3.2381 \(\int \frac{1}{\sqrt{1+\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=42 \[ \frac{6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt{\sqrt [3]{x}+1} \]

[Out]

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

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Rubi [A]  time = 0.0106895, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt{\sqrt [3]{x}+1} \]

Antiderivative was successfully verified.

[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 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 \frac{1}{\sqrt{1+\sqrt [3]{x}}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{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]  time = 0.0091384, size = 31, normalized size = 0.74 \[ \frac{2}{5} \sqrt{\sqrt [3]{x}+1} \left (3 x^{2/3}-4 \sqrt [3]{x}+8\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.005, size = 29, normalized size = 0.7 \begin{align*} -4\, \left ( \sqrt [3]{x}+1 \right ) ^{3/2}+{\frac{6}{5} \left ( \sqrt [3]{x}+1 \right ) ^{{\frac{5}{2}}}}+6\,\sqrt{\sqrt [3]{x}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05006, size = 38, normalized size = 0.9 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.49724, size = 69, normalized size = 1.64 \begin{align*} \frac{2}{5} \,{\left (3 \, x^{\frac{2}{3}} - 4 \, x^{\frac{1}{3}} + 8\right )} \sqrt{x^{\frac{1}{3}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 1.10344, size = 359, normalized size = 8.55 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.20967, size = 38, normalized size = 0.9 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)