Optimal. Leaf size=66 \[ \frac {1}{2} \left (\sqrt {x}+3 \sqrt {x+1}\right ) \sqrt {\sqrt {x} \sqrt {x+1}-x}-\frac {3 \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )}{2 \sqrt {2}} \]
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Rubi [F] time = 0.14, 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+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}
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Mathematica [B] time = 0.53, size = 180, normalized size = 2.73 \begin {gather*} -\frac {(x+1) \left (2 x-2 \sqrt {x+1} \sqrt {x}+1\right )^2 \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x} \left (-2 x+2 \sqrt {x+1} \sqrt {x}-3\right )+3 \sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2} \log \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x}+\sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2}\right )\right )}{4 \left (\sqrt {x+1}-\sqrt {x}\right )^3 \left (x-\sqrt {x+1} \sqrt {x}+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.00, size = 130, normalized size = 1.97 \begin {gather*} \frac {x^{3/2}+3 (x+1)^{3/2}-3 \sqrt {x+1}}{2 \sqrt {\sqrt {x} \sqrt {x+1}-x} x+2 \sqrt {x+1} \sqrt {\sqrt {x} \sqrt {x+1}-x} \sqrt {x}}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\sqrt {x} \sqrt {x+1}-x}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x +\sqrt {x +1}\, \sqrt {x}}}{\sqrt {x +1}}\, 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 {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}}\,{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 = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x}\,\sqrt {x+1}-x}}{\sqrt {x+1}} \,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 {\sqrt {x} \sqrt {x + 1} - x}}{\sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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