Optimal. Leaf size=130 \[ \frac {\sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-384 x^2-136 x+255\right )}{960 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {1}{960} \left (384 x^2+568 x-85\right )-\frac {17 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{64 \sqrt {2} x}\right ) \]
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Rubi [F] time = 1.63, 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 {x \sqrt {x+x^2}}{\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 {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {x^{3/2} \sqrt {1+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 {x^4 \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 = 0.43, size = 137, normalized size = 1.05 \begin {gather*} \frac {\left (x+\sqrt {x (x+1)}\right )^2 \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right ) \left (17 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1+\frac {1}{2 \left (x+\sqrt {x (x+1)}\right )}\right )+10 \left (8 x^2+\left (8 \sqrt {x (x+1)}+11\right ) x+7 \sqrt {x (x+1)}\right )\right )}{80 \sqrt {x (x+1)} \left (2 x+2 \sqrt {x (x+1)}+1\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.64, size = 130, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-384 x^2-136 x+255\right )}{960 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {1}{960} \left (384 x^2+568 x-85\right )-\frac {17 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{64 \sqrt {2} x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 118, normalized size = 0.91 \begin {gather*} \frac {255 \, \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 \, {\left (384 \, x^{3} + 568 \, x^{2} - {\left (384 \, x^{2} + 136 \, x - 255\right )} \sqrt {x^{2} + x} - 85 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{3840 \, 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} x}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx \end {gather*}
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} x}{\sqrt {x^{2} + \sqrt {x^{2} + 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 {x\,\sqrt {x^2+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 {x \sqrt {x \left (x + 1\right )}}{\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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