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

Optimal. Leaf size=988 \[ \text{result too large to display} \]

[Out]

(-45*a^2*(a + 2*b*x^(1/3))*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3))/(14*2^(1/3)*b^3*(1 - Sqrt[3] - 2^(2/3)*
(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3)) - (45*a*(a + b*x^(1/3))*x^(1/3))/(2
8*b^2*(a*x^(1/3) + b*x^(2/3))^(1/3)) + (9*(a + b*x^(1/3))*x^(2/3))/(7*b*(a*x^(1/3) + b*x^(2/3))^(1/3)) - (45*3
^(1/4)*Sqrt[2 + Sqrt[3]]*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((b*
(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1 - Sqrt[3] - 2^
(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2]*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)*EllipticE[ArcS
in[(1 + Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1
/3))*x^(1/3))/a^2))^(1/3))], -7 + 4*Sqrt[3]])/(28*2^(1/3)*b^3*Sqrt[-((1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3
))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)]*(a + 2*b*x^(1/3))*(a*x^
(1/3) + b*x^(2/3))^(1/3)) + (15*3^(3/4)*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*Sqrt[(1 +
2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1
- Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2]*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)
*EllipticF[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-
((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))], -7 + 4*Sqrt[3]])/(7*2^(5/6)*b^3*Sqrt[-((1 - 2^(2/3)*(-((b*(a + b*x
^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)]*(a + 2*b*
x^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 2.24545, antiderivative size = 988, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {2011, 341, 50, 61, 622, 619, 235, 304, 219, 1879} \[ -\frac{45 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1\right )^2}} \sqrt [3]{-\frac{b \left (\sqrt [3]{x} a+b x^{2/3}\right )}{a^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) a^4}{28 \sqrt [3]{2} b^3 \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{\sqrt [3]{x} a+b x^{2/3}}}+\frac{15\ 3^{3/4} \left (1-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1\right )^2}} \sqrt [3]{-\frac{b \left (\sqrt [3]{x} a+b x^{2/3}\right )}{a^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) a^4}{7\ 2^{5/6} b^3 \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{\sqrt [3]{x} a+b x^{2/3}}}-\frac{45 \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{-\frac{b \left (\sqrt [3]{x} a+b x^{2/3}\right )}{a^2}} a^2}{14 \sqrt [3]{2} b^3 \left (-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1\right ) \sqrt [3]{\sqrt [3]{x} a+b x^{2/3}}}-\frac{45 \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x} a}{28 b^2 \sqrt [3]{\sqrt [3]{x} a+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{\sqrt [3]{x} a+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^(1/3) + b*x^(2/3))^(-1/3),x]

[Out]

(-45*a^2*(a + 2*b*x^(1/3))*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3))/(14*2^(1/3)*b^3*(1 - Sqrt[3] - 2^(2/3)*
(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3)) - (45*a*(a + b*x^(1/3))*x^(1/3))/(2
8*b^2*(a*x^(1/3) + b*x^(2/3))^(1/3)) + (9*(a + b*x^(1/3))*x^(2/3))/(7*b*(a*x^(1/3) + b*x^(2/3))^(1/3)) - (45*3
^(1/4)*Sqrt[2 + Sqrt[3]]*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((b*
(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1 - Sqrt[3] - 2^
(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2]*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)*EllipticE[ArcS
in[(1 + Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1
/3))*x^(1/3))/a^2))^(1/3))], -7 + 4*Sqrt[3]])/(28*2^(1/3)*b^3*Sqrt[-((1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3
))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)]*(a + 2*b*x^(1/3))*(a*x^
(1/3) + b*x^(2/3))^(1/3)) + (15*3^(3/4)*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*Sqrt[(1 +
2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1
- Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2]*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)
*EllipticF[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-
((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))], -7 + 4*Sqrt[3]])/(7*2^(5/6)*b^3*Sqrt[-((1 - 2^(2/3)*(-((b*(a + b*x
^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - Sqrt[3] - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)]*(a + 2*b*
x^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3))

