3.828 \(\int \sqrt{x} \sqrt{\sqrt{x}+x} \, dx\)

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]

(5*(1 + 2*Sqrt[x])*Sqrt[Sqrt[x] + x])/32 - (5*(Sqrt[x] + x)^(3/2))/12 + (Sqrt[x]
*(Sqrt[x] + x)^(3/2))/2 - (5*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]])/32

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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]

(5*(1 + 2*Sqrt[x])*Sqrt[Sqrt[x] + x])/32 - (5*(Sqrt[x] + x)^(3/2))/12 + (Sqrt[x]
*(Sqrt[x] + x)^(3/2))/2 - (5*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]])/32

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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]

sqrt(x)*(sqrt(x) + x)**(3/2)/2 - 5*(sqrt(x) + x)**(3/2)/12 + 5*sqrt(sqrt(x) + x)
*(2*sqrt(x) + 1)/32 - 5*atanh(sqrt(x)/sqrt(sqrt(x) + x))/32

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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]

(Sqrt[Sqrt[x] + x]*(15 - 10*Sqrt[x] + 8*x + 48*x^(3/2)))/96 - (5*Log[1 + 2*Sqrt[
x] + 2*Sqrt[Sqrt[x] + x]])/64

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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]

1/2*x^(1/2)*(x+x^(1/2))^(3/2)-5/12*(x+x^(1/2))^(3/2)+5/32*(1+2*x^(1/2))*(x+x^(1/
2))^(1/2)-5/64*ln(1/2+x^(1/2)+(x+x^(1/2))^(1/2))

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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]

1/96*(15*(sqrt(x) + 1)^(7/2)/x^(7/4) - 55*(sqrt(x) + 1)^(5/2)/x^(5/4) + 73*(sqrt
(x) + 1)^(3/2)/x^(3/4) + 15*sqrt(sqrt(x) + 1)/x^(1/4))/((sqrt(x) + 1)^4/x^2 - 4*
(sqrt(x) + 1)^3/x^(3/2) + 6*(sqrt(x) + 1)^2/x - 4*(sqrt(x) + 1)/sqrt(x) + 1) - 5
/64*log(sqrt(sqrt(x) + 1)/x^(1/4) + 1) + 5/64*log(sqrt(sqrt(x) + 1)/x^(1/4) - 1)

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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]

1/96*(2*(24*x - 5)*sqrt(x) + 8*x + 15)*sqrt(x + sqrt(x)) + 5/128*log(4*sqrt(x +
sqrt(x))*(2*sqrt(x) + 1) - 8*x - 8*sqrt(x) - 1)

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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]

Integral(sqrt(x)*sqrt(sqrt(x) + x), x)

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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]

Timed out