Optimal. Leaf size=93 \[ \frac {\sqrt {x+\sqrt {x+1}} \left (128 x^2+328 x+563\right )}{3840}+\frac {\sqrt {x+1} \left (640 x^2-872 x+975\right ) \sqrt {x+\sqrt {x+1}}}{1920}+\frac {385}{512} \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]
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Rubi [A] time = 0.41, antiderivative size = 147, normalized size of antiderivative = 1.58, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1661, 640, 612, 621, 206} \begin {gather*} \frac {1}{3} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}-\frac {3}{10} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}-\frac {39}{80} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}+\frac {33}{160} \left (x+\sqrt {x+1}\right )^{3/2}+\frac {77}{256} \left (2 \sqrt {x+1}+1\right ) \sqrt {x+\sqrt {x+1}}-\frac {385}{512} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {-1+x+x^2} \left (6-9 x^2-\frac {9 x^3}{2}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{15} \operatorname {Subst}\left (\int \left (30-9 x-\frac {117 x^2}{4}\right ) \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {39}{80} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{60} \operatorname {Subst}\left (\int \left (\frac {363}{4}+\frac {297 x}{8}\right ) \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {33}{160} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {39}{80} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {77}{64} \operatorname {Subst}\left (\int \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {33}{160} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {39}{80} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {77}{256} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {385}{512} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {33}{160} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {39}{80} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {77}{256} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {385}{256} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=\frac {33}{160} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {39}{80} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {3}{10} (1+x) \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{3} (1+x)^{3/2} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {77}{256} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {385}{512} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 92, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {x+\sqrt {x+1}} \left (128 \left (10 \sqrt {x+1}+1\right ) x^2-8 \left (218 \sqrt {x+1}-41\right ) x+1950 \sqrt {x+1}+563\right )-5775 \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )}{7680} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 90, normalized size = 0.97 \begin {gather*} \frac {\sqrt {x+\sqrt {1+x}} \left (363+4974 \sqrt {1+x}+72 (1+x)-4304 (1+x)^{3/2}+128 (1+x)^2+1280 (1+x)^{5/2}\right )}{3840}+\frac {385}{512} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 74, normalized size = 0.80 \begin {gather*} \frac {1}{3840} \, {\left (128 \, x^{2} + 2 \, {\left (640 \, x^{2} - 872 \, x + 975\right )} \sqrt {x + 1} + 328 \, x + 563\right )} \sqrt {x + \sqrt {x + 1}} + \frac {385}{1024} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 80, normalized size = 0.86 \begin {gather*} \frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, \sqrt {x + 1} {\left (10 \, \sqrt {x + 1} + 1\right )} - 269\right )} \sqrt {x + 1} + 9\right )} \sqrt {x + 1} + 2487\right )} \sqrt {x + 1} + 363\right )} \sqrt {x + \sqrt {x + 1}} + \frac {385}{512} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 98, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\left (1+x \right )^{\frac {3}{2}} \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (1+x \right ) \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{10}-\frac {39 \sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{80}+\frac {33 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{160}+\frac {77 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{256}-\frac {385 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{512}\) | \(98\) |
default | \(\frac {\left (1+x \right )^{\frac {3}{2}} \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (1+x \right ) \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{10}-\frac {39 \sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{80}+\frac {33 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{160}+\frac {77 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{256}-\frac {385 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{512}\) | \(98\) |
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 + \sqrt {x + 1}} x^{2}}{\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 = -0.01 \begin {gather*} \int \frac {x^2\,\sqrt {x+\sqrt {x+1}}}{\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 {x^{2} \sqrt {x + \sqrt {x + 1}}}{\sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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