Optimal. Leaf size=20 \[ \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \]
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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)^{3/2}}{3 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 20, normalized size = 1.00 \begin {gather*} \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \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. \(29\) vs.
\(2(14)=28\).
time = 0.16, size = 30, normalized size = 1.50
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (1-x \right )^{\frac {3}{2}}}\) | \(15\) |
default | \(\frac {2 \sqrt {1+x}}{3 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\) | \(30\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+2 x +1\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(49\) |
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. 38 vs.
\(2 (14) = 28\).
time = 0.27, size = 38, normalized size = 1.90 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\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. 33 vs.
\(2 (14) = 28\).
time = 1.08, size = 33, normalized size = 1.65 \begin {gather*} \frac {x^{2} + {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, 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 = 0.84, size = 60, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \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 = 0.99, size = 19, normalized size = 0.95 \begin {gather*} \frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 34, normalized size = 1.70 \begin {gather*} \frac {\left (\frac {x\,\sqrt {x+1}}{3}+\frac {\sqrt {x+1}}{3}\right )\,\sqrt {1-x}}{x^2-2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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