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

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 50

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

Rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/(a*c + (b*
c + a*d)*x + b*d*x^2)^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c -
a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4] && AtomQ[b*c + a*d]

Rule 622

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(b*x + c*x^2)^p/(-((c*(b*x + c*x^2))/b^2))^p, Int[(-((
c*x)/b) - (c^2*x^2)/b^2)^p, x], x] /; FreeQ[{b, c}, x] && RationalQ[p] && 3 <= Denominator[p] <= 4

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx &=\frac{\left (\sqrt [3]{a+b \sqrt [3]{x}} \sqrt [9]{x}\right ) \int \frac{1}{\sqrt [3]{a+b \sqrt [3]{x}} \sqrt [9]{x}} \, dx}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=\frac{\left (3 \sqrt [3]{a+b \sqrt [3]{x}} \sqrt [9]{x}\right ) \operatorname{Subst}\left (\int \frac{x^{5/3}}{\sqrt [3]{a+b x}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{\left (15 a \sqrt [3]{a+b \sqrt [3]{x}} \sqrt [9]{x}\right ) \operatorname{Subst}\left (\int \frac{x^{2/3}}{\sqrt [3]{a+b x}} \, dx,x,\sqrt [3]{x}\right )}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{\left (15 a^2 \sqrt [3]{a+b \sqrt [3]{x}} \sqrt [9]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \sqrt [3]{a+b x}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{\left (15 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a x+b x^2}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{\left (15 a^2 \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{-\frac{b x}{a}-\frac{b^2 x^2}{a^2}}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{\left (15 a^4 \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-\frac{a^2 x^2}{b^2}}} \, dx,x,-\frac{b \left (a+2 b \sqrt [3]{x}\right )}{a^2}\right )}{14 \sqrt [3]{2} b^4 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{\left (45 a^4 \sqrt{-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}} \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )}{28 \sqrt [3]{2} b^3 \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{\left (45 a^4 \sqrt{-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}} \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )}{28 \sqrt [3]{2} b^3 \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{\left (45 \sqrt{2+\sqrt{3}} a^4 \sqrt{-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}} \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )}{14\ 2^{5/6} b^3 \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ &=-\frac{45 a^2 \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}}}{14 \sqrt [3]{2} b^3 \left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{45 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{28 b^2 \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{7 b \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}-\frac{45 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^4 \left (1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right ) \sqrt{\frac{1+\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}+\left (1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}\right )|-7+4 \sqrt{3}\right )}{28 \sqrt [3]{2} b^3 \sqrt{-\frac{1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac{15\ 3^{3/4} a^4 \left (1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right ) \sqrt{\frac{1+\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}+\left (1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \sqrt [3]{-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}\right )|-7+4 \sqrt{3}\right )}{7\ 2^{5/6} b^3 \sqrt{-\frac{1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}\\ \end{align*}

Mathematica [C]  time = 0.0185253, size = 61, normalized size = 0.06 \[ \frac{9 x \sqrt [3]{\frac{b \sqrt [3]{x}}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{11}{3};-\frac{b \sqrt [3]{x}}{a}\right )}{8 \sqrt [3]{\sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*(1 + (b*x^(1/3))/a)^(1/3)*x*Hypergeometric2F1[1/3, 8/3, 11/3, -((b*x^(1/3))/a)])/(8*((a + b*x^(1/3))*x^(1/3
))^(1/3))

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Maple [F]  time = 0.203, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{a\sqrt [3]{x}+b{x}^{{\frac{2}{3}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^(1/3)+b*x^(2/3))^(1/3),x)

[Out]

int(1/(a*x^(1/3)+b*x^(2/3))^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{2}{3}} + a x^{\frac{1}{3}}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{a \sqrt [3]{x} + b x^{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*x**(1/3) + b*x**(2/3))**(-1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{2}{3}} + a x^{\frac{1}{3}}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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