Optimal. Leaf size=109 \[ \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {\sqrt {2} \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{x}+\frac {2 (x-2)}{x}\right )-\frac {2 \sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )}}{x} \]
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Rubi [F] time = 1.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {x+x^2}}{x \sqrt {x^2+x \sqrt {x+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {x+x^2}}{x \sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {\sqrt {1+x}}{\sqrt {x} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x} \sqrt {1+x}}\\ &=\frac {\left (2 \sqrt {x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [C] time = 0.48, size = 97, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {x (x+1)} \left (\left (x+\sqrt {x (x+1)}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{2 \left (x+\sqrt {x (x+1)}\right )}\right )+4 x+4 \sqrt {x (x+1)}+2\right )}{\sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.09, size = 109, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {2 (-2+x)}{x}+\frac {\sqrt {2} \sqrt {-x+\sqrt {x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 96, normalized size = 0.88 \begin {gather*} \frac {\sqrt {2} x \log \left (\frac {4 \, x^{2} + 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (x - \sqrt {x^{2} + x} - 2\right )}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+x}}{x \sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} x}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x^2+x}}{x\,\sqrt {x^2+x\,\sqrt {x^2+x}}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x \left (x + 1\right )}}{x \sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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