\(\int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx\) [2381]

Optimal result

Integrand size = 11, antiderivative size = 42 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=6 \sqrt {1+\sqrt [3]{x}}-4 \left (1+\sqrt [3]{x}\right )^{3/2}+\frac {6}{5} \left (1+\sqrt [3]{x}\right )^{5/2} \]

[Out]

-4*(1+x^(1/3))^(3/2)+6/5*(1+x^(1/3))^(5/2)+6*(1+x^(1/3))^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt {\sqrt [3]{x}+1} \]

[In]

Int[1/Sqrt[1 + x^(1/3)],x]

[Out]

6*Sqrt[1 + x^(1/3)] - 4*(1 + x^(1/3))^(3/2) + (6*(1 + x^(1/3))^(5/2))/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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}-2 \sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = 6 \sqrt {1+\sqrt [3]{x}}-4 \left (1+\sqrt [3]{x}\right )^{3/2}+\frac {6}{5} \left (1+\sqrt [3]{x}\right )^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {2}{5} \sqrt {1+\sqrt [3]{x}} \left (8-4 \sqrt [3]{x}+3 x^{2/3}\right ) \]

[In]

Integrate[1/Sqrt[1 + x^(1/3)],x]

[Out]

(2*Sqrt[1 + x^(1/3)]*(8 - 4*x^(1/3) + 3*x^(2/3)))/5

Maple [A] (verified)

Time = 3.61 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(-4 \left (1+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+\frac {6 \left (1+x^{\frac {1}{3}}\right )^{\frac {5}{2}}}{5}+6 \sqrt {1+x^{\frac {1}{3}}}\)	\(29\)
default	\(-4 \left (1+x^{\frac {1}{3}}\right )^{\frac {3}{2}}+\frac {6 \left (1+x^{\frac {1}{3}}\right )^{\frac {5}{2}}}{5}+6 \sqrt {1+x^{\frac {1}{3}}}\)	\(29\)
meijerg	\(\frac {-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (6 x^{\frac {2}{3}}-8 x^{\frac {1}{3}}+16\right ) \sqrt {1+x^{\frac {1}{3}}}}{5}}{\sqrt {\pi }}\)	\(36\)

[In]

int(1/(1+x^(1/3))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-4*(1+x^(1/3))^(3/2)+6/5*(1+x^(1/3))^(5/2)+6*(1+x^(1/3))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {2}{5} \, {\left (3 \, x^{\frac {2}{3}} - 4 \, x^{\frac {1}{3}} + 8\right )} \sqrt {x^{\frac {1}{3}} + 1} \]

[In]

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

[Out]

2/5*(3*x^(2/3) - 4*x^(1/3) + 8)*sqrt(x^(1/3) + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (36) = 72\).

Time = 0.76 (sec) , antiderivative size = 359, normalized size of antiderivative = 8.55 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6 x^{\frac {14}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{\frac {13}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {30 x^{\frac {11}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {11}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {40 x^{\frac {10}{3}} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {48 x^{\frac {10}{3}}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {10 x^{4} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{4}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} + \frac {16 x^{3} \sqrt {\sqrt [3]{x} + 1}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} - \frac {16 x^{3}}{15 x^{\frac {11}{3}} + 15 x^{\frac {10}{3}} + 5 x^{4} + 5 x^{3}} \]

[In]

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

[Out]

6*x**(14/3)*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 10*x**(13/3)*sqrt(x**(1/3) +
1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 30*x**(11/3)*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10
/3) + 5*x**4 + 5*x**3) - 48*x**(11/3)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 40*x**(10/3)*sqrt(x**(
1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) - 48*x**(10/3)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4
+ 5*x**3) + 10*x**4*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) - 16*x**4/(15*x**(11/3
) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 16*x**3*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**
3) - 16*x**3/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]

[In]

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

[Out]

6/5*(x^(1/3) + 1)^(5/2) - 4*(x^(1/3) + 1)^(3/2) + 6*sqrt(x^(1/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=\frac {6}{5} \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {x^{\frac {1}{3}} + 1} \]

[In]

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

[Out]

6/5*(x^(1/3) + 1)^(5/2) - 4*(x^(1/3) + 1)^(3/2) + 6*sqrt(x^(1/3) + 1)

Mupad [B] (verification not implemented)

Time = 5.91 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt {1+\sqrt [3]{x}}} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},3;\ 4;\ -x^{1/3}\right ) \]

[In]

int(1/(x^(1/3) + 1)^(1/2),x)

[Out]

x*hypergeom([1/2, 3], 4, -x^(1/3))