Optimal. Leaf size=66 \[ \frac {1}{2} \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-\frac {3 \sin ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right )}{2 \sqrt {2}} \]
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Rubi [F]
time = 0.09, 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 \text {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 [A]
time = 0.64, size = 91, normalized size = 1.38 \begin {gather*} \frac {1}{4} \left (2 \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-x +\sqrt {x}\, \sqrt {1+x}}}{\sqrt {1+x}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs.
\(2 (46) = 92\).
time = 190.19, size = 127, normalized size = 1.92 \begin {gather*} \frac {1}{2} \, \sqrt {\sqrt {x + 1} \sqrt {x} - x} {\left (3 \, \sqrt {x + 1} + \sqrt {x}\right )} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {2 \, {\left (13230146471646497941753920 \, \sqrt {2} {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {x} + 472818412040 \, \sqrt {2} {\left (111925814517792 \, x^{2} + 83944360888344 \, x + 621904154881\right )} + 3497681703681 \, {\left (\sqrt {2} {\left (50987674019848 \, x - 621904154881\right )} \sqrt {x + 1} + \sqrt {2} {\left (60938140497944 \, x - 24871932855043\right )} \sqrt {x}\right )} \sqrt {\sqrt {x + 1} \sqrt {x} - x}\right )}}{139214258174109988596285504 \, x^{2} - 678551053586366160645570576 \, x + 386764777858250836124161}\right ) \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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