Integrand size = 17, antiderivative size = 101 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 (1+x)^{3/2}}{3465 (1-x)^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac {8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac {4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac {4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{3/2}}{11 (1-x)^{11/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4}{11} \int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4}{33} \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8}{231} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx}{1155} \\ & = \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 (1+x)^{3/2}}{3465 (1-x)^{3/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{3/2} \left (547-364 x+180 x^2-56 x^3+8 x^4\right )}{3465 (1-x)^{11/2}} \]
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Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {3}{2}} \left (8 x^{4}-56 x^{3}+180 x^{2}-364 x +547\right )}{3465 \left (1-x \right )^{\frac {11}{2}}}\) | \(35\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{6}-40 x^{5}+76 x^{4}-60 x^{3}-x^{2}+730 x +547\right )}{3465 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(71\) |
default | \(\frac {2 \sqrt {1+x}}{11 \left (1-x \right )^{\frac {11}{2}}}-\frac {\sqrt {1+x}}{99 \left (1-x \right )^{\frac {9}{2}}}-\frac {4 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {4 \sqrt {1+x}}{1155 \left (1-x \right )^{\frac {5}{2}}}-\frac {8 \sqrt {1+x}}{3465 \left (1-x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{3465 \sqrt {1-x}}\) | \(86\) |
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Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {547 \, x^{6} - 3282 \, x^{5} + 8205 \, x^{4} - 10940 \, x^{3} + 8205 \, x^{2} + {\left (8 \, x^{5} - 48 \, x^{4} + 124 \, x^{3} - 184 \, x^{2} + 183 \, x + 547\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3282 \, x + 547}{3465 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (71) = 142\).
Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {2 \, \sqrt {-x^{2} + 1}}{11 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{99 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{1155 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{3465 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{3465 \, {\left (x - 1\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 10\right )} + 99\right )} {\left (x + 1\right )} - 231\right )} {\left (x + 1\right )} + 1155\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3465 \, {\left (x - 1\right )}^{6}} \]
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Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {61\,x\,\sqrt {x+1}}{1155}+\frac {547\,\sqrt {x+1}}{3465}-\frac {184\,x^2\,\sqrt {x+1}}{3465}+\frac {124\,x^3\,\sqrt {x+1}}{3465}-\frac {16\,x^4\,\sqrt {x+1}}{1155}+\frac {8\,x^5\,\sqrt {x+1}}{3465}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \]
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