Optimal. Leaf size=43 \[ \frac {2 (x+2)}{243 \sqrt {-x^2-4 x+5}}+\frac {x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {614, 613} \[ \frac {2 (x+2)}{243 \sqrt {-x^2-4 x+5}}+\frac {x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (5-4 x-x^2\right )^{5/2}} \, dx &=\frac {2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac {2}{27} \int \frac {1}{\left (5-4 x-x^2\right )^{3/2}} \, dx\\ &=\frac {2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac {2 (2+x)}{243 \sqrt {5-4 x-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.72 \[ -\frac {(x+2) \left (2 x^2+8 x-19\right )}{243 \left (-x^2-4 x+5\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 49, normalized size = 1.14 \[ -\frac {{\left (2 \, x^{3} + 12 \, x^{2} - 3 \, x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{4} + 8 \, x^{3} + 6 \, x^{2} - 40 \, x + 25\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 36, normalized size = 0.84 \[ -\frac {{\left ({\left (2 \, {\left (x + 6\right )} x - 3\right )} x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{2} + 4 \, x - 5\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.84 \[ \frac {\left (x +5\right ) \left (x -1\right ) \left (2 x^{3}+12 x^{2}-3 x -38\right )}{243 \left (-x^{2}-4 x +5\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 59, normalized size = 1.37 \[ \frac {2 \, x}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {4}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {x}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} + \frac {2}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 29, normalized size = 0.67 \[ -\frac {\left (4\,x+8\right )\,\left (8\,x^2+32\,x-76\right )}{3888\,{\left (-x^2-4\,x+5\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- x^{2} - 4 x + 5\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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