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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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])

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 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]

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 = 0.01, size = 28, normalized size = 0.80 \[ \frac {3}{14} \left (\sqrt {x-3}+1\right )^{4/3} \left (4 \sqrt {x-3}-3\right ) \]

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

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 21, normalized size = 0.60 \[ \frac {3}{14} \, {\left (4 \, x + \sqrt {x - 3} - 15\right )} {\left (\sqrt {x - 3} + 1\right )}^{\frac {1}{3}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.32, size = 23, normalized size = 0.66 \[ \frac {6}{7} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {7}{3}} - \frac {3}{2} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {4}{3}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 24, normalized size = 0.69 \[ -\frac {3 \left (1+\sqrt {x -3}\right )^{\frac {4}{3}}}{2}+\frac {6 \left (1+\sqrt {x -3}\right )^{\frac {7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 23, normalized size = 0.66 \[ \frac {6}{7} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {7}{3}} - \frac {3}{2} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {4}{3}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 3.51, size = 16, normalized size = 0.46 \[ \left (x-3\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},2;\ 3;\ -\sqrt {x-3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x - 3)^(1/2) + 1)^(1/3),x)

[Out]

(x - 3)*hypergeom([-1/3, 2], 3, -(x - 3)^(1/2))

________________________________________________________________________________________

sympy [B]  time = 1.17, size = 184, normalized size = 5.26 \[ \frac {12 \left (x - 3\right )^{\frac {7}{2}} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} - \frac {6 \left (x - 3\right )^{\frac {5}{2}} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {9 \left (x - 3\right )^{\frac {5}{2}}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {15 \left (x - 3\right )^{3} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} - \frac {9 \left (x - 3\right )^{2} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {9 \left (x - 3\right )^{2}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} \]

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)

________________________________________________________________________________________