3.3.70 \(\int (1+\sqrt {x})^8 \, dx\) [270]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {196, 45} \begin {gather*} \frac {1}{5} \left (\sqrt {x}+1\right )^{10}-\frac {2}{9} \left (\sqrt {x}+1\right )^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])^8,x]

[Out]

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

Rule 45

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])

Rule 196

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]

Rubi steps

\begin {align*} \int \left (1+\sqrt {x}\right )^8 \, dx &=2 \text {Subst}\left (\int x (1+x)^8 \, dx,x,\sqrt {x}\right )\\ &=2 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(27)=54\).
time = 0.01, size = 56, normalized size = 2.07 \begin {gather*} \frac {1}{45} \left (45 x+240 x^{3/2}+630 x^2+1008 x^{5/2}+1050 x^3+720 x^{7/2}+315 x^4+80 x^{9/2}+9 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])^8,x]

[Out]

(45*x + 240*x^(3/2) + 630*x^2 + 1008*x^(5/2) + 1050*x^3 + 720*x^(7/2) + 315*x^4 + 80*x^(9/2) + 9*x^5)/45

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(27)=54\).
time = 1.91, size = 42, normalized size = 1.56 \begin {gather*} x+\frac {16 x^{\frac {3}{2}}}{3}+14 x^2+\frac {112 x^{\frac {5}{2}}}{5}+\frac {70 x^3}{3}+16 x^{\frac {7}{2}}+7 x^4+\frac {16 x^{\frac {9}{2}}}{9}+\frac {x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(42\) vs. \(2(19)=38\).
time = 0.04, size = 43, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {x^{5}}{5}+\frac {16 x^{\frac {9}{2}}}{9}+7 x^{4}+16 x^{\frac {7}{2}}+\frac {70 x^{3}}{3}+\frac {112 x^{\frac {5}{2}}}{5}+14 x^{2}+\frac {16 x^{\frac {3}{2}}}{3}+x\)	\(43\)
default	\(\frac {x^{5}}{5}+\frac {16 x^{\frac {9}{2}}}{9}+7 x^{4}+16 x^{\frac {7}{2}}+\frac {70 x^{3}}{3}+\frac {112 x^{\frac {5}{2}}}{5}+14 x^{2}+\frac {16 x^{\frac {3}{2}}}{3}+x\)	\(43\)
trager	\(\frac {\left (3 x^{4}+108 x^{3}+458 x^{2}+668 x +683\right ) \left (-1+x \right )}{15}+\frac {16 x^{\frac {3}{2}} \left (5 x^{3}+45 x^{2}+63 x +15\right )}{45}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(1/2))^8,x,method=_RETURNVERBOSE)

[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 = 0.25, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{5} \, {\left (\sqrt {x} + 1\right )}^{10} - \frac {2}{9} \, {\left (\sqrt {x} + 1\right )}^{9} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
time = 0.36, size = 46, normalized size = 1.70 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\)
time = 0.15, size = 54, normalized size = 2.00 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
time = 0.00, size = 79, normalized size = 2.93 \begin {gather*} \frac {x^{5}}{5}+\frac {16}{9} \sqrt {x} x^{4}+\frac {28}{4} x^{4}+\frac {112}{7} \sqrt {x} x^{3}+\frac {70}{3} x^{3}+\frac {112}{5} \sqrt {x} x^{2}+\frac {28}{2} x^{2}+\frac {16}{3} \sqrt {x} x+\frac {2}{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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Mupad [B]
time = 0.03, size = 42, normalized size = 1.56 \begin {gather*} x+14\,x^2+\frac {70\,x^3}{3}+7\,x^4+\frac {16\,x^{3/2}}{3}+\frac {x^5}{5}+\frac {112\,x^{5/2}}{5}+16\,x^{7/2}+\frac {16\,x^{9/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2) + 1)^8,x)

[Out]

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

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