\(\int \frac {1}{(a \sqrt [3]{x}+b x^{2/3})^{2/3}} \, dx\) [438]

Optimal result

Integrand size = 19, antiderivative size = 487 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=-\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {6 \sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+2 \sqrt [3]{2} \left (-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{5 b^3 \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]

[Out]

-18/5*a*(a+b*x^(1/3))*x^(1/3)/b^2/(a*x^(1/3)+b*x^(2/3))^(2/3)+9/5*(a+b*x^(1/3))*x^(2/3)/b/(a*x^(1/3)+b*x^(2/3)
)^(2/3)+6/5*2^(1/3)*3^(3/4)*a^4*(1-2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3))*(-b*(a*x^(1/3)+b*x^(2/3))/a^2
)^(2/3)*EllipticF((1-2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3)+3^(1/2))/(1-2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3
)/a^2)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((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))/(1-2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3)-3^(1/2))^2)^
(1/2)/b^3/(a+2*b*x^(1/3))/(a*x^(1/3)+b*x^(2/3))^(2/3)/((-1+2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3))/(1-2^
(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3)-3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2036, 348, 52, 63, 636, 633, 242, 225} \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\frac {6 \sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt {3}+1\right )^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} \operatorname {EllipticF}\left (\arcsin \left (\frac {-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{5 b^3 \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt {3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac {18 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]

[In]

Int[(a*x^(1/3) + b*x^(2/3))^(-2/3),x]

[Out]

(-18*a*(a + b*x^(1/3))*x^(1/3))/(5*b^2*(a*x^(1/3) + b*x^(2/3))^(2/3)) + (9*(a + b*x^(1/3))*x^(2/3))/(5*b*(a*x^
(1/3) + b*x^(2/3))^(2/3)) + (6*2^(1/3)*3^(3/4)*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))^(2/3)*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]])/(5*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))^(2/3))

Rule 52

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 63

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 225

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

Rule 242

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

Rule 348

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 633

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 636

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 2036

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}} \, dx}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = \frac {\left (3 \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \text {Subst}\left (\int \frac {x^{4/3}}{(a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = \frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac {\left (12 a \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{x}}{(a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {\left (6 a^2 \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \text {Subst}\left (\int \frac {1}{x^{2/3} (a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {\left (6 a^2 \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {b x}{a}-\frac {b^2 x^2}{a^2}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac {\left (6 \sqrt [3]{2} a^4 \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a^2 x^2}{b^2}\right )^{2/3}} \, dx,x,-\frac {b \left (a+2 b \sqrt [3]{x}\right )}{a^2}\right )}{5 b^4 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac {\left (9 \sqrt [3]{2} a^4 \sqrt {-\frac {\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \text {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 )}{5 b^3 \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ & = -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {6 \sqrt [3]{2} 3^{3/4} \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}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} 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 )}{5 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 ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.13 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\frac {9 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {10}{3},-\frac {b \sqrt [3]{x}}{a}\right )}{7 \left (\left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}\right )^{2/3}} \]

[In]

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

[Out]

(9*(1 + (b*x^(1/3))/a)^(2/3)*x*Hypergeometric2F1[2/3, 7/3, 10/3, -((b*x^(1/3))/a)])/(7*((a + b*x^(1/3))*x^(1/3
))^(2/3))

Maple [F]

\[\int \frac {1}{\left (a \,x^{\frac {1}{3}}+b \,x^{\frac {2}{3}}\right )^{\frac {2}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\frac {9\,x\,{\left (\frac {b\,x^{1/3}}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {2}{3},\frac {7}{3};\ \frac {10}{3};\ -\frac {b\,x^{1/3}}{a}\right )}{7\,{\left (a\,x^{1/3}+b\,x^{2/3}\right )}^{2/3}} \]

[In]

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

[Out]

(9*x*((b*x^(1/3))/a + 1)^(2/3)*hypergeom([2/3, 7/3], 10/3, -(b*x^(1/3))/a))/(7*(a*x^(1/3) + b*x^(2/3))^(2/3))