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

Optimal. Leaf size=201 \[ x+6 \sqrt [6]{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

[Out]

6*x^(1/6) + x - (3*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5 + Sqrt[5])]])/5 - (3*Sqrt[
2*(5 - Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3
*(1 - Sqrt[5])*Log[2 + x^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 + Sqrt[5])*Log[2 + x^(1/6) + Sqrt[5]
*x^(1/6) + 2*x^(1/3)])/10

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Rubi [A]  time = 0.216354, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1584, 341, 302, 202, 634, 618, 204, 628, 31} \[ x+6 \sqrt [6]{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + x - (3*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5 + Sqrt[5])]])/5 - (3*Sqrt[
2*(5 - Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3
*(1 - Sqrt[5])*Log[2 + x^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 + Sqrt[5])*Log[2 + x^(1/6) + Sqrt[5]
*x^(1/6) + 2*x^(1/3)])/10

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 202

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x
]; (r*Int[1/(r - s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] &&
IGtQ[(n - 3)/2, 0] && NegQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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])

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{-\frac{1}{\sqrt [3]{x}}+\sqrt{x}} \, dx &=\int \frac{x^{5/6}}{-1+x^{5/6}} \, dx\\ &=6 \operatorname{Subst}\left (\int \frac{x^{10}}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (1+x^5+\frac{1}{-1+x^5}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x+6 \operatorname{Subst}\left (\int \frac{1}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x-\frac{6}{5} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac{12}{5} \operatorname{Subst}\left (\int \frac{1+\frac{1}{4} \left (1-\sqrt{5}\right ) x}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{12}{5} \operatorname{Subst}\left (\int \frac{1+\frac{1}{4} \left (1+\sqrt{5}\right ) x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{1}{10} \left (3 \left (1-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1-\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{1}{10} \left (3 \left (5-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{1}{10} \left (3 \left (1+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1+\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{1}{10} \left (3 \left (5+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )+\frac{1}{5} \left (3 \left (5-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5+\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1+\sqrt{5}\right )+2 \sqrt [6]{x}\right )+\frac{1}{5} \left (3 \left (5+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5-\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1-\sqrt{5}\right )+2 \sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1-\sqrt{5}+4 \sqrt [6]{x}}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (1+\sqrt{5}+4 \sqrt [6]{x}\right )\right )+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0067922, size = 29, normalized size = 0.14 \[ -6 \sqrt [6]{x} \text{Hypergeometric2F1}\left (\frac{1}{5},1,\frac{6}{5},x^{5/6}\right )+x+6 \sqrt [6]{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + x - 6*x^(1/6)*Hypergeometric2F1[1/5, 1, 6/5, x^(5/6)]

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Maple [A]  time = 0.013, size = 242, normalized size = 1.2 \begin{align*} x+6\,\sqrt [6]{x}-{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-6\,{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,\sqrt [6]{x}+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{6\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10-2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}+\sqrt{5} \right ) } \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }+{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }-6\,{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,\sqrt [6]{x}-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{6\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10+2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}-\sqrt{5} \right ) } \right ) }+{\frac{6}{5}\ln \left ( -1+\sqrt [6]{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+6*x^(1/6)-3/10*ln(2+x^(1/6)+2*x^(1/3)+x^(1/6)*5^(1/2))*5^(1/2)-3/10*ln(2+x^(1/6)+2*x^(1/3)+x^(1/6)*5^(1/2))-
6/(10-2*5^(1/2))^(1/2)*arctan((1+4*x^(1/6)+5^(1/2))/(10-2*5^(1/2))^(1/2))+6/5/(10-2*5^(1/2))^(1/2)*arctan((1+4
*x^(1/6)+5^(1/2))/(10-2*5^(1/2))^(1/2))*5^(1/2)-3/10*ln(2+x^(1/6)+2*x^(1/3)-x^(1/6)*5^(1/2))+3/10*ln(2+x^(1/6)
+2*x^(1/3)-x^(1/6)*5^(1/2))*5^(1/2)-6/(10+2*5^(1/2))^(1/2)*arctan((1+4*x^(1/6)-5^(1/2))/(10+2*5^(1/2))^(1/2))-
6/5/(10+2*5^(1/2))^(1/2)*arctan((1+4*x^(1/6)-5^(1/2))/(10+2*5^(1/2))^(1/2))*5^(1/2)+6/5*ln(-1+x^(1/6))

