Optimal. Leaf size=72 \[ -\frac {1}{18} \sqrt {x+1} x^{5/2}+\frac {5}{72} \sqrt {x+1} x^{3/2}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \sqrt {x+1} \sqrt {x}+\frac {5}{48} \sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5902, 12, 50, 54, 215} \[ -\frac {1}{18} \sqrt {x+1} x^{5/2}+\frac {5}{72} \sqrt {x+1} x^{3/2}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \sqrt {x+1} \sqrt {x}+\frac {5}{48} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 54
Rule 215
Rule 5902
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x^{5/2}}{2 \sqrt {1+x}} \, dx\\ &=\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{5/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {1}{18} x^{5/2} \sqrt {1+x}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {5}{36} \int \frac {x^{3/2}}{\sqrt {1+x}} \, dx\\ &=\frac {5}{72} x^{3/2} \sqrt {1+x}-\frac {1}{18} x^{5/2} \sqrt {1+x}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=-\frac {5}{48} \sqrt {x} \sqrt {1+x}+\frac {5}{72} x^{3/2} \sqrt {1+x}-\frac {1}{18} x^{5/2} \sqrt {1+x}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {5}{96} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=-\frac {5}{48} \sqrt {x} \sqrt {1+x}+\frac {5}{72} x^{3/2} \sqrt {1+x}-\frac {1}{18} x^{5/2} \sqrt {1+x}+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )+\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{48} \sqrt {x} \sqrt {1+x}+\frac {5}{72} x^{3/2} \sqrt {1+x}-\frac {1}{18} x^{5/2} \sqrt {1+x}+\frac {5}{48} \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} x^3 \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.60 \[ \frac {1}{144} \left (3 \left (16 x^3+5\right ) \sinh ^{-1}\left (\sqrt {x}\right )+\sqrt {x} \sqrt {x+1} \left (-8 x^2+10 x-15\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 40, normalized size = 0.56 \[ -\frac {1}{144} \, {\left (8 \, x^{2} - 10 \, x + 15\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{48} \, {\left (16 \, x^{3} + 5\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 50, normalized size = 0.69 \[ \frac {1}{3} \, x^{3} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{144} \, {\left (2 \, {\left (4 \, x - 5\right )} x + 15\right )} \sqrt {x + 1} \sqrt {x} - \frac {5}{48} \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.65 \[ \frac {5 \arcsinh \left (\sqrt {x}\right )}{48}+\frac {x^{3} \arcsinh \left (\sqrt {x}\right )}{3}+\frac {5 x^{\frac {3}{2}} \sqrt {1+x}}{72}-\frac {x^{\frac {5}{2}} \sqrt {1+x}}{18}-\frac {5 \sqrt {x}\, \sqrt {1+x}}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 46, normalized size = 0.64 \[ \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (\sqrt {x}\right ) - \frac {1}{18} \, \sqrt {x + 1} x^{\frac {5}{2}} + \frac {5}{72} \, \sqrt {x + 1} x^{\frac {3}{2}} - \frac {5}{48} \, \sqrt {x + 1} \sqrt {x} + \frac {5}{48} \, \operatorname {arsinh}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {asinh}\left (\sqrt {x}\right ) \,d x \]
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 \[ \int x^{2} \operatorname {asinh}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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