Integrand size = 14, antiderivative size = 47 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}-\frac {2 (2-3 x)}{361 \sqrt {5+4 x-3 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {628, 627} \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {2 (2-3 x)}{361 \sqrt {-3 x^2+4 x+5}}-\frac {2-3 x}{57 \left (-3 x^2+4 x+5\right )^{3/2}} \]
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Rule 627
Rule 628
Rubi steps \begin{align*} \text {integral}& = -\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}+\frac {2}{19} \int \frac {1}{\left (5+4 x-3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}-\frac {2 (2-3 x)}{361 \sqrt {5+4 x-3 x^2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {-98+99 x+108 x^2-54 x^3}{1083 \left (5+4 x-3 x^2\right )^{3/2}} \]
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Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {54 x^{3}-108 x^{2}-99 x +98}{1083 \left (-3 x^{2}+4 x +5\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {-6 x +4}{114 \left (-3 x^{2}+4 x +5\right )^{\frac {3}{2}}}-\frac {-6 x +4}{361 \sqrt {-3 x^{2}+4 x +5}}\) | \(40\) |
trager | \(-\frac {\left (54 x^{3}-108 x^{2}-99 x +98\right ) \sqrt {-3 x^{2}+4 x +5}}{1083 \left (3 x^{2}-4 x -5\right )^{2}}\) | \(42\) |
risch | \(\frac {54 x^{3}-108 x^{2}-99 x +98}{1083 \left (3 x^{2}-4 x -5\right ) \sqrt {-3 x^{2}+4 x +5}}\) | \(42\) |
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Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {{\left (54 \, x^{3} - 108 \, x^{2} - 99 \, x + 98\right )} \sqrt {-3 \, x^{2} + 4 \, x + 5}}{1083 \, {\left (9 \, x^{4} - 24 \, x^{3} - 14 \, x^{2} + 40 \, x + 25\right )}} \]
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\[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- 3 x^{2} + 4 x + 5\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {6 \, x}{361 \, \sqrt {-3 \, x^{2} + 4 \, x + 5}} - \frac {4}{361 \, \sqrt {-3 \, x^{2} + 4 \, x + 5}} + \frac {x}{19 \, {\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac {3}{2}}} - \frac {2}{57 \, {\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac {3}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {{\left (9 \, {\left (6 \, {\left (x - 2\right )} x - 11\right )} x + 98\right )} \sqrt {-3 \, x^{2} + 4 \, x + 5}}{1083 \, {\left (3 \, x^{2} - 4 \, x - 5\right )}^{2}} \]
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {\left (12\,x-8\right )\,\left (-72\,x^2+96\,x+196\right )}{17328\,{\left (-3\,x^2+4\,x+5\right )}^{3/2}} \]
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