3.2389 \(\int \frac{\sqrt{x}}{1+x^{2/3}} \, dx\)

Optimal. Leaf size=117 \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0771903, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {341, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x^(2/3)),x]

[Out]

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

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
 && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
 x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
 FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{1+x^{2/3}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{7/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{6 x^{5/6}}{5}-3 \operatorname{Subst}\left (\int \frac{x^{3/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+6 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+3 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )+3 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0243979, size = 117, normalized size = 1. \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + x^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 76, normalized size = 0.7 \begin{align*}{\frac{6}{5}{x}^{{\frac{5}{6}}}}-6\,\sqrt [6]{x}+{\frac{3\,\sqrt{2}}{2}\arctan \left ( -1+\sqrt [6]{x}\sqrt{2} \right ) }+{\frac{3\,\sqrt{2}}{4}\ln \left ({ \left ( 1+\sqrt [3]{x}+\sqrt [6]{x}\sqrt{2} \right ) \left ( 1+\sqrt [3]{x}-\sqrt [6]{x}\sqrt{2} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{2}\arctan \left ( 1+\sqrt [6]{x}\sqrt{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44915, size = 119, normalized size = 1.02 \begin{align*} \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57731, size = 424, normalized size = 3.62 \begin{align*} -3 \, \sqrt{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1} - \sqrt{2} x^{\frac{1}{6}} - 1\right ) - 3 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4} - \sqrt{2} x^{\frac{1}{6}} + 1\right ) + \frac{3}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*sqrt(2)*arctan(sqrt(2)*sqrt(sqrt(2)*x^(1/6) + x^(1/3) + 1) - sqrt(2)*x^(1/6) - 1) - 3*sqrt(2)*arctan(1/2*sq
rt(2)*sqrt(-4*sqrt(2)*x^(1/6) + 4*x^(1/3) + 4) - sqrt(2)*x^(1/6) + 1) + 3/4*sqrt(2)*log(4*sqrt(2)*x^(1/6) + 4*
x^(1/3) + 4) - 3/4*sqrt(2)*log(-4*sqrt(2)*x^(1/6) + 4*x^(1/3) + 4) + 6/5*x^(5/6) - 6*x^(1/6)

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Sympy [C]  time = 1.10253, size = 187, normalized size = 1.6 \begin{align*} \frac{27 x^{\frac{5}{6}} \Gamma \left (\frac{9}{4}\right )}{10 \Gamma \left (\frac{13}{4}\right )} - \frac{27 \sqrt [6]{x} \Gamma \left (\frac{9}{4}\right )}{2 \Gamma \left (\frac{13}{4}\right )} - \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{3 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} - \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{7 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*x**(5/6)*gamma(9/4)/(10*gamma(13/4)) - 27*x**(1/6)*gamma(9/4)/(2*gamma(13/4)) - 27*exp(-I*pi/4)*log(-x**(1/
6)*exp_polar(I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4)) + 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(3*I*pi/4) + 1
)*gamma(9/4)/(8*gamma(13/4)) + 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(5*I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4
)) - 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(7*I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4))

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Giac [A]  time = 1.24713, size = 119, normalized size = 1.02 \begin{align*} \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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