Optimal. Leaf size=82 \[ \frac{1}{2} \sqrt{x} \left (x+\sqrt{x}\right )^{3/2}-\frac{5}{12} \left (x+\sqrt{x}\right )^{3/2}+\frac{5}{32} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}}-\frac{5}{32} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
[Out]
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Rubi [A] time = 0.0859314, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{1}{2} \sqrt{x} \left (x+\sqrt{x}\right )^{3/2}-\frac{5}{12} \left (x+\sqrt{x}\right )^{3/2}+\frac{5}{32} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}}-\frac{5}{32} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*Sqrt[Sqrt[x] + x],x]
[Out]
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Rubi in Sympy [A] time = 4.77879, size = 71, normalized size = 0.87 \[ \frac{\sqrt{x} \left (\sqrt{x} + x\right )^{\frac{3}{2}}}{2} - \frac{5 \left (\sqrt{x} + x\right )^{\frac{3}{2}}}{12} + \frac{5 \sqrt{\sqrt{x} + x} \left (2 \sqrt{x} + 1\right )}{32} - \frac{5 \operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{\sqrt{x} + x}} \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(x+x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0395397, size = 62, normalized size = 0.76 \[ \frac{1}{96} \sqrt{x+\sqrt{x}} \left (48 x^{3/2}+8 x-10 \sqrt{x}+15\right )-\frac{5}{64} \log \left (2 \sqrt{x}+2 \sqrt{x+\sqrt{x}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*Sqrt[Sqrt[x] + x],x]
[Out]
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Maple [A] time = 0.005, size = 54, normalized size = 0.7 \[{\frac{1}{2}\sqrt{x} \left ( x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{5}{12} \left ( x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}+{\frac{5}{32} \left ( 1+2\,\sqrt{x} \right ) \sqrt{x+\sqrt{x}}}-{\frac{5}{64}\ln \left ({\frac{1}{2}}+\sqrt{x}+\sqrt{x+\sqrt{x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(x+x^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 0.750809, size = 180, normalized size = 2.2 \[ \frac{\frac{15 \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}}}{x^{\frac{7}{4}}} - \frac{55 \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}}}{x^{\frac{5}{4}}} + \frac{73 \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{4}}} + \frac{15 \, \sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}}}{96 \,{\left (\frac{{\left (\sqrt{x} + 1\right )}^{4}}{x^{2}} - \frac{4 \,{\left (\sqrt{x} + 1\right )}^{3}}{x^{\frac{3}{2}}} + \frac{6 \,{\left (\sqrt{x} + 1\right )}^{2}}{x} - \frac{4 \,{\left (\sqrt{x} + 1\right )}}{\sqrt{x}} + 1\right )}} - \frac{5}{64} \, \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} + 1\right ) + \frac{5}{64} \, \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x))*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.701001, size = 73, normalized size = 0.89 \[ \frac{1}{96} \,{\left (2 \,{\left (24 \, x - 5\right )} \sqrt{x} + 8 \, x + 15\right )} \sqrt{x + \sqrt{x}} + \frac{5}{128} \, \log \left (4 \, \sqrt{x + \sqrt{x}}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x))*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{\sqrt{x} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(x+x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x))*sqrt(x),x, algorithm="giac")
[Out]