Optimal. Leaf size=75 \[ \frac {1}{4} x^{3/2} (x+1)^{5/2}+\frac {5}{24} x^{3/2} (x+1)^{3/2}+\frac {5}{32} x^{3/2} \sqrt {x+1}+\frac {5}{64} \sqrt {x} \sqrt {x+1}-\frac {5}{64} \sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {50, 54, 215} \[ \frac {1}{4} x^{3/2} (x+1)^{5/2}+\frac {5}{24} x^{3/2} (x+1)^{3/2}+\frac {5}{32} x^{3/2} \sqrt {x+1}+\frac {5}{64} \sqrt {x} \sqrt {x+1}-\frac {5}{64} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \sqrt {x} (1+x)^{5/2} \, dx &=\frac {1}{4} x^{3/2} (1+x)^{5/2}+\frac {5}{8} \int \sqrt {x} (1+x)^{3/2} \, dx\\ &=\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}+\frac {5}{16} \int \sqrt {x} \sqrt {1+x} \, dx\\ &=\frac {5}{32} x^{3/2} \sqrt {1+x}+\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}+\frac {5}{64} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=\frac {5}{64} \sqrt {x} \sqrt {1+x}+\frac {5}{32} x^{3/2} \sqrt {1+x}+\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}-\frac {5}{128} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=\frac {5}{64} \sqrt {x} \sqrt {1+x}+\frac {5}{32} x^{3/2} \sqrt {1+x}+\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}-\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{64} \sqrt {x} \sqrt {1+x}+\frac {5}{32} x^{3/2} \sqrt {1+x}+\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}-\frac {5}{64} \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.55 \[ \frac {1}{192} \left (\sqrt {x} \sqrt {x+1} \left (48 x^3+136 x^2+118 x+15\right )-15 \sinh ^{-1}\left (\sqrt {x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 44, normalized size = 0.59 \[ \frac {1}{192} \, {\left (48 \, x^{3} + 136 \, x^{2} + 118 \, x + 15\right )} \sqrt {x + 1} \sqrt {x} + \frac {5}{128} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 90, normalized size = 1.20 \[ \frac {1}{192} \, {\left (2 \, {\left (4 \, {\left (6 \, x - 19\right )} {\left (x + 1\right )} + 163\right )} {\left (x + 1\right )} - 279\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{8} \, {\left (2 \, {\left (4 \, x - 9\right )} {\left (x + 1\right )} + 33\right )} \sqrt {x + 1} \sqrt {x} + \frac {3}{4} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \sqrt {x + 1} \sqrt {x} + \frac {5}{64} \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 70, normalized size = 0.93 \[ \frac {\left (x +1\right )^{\frac {7}{2}} \sqrt {x}}{4}-\frac {\left (x +1\right )^{\frac {5}{2}} \sqrt {x}}{24}-\frac {5 \left (x +1\right )^{\frac {3}{2}} \sqrt {x}}{96}-\frac {5 \sqrt {x +1}\, \sqrt {x}}{64}-\frac {5 \sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{128 \sqrt {x +1}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 113, normalized size = 1.51 \[ \frac {\frac {15 \, {\left (x + 1\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}} + \frac {73 \, {\left (x + 1\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}} - \frac {55 \, {\left (x + 1\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} + \frac {15 \, \sqrt {x + 1}}{\sqrt {x}}}{192 \, {\left (\frac {{\left (x + 1\right )}^{4}}{x^{4}} - \frac {4 \, {\left (x + 1\right )}^{3}}{x^{3}} + \frac {6 \, {\left (x + 1\right )}^{2}}{x^{2}} - \frac {4 \, {\left (x + 1\right )}}{x} + 1\right )}} - \frac {5}{128} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) + \frac {5}{128} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x}\,{\left (x+1\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.75, size = 190, normalized size = 2.53 \[ \begin {cases} - \frac {5 \operatorname {acosh}{\left (\sqrt {x + 1} \right )}}{64} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x}} - \frac {7 \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {x}} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{96 \sqrt {x}} - \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{192 \sqrt {x}} + \frac {5 \sqrt {x + 1}}{64 \sqrt {x}} & \text {for}\: \left |{x + 1}\right | > 1 \\\frac {5 i \operatorname {asin}{\left (\sqrt {x + 1} \right )}}{64} - \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {- x}} + \frac {7 i \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {- x}} + \frac {i \left (x + 1\right )^{\frac {5}{2}}}{96 \sqrt {- x}} + \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{192 \sqrt {- x}} - \frac {5 i \sqrt {x + 1}}{64 \sqrt {- x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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