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Maxima [B]  time = 1.47636, size = 396, normalized size = 1.97 \begin{align*} -\frac{3 \, \sqrt{5} \left (-1\right )^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{2 \, \sqrt{5} - 10}} - \frac{3 \, \sqrt{5} \left (-1\right )^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} - 10}} - \frac{6}{5} \, \left (-1\right )^{\frac{1}{5}} \log \left (\left (-1\right )^{\frac{1}{5}} + x^{\frac{1}{6}}\right ) + x - \frac{3 \,{\left (\sqrt{5} + 3\right )} \log \left (-x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{4}{5}} + \left (-1\right )^{\frac{4}{5}}\right )}} - \frac{3 \,{\left (\sqrt{5} - 3\right )} \log \left (x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{4}{5}} - \left (-1\right )^{\frac{4}{5}}\right )}} + 6 \, x^{\frac{1}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*sqrt(5)*(-1)^(1/5)*(sqrt(5) - 1)*log((sqrt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) -
4*x^(1/6))/(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4*x^(1/6)))/sqrt(2*sqrt(5) -
10) - 3/5*sqrt(5)*(-1)^(1/5)*(sqrt(5) + 1)*log((sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(
1/5) + 4*x^(1/6))/(sqrt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6)))/sqrt(-2*sq
rt(5) - 10) - 6/5*(-1)^(1/5)*log((-1)^(1/5) + x^(1/6)) + x - 3/5*(sqrt(5) + 3)*log(-x^(1/6)*(sqrt(5)*(-1)^(1/5
) + (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(4/5) + (-1)^(4/5)) - 3/5*(sqrt(5) - 3)*log(x^(1/6)*
(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(4/5) - (-1)^(4/5)) + 6*x^(1/6)

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Fricas [B]  time = 14.3074, size = 1858, normalized size = 9.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/10*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*log(3/2*sqrt(2)*sqrt(sqrt(5) - 5) + 3/2*sqrt(5) + 6*x^(1/6) +
3/2) + 3/10*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)*log(-3/2*sqrt(2)*sqrt(sqrt(5) - 5) + 3/2*sqrt(5) + 6*x^(
1/6) + 3/2) + 1/10*(3*sqrt(5) - sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqr
t(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(-3*sqrt(5) + sqrt(-27/4*(sqrt(2)*sqrt(sqrt(
5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) + 12
*x^(1/6) + 3) + 1/10*(3*sqrt(5) + sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(s
qrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(
5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(-3*sqrt(5) - sqrt(-27/4*(sqrt(2)*sqrt(sqr
t(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5
) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) +
12*x^(1/6) + 3) + x + 6*x^(1/6) + 6/5*log(x^(1/6) - 1)

