Optimal. Leaf size=78 \[ -\frac {2 \sqrt {x+1}}{35 x^{3/2}}+\frac {3 \sqrt {x+1}}{70 x^{5/2}}-\frac {\sqrt {x+1}}{28 x^{7/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}+\frac {4 \sqrt {x+1}}{35 \sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5902, 12, 45, 37} \[ -\frac {2 \sqrt {x+1}}{35 x^{3/2}}+\frac {3 \sqrt {x+1}}{70 x^{5/2}}-\frac {\sqrt {x+1}}{28 x^{7/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}+\frac {4 \sqrt {x+1}}{35 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 45
Rule 5902
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\sqrt {x}\right )}{x^5} \, dx &=-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}+\frac {1}{4} \int \frac {1}{2 x^{9/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}+\frac {1}{8} \int \frac {1}{x^{9/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{28 x^{7/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}-\frac {3}{28} \int \frac {1}{x^{7/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{28 x^{7/2}}+\frac {3 \sqrt {1+x}}{70 x^{5/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}+\frac {3}{35} \int \frac {1}{x^{5/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{28 x^{7/2}}+\frac {3 \sqrt {1+x}}{70 x^{5/2}}-\frac {2 \sqrt {1+x}}{35 x^{3/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}-\frac {2}{35} \int \frac {1}{x^{3/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{28 x^{7/2}}+\frac {3 \sqrt {1+x}}{70 x^{5/2}}-\frac {2 \sqrt {1+x}}{35 x^{3/2}}+\frac {4 \sqrt {1+x}}{35 \sqrt {x}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.56 \[ \frac {\sqrt {x} \sqrt {x+1} \left (16 x^3-8 x^2+6 x-5\right )-35 \sinh ^{-1}\left (\sqrt {x}\right )}{140 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 42, normalized size = 0.54 \[ \frac {{\left (16 \, x^{3} - 8 \, x^{2} + 6 \, x - 5\right )} \sqrt {x + 1} \sqrt {x} - 35 \, \log \left (\sqrt {x + 1} + \sqrt {x}\right )}{140 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 82, normalized size = 1.05 \[ -\frac {\log \left (\sqrt {x + 1} + \sqrt {x}\right )}{4 \, x^{4}} + \frac {8 \, {\left (35 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{6} - 21 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{4} + 7 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}}{35 \, {\left ({\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.65 \[ -\frac {\arcsinh \left (\sqrt {x}\right )}{4 x^{4}}-\frac {\sqrt {1+x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1+x}}{70 x^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1+x}}{35 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 50, normalized size = 0.64 \[ \frac {4 \, \sqrt {x + 1}}{35 \, \sqrt {x}} - \frac {2 \, \sqrt {x + 1}}{35 \, x^{\frac {3}{2}}} + \frac {3 \, \sqrt {x + 1}}{70 \, x^{\frac {5}{2}}} - \frac {\sqrt {x + 1}}{28 \, x^{\frac {7}{2}}} - \frac {\operatorname {arsinh}\left (\sqrt {x}\right )}{4 \, x^{4}} \]
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 \[ \int \frac {\mathrm {asinh}\left (\sqrt {x}\right )}{x^5} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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