3.963 \(\int \sqrt [3]{1+\sqrt{-3+x}} \, dx\)

Optimal. Leaf size=35 \[ \frac{6}{7} \left (\sqrt{x-3}+1\right )^{7/3}-\frac{3}{2} \left (\sqrt{x-3}+1\right )^{4/3} \]

[Out]

(-3*(1 + Sqrt[-3 + x])^(4/3))/2 + (6*(1 + Sqrt[-3 + x])^(7/3))/7

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Rubi [A]  time = 0.0111444, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 190, 43} \[ \frac{6}{7} \left (\sqrt{x-3}+1\right )^{7/3}-\frac{3}{2} \left (\sqrt{x-3}+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[-3 + x])^(1/3),x]

[Out]

(-3*(1 + Sqrt[-3 + x])^(4/3))/2 + (6*(1 + Sqrt[-3 + x])^(7/3))/7

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

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 \sqrt [3]{1+\sqrt{-3+x}} \, dx &=\operatorname{Subst}\left (\int \sqrt [3]{1+\sqrt{x}} \, dx,x,-3+x\right )\\ &=2 \operatorname{Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,\sqrt{-3+x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,\sqrt{-3+x}\right )\\ &=-\frac{3}{2} \left (1+\sqrt{-3+x}\right )^{4/3}+\frac{6}{7} \left (1+\sqrt{-3+x}\right )^{7/3}\\ \end{align*}

Mathematica [A]  time = 0.0098442, size = 28, normalized size = 0.8 \[ \frac{3}{14} \left (\sqrt{x-3}+1\right )^{4/3} \left (4 \sqrt{x-3}-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[-3 + x])^(1/3),x]

[Out]

(3*(1 + Sqrt[-3 + x])^(4/3)*(-3 + 4*Sqrt[-3 + x]))/14

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Maple [A]  time = 0.002, size = 24, normalized size = 0.7 \begin{align*} -{\frac{3}{2} \left ( 1+\sqrt{-3+x} \right ) ^{{\frac{4}{3}}}}+{\frac{6}{7} \left ( 1+\sqrt{-3+x} \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+(-3+x)^(1/2))^(1/3),x)

[Out]

-3/2*(1+(-3+x)^(1/2))^(4/3)+6/7*(1+(-3+x)^(1/2))^(7/3)

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Maxima [A]  time = 0.996148, size = 31, normalized size = 0.89 \begin{align*} \frac{6}{7} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{7}{3}} - \frac{3}{2} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(-3+x)^(1/2))^(1/3),x, algorithm="maxima")

[Out]

6/7*(sqrt(x - 3) + 1)^(7/3) - 3/2*(sqrt(x - 3) + 1)^(4/3)

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Fricas [A]  time = 1.49347, size = 74, normalized size = 2.11 \begin{align*} \frac{3}{14} \,{\left (4 \, x + \sqrt{x - 3} - 15\right )}{\left (\sqrt{x - 3} + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(-3+x)^(1/2))^(1/3),x, algorithm="fricas")

[Out]

3/14*(4*x + sqrt(x - 3) - 15)*(sqrt(x - 3) + 1)^(1/3)

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Sympy [B]  time = 1.02531, size = 184, normalized size = 5.26 \begin{align*} \frac{12 \left (x - 3\right )^{\frac{7}{2}} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} - \frac{6 \left (x - 3\right )^{\frac{5}{2}} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{9 \left (x - 3\right )^{\frac{5}{2}}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{15 \left (x - 3\right )^{3} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} - \frac{9 \left (x - 3\right )^{2} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{9 \left (x - 3\right )^{2}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(-3+x)**(1/2))**(1/3),x)

[Out]

12*(x - 3)**(7/2)*(sqrt(x - 3) + 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) - 6*(x - 3)**(5/2)*(sqrt(x - 3)
+ 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) + 9*(x - 3)**(5/2)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) + 15*(
x - 3)**3*(sqrt(x - 3) + 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) - 9*(x - 3)**2*(sqrt(x - 3) + 1)**(1/3)
/(14*(x - 3)**(5/2) + 14*(x - 3)**2) + 9*(x - 3)**2/(14*(x - 3)**(5/2) + 14*(x - 3)**2)

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Giac [A]  time = 1.23988, size = 31, normalized size = 0.89 \begin{align*} \frac{6}{7} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{7}{3}} - \frac{3}{2} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(-3+x)^(1/2))^(1/3),x, algorithm="giac")

[Out]

6/7*(sqrt(x - 3) + 1)^(7/3) - 3/2*(sqrt(x - 3) + 1)^(4/3)