Optimal. Leaf size=101 \[ \frac {8 \sqrt {x+1}}{315 \sqrt {1-x}}+\frac {8 \sqrt {x+1}}{315 (1-x)^{3/2}}+\frac {4 \sqrt {x+1}}{105 (1-x)^{5/2}}+\frac {4 \sqrt {x+1}}{63 (1-x)^{7/2}}+\frac {\sqrt {x+1}}{9 (1-x)^{9/2}} \]
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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} \[ \frac {8 \sqrt {x+1}}{315 \sqrt {1-x}}+\frac {8 \sqrt {x+1}}{315 (1-x)^{3/2}}+\frac {4 \sqrt {x+1}}{105 (1-x)^{5/2}}+\frac {4 \sqrt {x+1}}{63 (1-x)^{7/2}}+\frac {\sqrt {x+1}}{9 (1-x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{11/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4}{9} \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4}{21} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8}{105} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8 \sqrt {1+x}}{315 (1-x)^{3/2}}+\frac {8}{315} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8 \sqrt {1+x}}{315 (1-x)^{3/2}}+\frac {8 \sqrt {1+x}}{315 \sqrt {1-x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.40 \[ \frac {\sqrt {x+1} \left (8 x^4-40 x^3+84 x^2-100 x+83\right )}{315 (1-x)^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 86, normalized size = 0.85 \[ \frac {83 \, x^{5} - 415 \, x^{4} + 830 \, x^{3} - 830 \, x^{2} - {\left (8 \, x^{4} - 40 \, x^{3} + 84 \, x^{2} - 100 \, x + 83\right )} \sqrt {x + 1} \sqrt {-x + 1} + 415 \, x - 83}{315 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 42, normalized size = 0.42 \[ -\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 8\right )} + 63\right )} {\left (x + 1\right )} - 105\right )} {\left (x + 1\right )} + 315\right )} \sqrt {x + 1} \sqrt {-x + 1}}{315 \, {\left (x - 1\right )}^{5}} \]
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 \[ \frac {\sqrt {x +1}\, \left (8 x^{4}-40 x^{3}+84 x^{2}-100 x +83\right )}{315 \left (-x +1\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 131, normalized size = 1.30 \[ -\frac {\sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 80, normalized size = 0.79 \[ \frac {17\,x\,\sqrt {1-x}-83\,\sqrt {1-x}+16\,x^2\,\sqrt {1-x}-44\,x^3\,\sqrt {1-x}+32\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{315\,{\left (x-1\right )}^5\,\sqrt {x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 58.39, size = 933, normalized size = 9.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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