\right )}{9\,x^4} \]
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