3.270 \(\int \left (1+\sqrt{x}\right )^8 \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{5} \left (\sqrt{x}+1\right )^{10}-\frac{2}{9} \left (\sqrt{x}+1\right )^9 \]

[Out]

(-2*(1 + Sqrt[x])^9)/9 + (1 + Sqrt[x])^10/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0191267, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{5} \left (\sqrt{x}+1\right )^{10}-\frac{2}{9} \left (\sqrt{x}+1\right )^9 \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[x])^8,x]

[Out]

(-2*(1 + Sqrt[x])^9)/9 + (1 + Sqrt[x])^10/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.07644, size = 20, normalized size = 0.74 \[ \frac{\left (\sqrt{x} + 1\right )^{10}}{5} - \frac{2 \left (\sqrt{x} + 1\right )^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**(1/2))**8,x)

[Out]

(sqrt(x) + 1)**10/5 - 2*(sqrt(x) + 1)**9/9

_______________________________________________________________________________________

Mathematica [B]  time = 0.00992939, size = 60, normalized size = 2.22 \[ \frac{16 x^{9/2}}{9}+16 x^{7/2}+\frac{112 x^{5/2}}{5}+\frac{16 x^{3/2}}{3}+\frac{x^5}{5}+7 x^4+\frac{70 x^3}{3}+14 x^2+x \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[x])^8,x]

[Out]

x + (16*x^(3/2))/3 + 14*x^2 + (112*x^(5/2))/5 + (70*x^3)/3 + 16*x^(7/2) + 7*x^4
+ (16*x^(9/2))/9 + x^5/5

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 43, normalized size = 1.6 \[{\frac{{x}^{5}}{5}}+{\frac{16}{9}{x}^{{\frac{9}{2}}}}+7\,{x}^{4}+16\,{x}^{7/2}+{\frac{70\,{x}^{3}}{3}}+{\frac{112}{5}{x}^{{\frac{5}{2}}}}+14\,{x}^{2}+{\frac{16}{3}{x}^{{\frac{3}{2}}}}+x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/2))^8,x)

[Out]

1/5*x^5+16/9*x^(9/2)+7*x^4+16*x^(7/2)+70/3*x^3+112/5*x^(5/2)+14*x^2+16/3*x^(3/2)
+x

_______________________________________________________________________________________

Maxima [A]  time = 1.33507, size = 26, normalized size = 0.96 \[ \frac{1}{5} \,{\left (\sqrt{x} + 1\right )}^{10} - \frac{2}{9} \,{\left (\sqrt{x} + 1\right )}^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)^8,x, algorithm="maxima")

[Out]

1/5*(sqrt(x) + 1)^10 - 2/9*(sqrt(x) + 1)^9

_______________________________________________________________________________________

Fricas [A]  time = 0.19428, size = 62, normalized size = 2.3 \[ \frac{1}{5} \, x^{5} + 7 \, x^{4} + \frac{70}{3} \, x^{3} + 14 \, x^{2} + \frac{16}{45} \,{\left (5 \, x^{4} + 45 \, x^{3} + 63 \, x^{2} + 15 \, x\right )} \sqrt{x} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)^8,x, algorithm="fricas")

[Out]

1/5*x^5 + 7*x^4 + 70/3*x^3 + 14*x^2 + 16/45*(5*x^4 + 45*x^3 + 63*x^2 + 15*x)*sqr
t(x) + x

_______________________________________________________________________________________

Sympy [A]  time = 1.20865, size = 54, normalized size = 2. \[ \frac{16 x^{\frac{9}{2}}}{9} + 16 x^{\frac{7}{2}} + \frac{112 x^{\frac{5}{2}}}{5} + \frac{16 x^{\frac{3}{2}}}{3} + \frac{x^{5}}{5} + 7 x^{4} + \frac{70 x^{3}}{3} + 14 x^{2} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**(1/2))**8,x)

[Out]

16*x**(9/2)/9 + 16*x**(7/2) + 112*x**(5/2)/5 + 16*x**(3/2)/3 + x**5/5 + 7*x**4 +
70*x**3/3 + 14*x**2 + x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.235854, size = 57, normalized size = 2.11 \[ \frac{1}{5} \, x^{5} + \frac{16}{9} \, x^{\frac{9}{2}} + 7 \, x^{4} + 16 \, x^{\frac{7}{2}} + \frac{70}{3} \, x^{3} + \frac{112}{5} \, x^{\frac{5}{2}} + 14 \, x^{2} + \frac{16}{3} \, x^{\frac{3}{2}} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)^8,x, algorithm="giac")

[Out]

1/5*x^5 + 16/9*x^(9/2) + 7*x^4 + 16*x^(7/2) + 70/3*x^3 + 112/5*x^(5/2) + 14*x^2
+ 16/3*x^(3/2) + x