Optimal. Leaf size=101 \[ \frac {8 (x+1)^{7/2}}{45045 (1-x)^{7/2}}+\frac {8 (x+1)^{7/2}}{6435 (1-x)^{9/2}}+\frac {4 (x+1)^{7/2}}{715 (1-x)^{11/2}}+\frac {4 (x+1)^{7/2}}{195 (1-x)^{13/2}}+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac {8 (x+1)^{7/2}}{45045 (1-x)^{7/2}}+\frac {8 (x+1)^{7/2}}{6435 (1-x)^{9/2}}+\frac {4 (x+1)^{7/2}}{715 (1-x)^{11/2}}+\frac {4 (x+1)^{7/2}}{195 (1-x)^{13/2}}+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx &=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4}{15} \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4}{65} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8}{715} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac {8 \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx}{6435}\\ &=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{45045 (1-x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.40 \[ \frac {(x+1)^{7/2} \left (8 x^4-88 x^3+468 x^2-1628 x+4243\right )}{45045 (1-x)^{15/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 130, normalized size = 1.29 \[ \frac {4243 \, x^{8} - 33944 \, x^{7} + 118804 \, x^{6} - 237608 \, x^{5} + 297010 \, x^{4} - 237608 \, x^{3} + 118804 \, x^{2} + {\left (8 \, x^{7} - 64 \, x^{6} + 228 \, x^{5} - 480 \, x^{4} + 675 \, x^{3} + 8313 \, x^{2} + 11101 \, x + 4243\right )} \sqrt {x + 1} \sqrt {-x + 1} - 33944 \, x + 4243}{45045 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 42, normalized size = 0.42 \[ \frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 14\right )} + 195\right )} {\left (x + 1\right )} - 715\right )} {\left (x + 1\right )} + 6435\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{45045 \, {\left (x - 1\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.35 \[ \frac {\left (x +1\right )^{\frac {7}{2}} \left (8 x^{4}-88 x^{3}+468 x^{2}-1628 x +4243\right )}{45045 \left (-x +1\right )^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 386, normalized size = 3.82 \[ \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{5 \, {\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} + \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{390 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{715 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1287 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{9009 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{45045 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{45045 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 124, normalized size = 1.23 \[ \frac {\sqrt {1-x}\,\left (\frac {11101\,x\,\sqrt {x+1}}{45045}+\frac {4243\,\sqrt {x+1}}{45045}+\frac {2771\,x^2\,\sqrt {x+1}}{15015}+\frac {15\,x^3\,\sqrt {x+1}}{1001}-\frac {32\,x^4\,\sqrt {x+1}}{3003}+\frac {76\,x^5\,\sqrt {x+1}}{15015}-\frac {64\,x^6\,\sqrt {x+1}}{45045}+\frac {8\,x^7\,\sqrt {x+1}}{45045}\right )}{x^8-8\,x^7+28\,x^6-56\,x^5+70\,x^4-56\,x^3+28\,x^2-8\,x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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