Optimal. Leaf size=56 \[ -\frac {1}{8} \sqrt {x+1} x^{3/2}+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {3}{16} \sqrt {x+1} \sqrt {x}-\frac {3}{16} \sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5902, 12, 50, 54, 215} \[ -\frac {1}{8} \sqrt {x+1} x^{3/2}+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {3}{16} \sqrt {x+1} \sqrt {x}-\frac {3}{16} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 54
Rule 215
Rule 5902
Rubi steps
\begin {align*} \int x \sinh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {x^{3/2}}{2 \sqrt {1+x}} \, dx\\ &=\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {3}{16} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=\frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {3}{32} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=\frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}-\frac {3}{16} \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} x^2 \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.66 \[ \frac {1}{16} \left (\left (8 x^2-3\right ) \sinh ^{-1}\left (\sqrt {x}\right )+\sqrt {x} \sqrt {x+1} (3-2 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 35, normalized size = 0.62 \[ -\frac {1}{16} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{16} \, {\left (8 \, x^{2} - 3\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 48, normalized size = 0.86 \[ \frac {1}{2} \, x^{2} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{16} \, \sqrt {x^{2} + x} {\left (2 \, x - 3\right )} + \frac {3}{32} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.66 \[ -\frac {3 \arcsinh \left (\sqrt {x}\right )}{16}+\frac {x^{2} \arcsinh \left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {1+x}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 36, normalized size = 0.64 \[ \frac {1}{2} \, x^{2} \operatorname {arsinh}\left (\sqrt {x}\right ) - \frac {1}{8} \, \sqrt {x + 1} x^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {x + 1} \sqrt {x} - \frac {3}{16} \, \operatorname {arsinh}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {asinh}\left (\sqrt {x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {asinh}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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