Integrand size = 44, antiderivative size = 23 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {\left (-4+e^{25}\right ) x}{6075 \left (x+\frac {1}{4} (5+x)^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 1694, 1828, 8} \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4 \left (4-e^{25}\right ) x}{6075 \left (24-(x+7)^2\right )} \]
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \frac {1}{243} \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{15625+17500 x+6150 x^2+700 x^3+25 x^4} \, dx \\ & = \frac {1}{243} \text {Subst}\left (\int \frac {4 \left (4-e^{25}\right ) \left (24-14 x+x^2\right )}{25 \left (24-x^2\right )^2} \, dx,x,7+x\right ) \\ & = \frac {\left (4 \left (4-e^{25}\right )\right ) \text {Subst}\left (\int \frac {24-14 x+x^2}{\left (24-x^2\right )^2} \, dx,x,7+x\right )}{6075} \\ & = \frac {4 \left (4-e^{25}\right ) x}{6075 \left (24-(7+x)^2\right )}+\frac {\left (-4+e^{25}\right ) \text {Subst}(\int 0 \, dx,x,7+x)}{72900} \\ & = \frac {4 \left (4-e^{25}\right ) x}{6075 \left (24-(7+x)^2\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4 \left (-4+e^{25}\right ) x}{6075 \left (25+14 x+x^2\right )} \]
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Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {4 x \left ({\mathrm e}^{25}-4\right )}{6075 \left (x^{2}+14 x +25\right )}\) | \(18\) |
norman | \(\frac {\left (-\frac {16}{6075}+\frac {4 \,{\mathrm e}^{25}}{6075}\right ) x}{x^{2}+14 x +25}\) | \(19\) |
default | \(\frac {\left (4 \,{\mathrm e}^{25}-16\right ) x}{6075 x^{2}+85050 x +151875}\) | \(20\) |
risch | \(\frac {x \left (\frac {4 \,{\mathrm e}^{25}}{25}-\frac {16}{25}\right )}{243 x^{2}+3402 x +6075}\) | \(20\) |
parallelrisch | \(\frac {4 x \,{\mathrm e}^{25}-16 x}{6075 x^{2}+85050 x +151875}\) | \(22\) |
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Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4 \, {\left (x e^{25} - 4 \, x\right )}}{6075 \, {\left (x^{2} + 14 \, x + 25\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {x \left (-16 + 4 e^{25}\right )}{6075 x^{2} + 85050 x + 151875} \]
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Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4 \, x {\left (e^{25} - 4\right )}}{6075 \, {\left (x^{2} + 14 \, x + 25\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4 \, {\left (x e^{25} - 4 \, x\right )}}{6075 \, {\left (x^{2} + 14 \, x + 25\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-400+16 x^2+e^{25} \left (100-4 x^2\right )}{243 \left (15625+17500 x+6150 x^2+700 x^3+25 x^4\right )} \, dx=\frac {4\,x\,\left ({\mathrm {e}}^{25}-4\right )}{6075\,\left (x^2+14\,x+25\right )} \]
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