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Sympy [B]  time = 36.2715, size = 561, normalized size = 2.79 \begin{align*} - \frac{60 \sqrt [6]{x}}{-10 + 10 \sqrt{5}} + \frac{60 \sqrt{5} \sqrt [6]{x}}{-10 + 10 \sqrt{5}} - \frac{10 x}{-10 + 10 \sqrt{5}} + \frac{10 \sqrt{5} x}{-10 + 10 \sqrt{5}} - \frac{12 \log{\left (\sqrt [6]{x} - 1 \right )}}{-10 + 10 \sqrt{5}} + \frac{12 \sqrt{5} \log{\left (\sqrt [6]{x} - 1 \right )}}{-10 + 10 \sqrt{5}} - \frac{12 \log{\left (8 \sqrt [6]{x} + 8 \sqrt{5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{-10 + 10 \sqrt{5}} - \frac{6 \sqrt{5} \log{\left (- 8 \sqrt{5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{-10 + 10 \sqrt{5}} + \frac{18 \log{\left (- 8 \sqrt{5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{-10 + 10 \sqrt{5}} - \frac{6 \sqrt{10} \sqrt{5 - \sqrt{5}} \operatorname{atan}{\left (\frac{2 \sqrt{2} \sqrt [6]{x}}{\sqrt{5 - \sqrt{5}}} + \frac{\sqrt{2}}{2 \sqrt{5 - \sqrt{5}}} + \frac{\sqrt{10}}{2 \sqrt{5 - \sqrt{5}}} \right )}}{-10 + 10 \sqrt{5}} + \frac{6 \sqrt{2} \sqrt{5 - \sqrt{5}} \operatorname{atan}{\left (\frac{2 \sqrt{2} \sqrt [6]{x}}{\sqrt{5 - \sqrt{5}}} + \frac{\sqrt{2}}{2 \sqrt{5 - \sqrt{5}}} + \frac{\sqrt{10}}{2 \sqrt{5 - \sqrt{5}}} \right )}}{-10 + 10 \sqrt{5}} - \frac{6 \sqrt{10} \sqrt{\sqrt{5} + 5} \operatorname{atan}{\left (\frac{2 \sqrt{2} \sqrt [6]{x}}{\sqrt{\sqrt{5} + 5}} - \frac{\sqrt{10}}{2 \sqrt{\sqrt{5} + 5}} + \frac{\sqrt{2}}{2 \sqrt{\sqrt{5} + 5}} \right )}}{-10 + 10 \sqrt{5}} + \frac{6 \sqrt{2} \sqrt{\sqrt{5} + 5} \operatorname{atan}{\left (\frac{2 \sqrt{2} \sqrt [6]{x}}{\sqrt{\sqrt{5} + 5}} - \frac{\sqrt{10}}{2 \sqrt{\sqrt{5} + 5}} + \frac{\sqrt{2}}{2 \sqrt{\sqrt{5} + 5}} \right )}}{-10 + 10 \sqrt{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-60*x**(1/6)/(-10 + 10*sqrt(5)) + 60*sqrt(5)*x**(1/6)/(-10 + 10*sqrt(5)) - 10*x/(-10 + 10*sqrt(5)) + 10*sqrt(5
)*x/(-10 + 10*sqrt(5)) - 12*log(x**(1/6) - 1)/(-10 + 10*sqrt(5)) + 12*sqrt(5)*log(x**(1/6) - 1)/(-10 + 10*sqrt
(5)) - 12*log(8*x**(1/6) + 8*sqrt(5)*x**(1/6) + 16*x**(1/3) + 16)/(-10 + 10*sqrt(5)) - 6*sqrt(5)*log(-8*sqrt(5
)*x**(1/6) + 8*x**(1/6) + 16*x**(1/3) + 16)/(-10 + 10*sqrt(5)) + 18*log(-8*sqrt(5)*x**(1/6) + 8*x**(1/6) + 16*
x**(1/3) + 16)/(-10 + 10*sqrt(5)) - 6*sqrt(10)*sqrt(5 - sqrt(5))*atan(2*sqrt(2)*x**(1/6)/sqrt(5 - sqrt(5)) + s
qrt(2)/(2*sqrt(5 - sqrt(5))) + sqrt(10)/(2*sqrt(5 - sqrt(5))))/(-10 + 10*sqrt(5)) + 6*sqrt(2)*sqrt(5 - sqrt(5)
)*atan(2*sqrt(2)*x**(1/6)/sqrt(5 - sqrt(5)) + sqrt(2)/(2*sqrt(5 - sqrt(5))) + sqrt(10)/(2*sqrt(5 - sqrt(5))))/
(-10 + 10*sqrt(5)) - 6*sqrt(10)*sqrt(sqrt(5) + 5)*atan(2*sqrt(2)*x**(1/6)/sqrt(sqrt(5) + 5) - sqrt(10)/(2*sqrt
(sqrt(5) + 5)) + sqrt(2)/(2*sqrt(sqrt(5) + 5)))/(-10 + 10*sqrt(5)) + 6*sqrt(2)*sqrt(sqrt(5) + 5)*atan(2*sqrt(2
)*x**(1/6)/sqrt(sqrt(5) + 5) - sqrt(10)/(2*sqrt(sqrt(5) + 5)) + sqrt(2)/(2*sqrt(sqrt(5) + 5)))/(-10 + 10*sqrt(
5))

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Giac [A]  time = 1.40893, size = 189, normalized size = 0.94 \begin{align*} -\frac{3}{5} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{\sqrt{5} - 4 \, x^{\frac{1}{6}} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{3}{5} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{\sqrt{5} + 4 \, x^{\frac{1}{6}} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{3}{10} \, \sqrt{5} \log \left (\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} + 1\right )} + x^{\frac{1}{3}} + 1\right ) + \frac{3}{10} \, \sqrt{5} \log \left (-\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} - 1\right )} + x^{\frac{1}{3}} + 1\right ) + x + 6 \, x^{\frac{1}{6}} - \frac{3}{10} \, \log \left (x^{\frac{2}{3}} + \sqrt{x} + x^{\frac{1}{3}} + x^{\frac{1}{6}} + 1\right ) + \frac{6}{5} \, \log \left ({\left | x^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*sqrt(2*sqrt(5) + 10)*arctan(-(sqrt(5) - 4*x^(1/6) - 1)/sqrt(2*sqrt(5) + 10)) - 3/5*sqrt(-2*sqrt(5) + 10)*
arctan((sqrt(5) + 4*x^(1/6) + 1)/sqrt(-2*sqrt(5) + 10)) - 3/10*sqrt(5)*log(1/2*x^(1/6)*(sqrt(5) + 1) + x^(1/3)
+ 1) + 3/10*sqrt(5)*log(-1/2*x^(1/6)*(sqrt(5) - 1) + x^(1/3) + 1) + x + 6*x^(1/6) - 3/10*log(x^(2/3) + sqrt(x
) + x^(1/3) + x^(1/6) + 1) + 6/5*log(abs(x^(1/6) - 1))