Optimal. Leaf size=20 \[ \frac {(1+x)^{7/2}}{7 (1-x)^{7/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37}
\begin {gather*} \frac {(x+1)^{7/2}}{7 (1-x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx &=\frac {(1+x)^{7/2}}{7 (1-x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 20, normalized size = 1.00 \begin {gather*} \frac {(1+x)^{7/2}}{7 (1-x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs.
\(2(14)=28\).
time = 0.14, size = 85, normalized size = 4.25
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {7}{2}}}{7 \left (1-x \right )^{\frac {7}{2}}}\) | \(15\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{4}+4 x^{3}+6 x^{2}+4 x +1\right )}{7 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(59\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{\left (1-x \right )^{\frac {7}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{2 \left (1-x \right )^{\frac {7}{2}}}+\frac {15 \sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}-\frac {3 \sqrt {1+x}}{14 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{7 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{7 \sqrt {1-x}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs.
\(2 (14) = 28\).
time = 0.28, size = 171, normalized size = 8.55 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {15 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {3 \, \sqrt {-x^{2} + 1}}{14 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (14) = 28\).
time = 1.12, size = 66, normalized size = 3.30 \begin {gather*} \frac {x^{4} - 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - 4 \, x + 1}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 15.69, size = 114, normalized size = 5.70 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {7}{2}}}{7 \sqrt {x - 1} \left (x + 1\right )^{3} - 42 \sqrt {x - 1} \left (x + 1\right )^{2} + 84 \sqrt {x - 1} \left (x + 1\right ) - 56 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {7}{2}}}{7 \sqrt {1 - x} \left (x + 1\right )^{3} - 42 \sqrt {1 - x} \left (x + 1\right )^{2} + 84 \sqrt {1 - x} \left (x + 1\right ) - 56 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 19, normalized size = 0.95 \begin {gather*} \frac {{\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{7 \, {\left (x - 1\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 64, normalized size = 3.20 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {3\,x\,\sqrt {x+1}}{7}+\frac {\sqrt {x+1}}{7}+\frac {3\,x^2\,\sqrt {x+1}}{7}+\frac {x^3\,\sqrt {x+1}}{7}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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