Optimal. Leaf size=76 \[ \frac{512 \left (2 \sqrt{x}+1\right )}{405 \sqrt{x+\sqrt{x}+1}}+\frac{64 \left (2 \sqrt{x}+1\right )}{135 \left (x+\sqrt{x}+1\right )^{3/2}}+\frac{4 \left (2 \sqrt{x}+1\right )}{15 \left (x+\sqrt{x}+1\right )^{5/2}} \]
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Rubi [A] time = 0.049974, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{512 \left (2 \sqrt{x}+1\right )}{405 \sqrt{x+\sqrt{x}+1}}+\frac{64 \left (2 \sqrt{x}+1\right )}{135 \left (x+\sqrt{x}+1\right )^{3/2}}+\frac{4 \left (2 \sqrt{x}+1\right )}{15 \left (x+\sqrt{x}+1\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]
[Out]
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Rubi in Sympy [A] time = 2.48128, size = 65, normalized size = 0.86 \[ \frac{64 \left (2 \sqrt{x} + 1\right )}{135 \left (\sqrt{x} + x + 1\right )^{\frac{3}{2}}} + \frac{4 \left (2 \sqrt{x} + 1\right )}{15 \left (\sqrt{x} + x + 1\right )^{\frac{5}{2}}} + \frac{256 \left (4 \sqrt{x} + 2\right )}{405 \sqrt{\sqrt{x} + x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(1/2)/(1+x+x**(1/2))**(7/2),x)
[Out]
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Mathematica [A] time = 0.0275061, size = 49, normalized size = 0.64 \[ \frac{4 \left (2 \sqrt{x}+1\right ) \left (256 x^{3/2}+128 x^2+432 x+304 \sqrt{x}+203\right )}{405 \left (x+\sqrt{x}+1\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]
[Out]
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Maple [A] time = 0.003, size = 53, normalized size = 0.7 \[{\frac{4}{15} \left ( 1+2\,\sqrt{x} \right ) \left ( 1+x+\sqrt{x} \right ) ^{-{\frac{5}{2}}}}+{\frac{64}{135} \left ( 1+2\,\sqrt{x} \right ) \left ( 1+x+\sqrt{x} \right ) ^{-{\frac{3}{2}}}}+{\frac{512}{405} \left ( 1+2\,\sqrt{x} \right ){\frac{1}{\sqrt{1+x+\sqrt{x}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(1/2)/(1+x+x^(1/2))^(7/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x + \sqrt{x} + 1\right )}^{\frac{7}{2}} \sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + sqrt(x) + 1)^(7/2)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300592, size = 128, normalized size = 1.68 \[ -\frac{4 \,{\left (128 \, x^{5} + 272 \, x^{4} + 455 \, x^{3} + 232 \, x^{2} -{\left (256 \, x^{5} + 736 \, x^{4} + 1366 \, x^{3} + 1427 \, x^{2} + 839 \, x + 101\right )} \sqrt{x} - 128 \, x - 203\right )} \sqrt{x + \sqrt{x} + 1}}{405 \,{\left (x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + sqrt(x) + 1)^(7/2)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(1/2)/(1+x+x**(1/2))**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268933, size = 61, normalized size = 0.8 \[ \frac{4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \, \sqrt{x}{\left (2 \, \sqrt{x} + 5\right )} + 35\right )} \sqrt{x} + 65\right )} \sqrt{x} + 355\right )} \sqrt{x} + 203\right )}}{405 \,{\left (x + \sqrt{x} + 1\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + sqrt(x) + 1)^(7/2)*sqrt(x)),x, algorithm="giac")
[Out]