Optimal. Leaf size=101 \[ \frac {8 (x+1)^{5/2}}{15015 (1-x)^{5/2}}+\frac {8 (x+1)^{5/2}}{3003 (1-x)^{7/2}}+\frac {4 (x+1)^{5/2}}{429 (1-x)^{9/2}}+\frac {4 (x+1)^{5/2}}{143 (1-x)^{11/2}}+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}} \]
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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} \[ \frac {8 (x+1)^{5/2}}{15015 (1-x)^{5/2}}+\frac {8 (x+1)^{5/2}}{3003 (1-x)^{7/2}}+\frac {4 (x+1)^{5/2}}{429 (1-x)^{9/2}}+\frac {4 (x+1)^{5/2}}{143 (1-x)^{11/2}}+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx &=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4}{13} \int \frac {(1+x)^{3/2}}{(1-x)^{13/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {12}{143} \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8}{429} \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx}{3003}\\ &=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 (1+x)^{5/2}}{15015 (1-x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.40 \[ \frac {(x+1)^{5/2} \left (8 x^4-72 x^3+308 x^2-852 x+1763\right )}{15015 (1-x)^{13/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 116, normalized size = 1.15 \[ \frac {1763 \, x^{7} - 12341 \, x^{6} + 37023 \, x^{5} - 61705 \, x^{4} + 61705 \, x^{3} - 37023 \, x^{2} - {\left (8 \, x^{6} - 56 \, x^{5} + 172 \, x^{4} - 308 \, x^{3} + 367 \, x^{2} + 2674 \, x + 1763\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12341 \, x - 1763}{15015 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 42, normalized size = 0.42 \[ -\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 12\right )} + 143\right )} {\left (x + 1\right )} - 429\right )} {\left (x + 1\right )} + 3003\right )} {\left (x + 1\right )}^{\frac {5}{2}} \sqrt {-x + 1}}{15015 \, {\left (x - 1\right )}^{7}} \]
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 \[ \frac {\left (x +1\right )^{\frac {5}{2}} \left (8 x^{4}-72 x^{3}+308 x^{2}-852 x +1763\right )}{15015 \left (-x +1\right )^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 269, normalized size = 2.66 \[ \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{5 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {6 \, \sqrt {-x^{2} + 1}}{65 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {3 \, \sqrt {-x^{2} + 1}}{715 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{429 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{5005 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 110, normalized size = 1.09 \[ -\frac {\sqrt {1-x}\,\left (\frac {382\,x\,\sqrt {x+1}}{2145}+\frac {1763\,\sqrt {x+1}}{15015}+\frac {367\,x^2\,\sqrt {x+1}}{15015}-\frac {4\,x^3\,\sqrt {x+1}}{195}+\frac {172\,x^4\,\sqrt {x+1}}{15015}-\frac {8\,x^5\,\sqrt {x+1}}{2145}+\frac {8\,x^6\,\sqrt {x+1}}{15015}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \]
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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