3.2382 \(\int \frac{1}{(1+\sqrt [3]{x}) x^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{6}{\sqrt [6]{x}}-\frac{2}{\sqrt{x}}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

-2/Sqrt[x] + 6/x^(1/6) + 6*ArcTan[x^(1/6)]

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Rubi [A]  time = 0.0081758, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {341, 51, 63, 203} \[ \frac{6}{\sqrt [6]{x}}-\frac{2}{\sqrt{x}}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^(1/3))*x^(3/2)),x]

[Out]

-2/Sqrt[x] + 6/x^(1/6) + 6*ArcTan[x^(1/6)]

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.0044194, size = 22, normalized size = 0.96 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\sqrt [3]{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^(1/3))*x^(3/2)),x]

[Out]

(-2*Hypergeometric2F1[-3/2, 1, -1/2, -x^(1/3)])/Sqrt[x]

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Maple [A]  time = 0.006, size = 18, normalized size = 0.8 \begin{align*} 6\,{\frac{1}{\sqrt [6]{x}}}+6\,\arctan \left ( \sqrt [6]{x} \right ) -2\,{\frac{1}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/x^(1/6)+6*arctan(x^(1/6))-2/x^(1/2)

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Maxima [A]  time = 1.45134, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \,{\left (3 \, x^{\frac{1}{3}} - 1\right )}}{\sqrt{x}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42815, size = 66, normalized size = 2.87 \begin{align*} \frac{2 \,{\left (3 \, x \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, x^{\frac{5}{6}} - \sqrt{x}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.01563, size = 20, normalized size = 0.87 \begin{align*} 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} - \frac{2}{\sqrt{x}} + \frac{6}{\sqrt [6]{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*atan(x**(1/6)) - 2/sqrt(x) + 6/x**(1/6)

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Giac [A]  time = 1.14844, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \,{\left (3 \, x^{\frac{1}{3}} - 1\right )}}{\sqrt{x}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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