Optimal. Leaf size=33 \[ \frac {4 x^{3/2}}{15 (x+1)^{3/2}}+\frac {2 x^{3/2}}{5 (x+1)^{5/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {45, 37} \[ \frac {4 x^{3/2}}{15 (x+1)^{3/2}}+\frac {2 x^{3/2}}{5 (x+1)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(1+x)^{7/2}} \, dx &=\frac {2 x^{3/2}}{5 (1+x)^{5/2}}+\frac {2}{5} \int \frac {\sqrt {x}}{(1+x)^{5/2}} \, dx\\ &=\frac {2 x^{3/2}}{5 (1+x)^{5/2}}+\frac {4 x^{3/2}}{15 (1+x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.64 \[ \frac {2 x^{3/2} (2 x+5)}{15 (x+1)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 50, normalized size = 1.52 \[ \frac {2 \, {\left (2 \, x^{3} + {\left (2 \, x^{2} + 5 \, x\right )} \sqrt {x + 1} \sqrt {x} + 6 \, x^{2} + 6 \, x + 2\right )}}{15 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 66, normalized size = 2.00 \[ \frac {8 \, {\left (15 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{6} - 5 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{4} + 5 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} + 1\right )}}{15 \, {\left ({\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.48 \[ \frac {2 \left (2 x +5\right ) x^{\frac {3}{2}}}{15 \left (x +1\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 20, normalized size = 0.61 \[ \frac {2 \, x^{\frac {5}{2}} {\left (\frac {5 \, {\left (x + 1\right )}}{x} - 3\right )}}{15 \, {\left (x + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 15, normalized size = 0.45 \[ \frac {2\,x^{3/2}\,\left (2\,x+5\right )}{15\,{\left (x+1\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.66, size = 165, normalized size = 5.00 \[ \begin {cases} \frac {4 i \sqrt {-1 + \frac {1}{x + 1}}}{15} + \frac {2 i \sqrt {-1 + \frac {1}{x + 1}}}{15 \left (x + 1\right )} - \frac {2 i \sqrt {-1 + \frac {1}{x + 1}}}{5 \left (x + 1\right )^{2}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > 1 \\\frac {4 \sqrt {1 - \frac {1}{x + 1}} \left (x + 1\right )^{2}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac {2 \sqrt {1 - \frac {1}{x + 1}} \left (x + 1\right )}{- 15 x + 15 \left (x + 1\right )^{2} - 15} - \frac {8 \sqrt {1 - \frac {1}{x + 1}}}{- 15 x + 15 \left (x + 1\right )^{2} - 15} + \frac {6 \sqrt {1 - \frac {1}{x + 1}}}{\left (x + 1\right ) \left (- 15 x + 15 \left (x + 1\right )^{2} - 15\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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