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

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

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Rubi [A]  time = 0.0062772, 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, Rules used = {190, 43} \[ \frac{1}{5} \left (\sqrt{x}+1\right )^{10}-\frac{2}{9} \left (\sqrt{x}+1\right )^9 \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])^8,x]

[Out]

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

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (1+\sqrt{x}\right )^8 \, dx &=2 \operatorname{Subst}\left (\int x (1+x)^8 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-(1+x)^8+(1+x)^9\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2}{9} \left (1+\sqrt{x}\right )^9+\frac{1}{5} \left (1+\sqrt{x}\right )^{10}\\ \end{align*}

Mathematica [A]  time = 0.0172791, size = 22, normalized size = 0.81 \[ \frac{1}{45} \left (\sqrt{x}+1\right )^9 \left (9 \sqrt{x}-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])^8,x]

[Out]

((1 + Sqrt[x])^9*(-1 + 9*Sqrt[x]))/45

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Maple [B]  time = 0.001, size = 43, normalized size = 1.6 \begin{align*}{\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)+1)^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

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Maxima [A]  time = 0.929772, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{5} \,{\left (\sqrt{x} + 1\right )}^{10} - \frac{2}{9} \,{\left (\sqrt{x} + 1\right )}^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^8,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.81559, size = 122, normalized size = 4.52 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^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)*sqrt(x) + x

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Sympy [B]  time = 0.808166, size = 54, normalized size = 2. \begin{align*} \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 \end{align*}

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

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Giac [B]  time = 1.09836, size = 57, normalized size = 2.11 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^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