\(\int \frac {1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx\) [437]

Optimal result

Integrand size = 19, antiderivative size = 988 \[ \int \frac {1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx=-\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}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{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 (1-2^{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}} \sqrt [3]{-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}} E\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 )}{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 (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 ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}}+\frac {15\ 3^{3/4} 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}} \sqrt [3]{-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}} \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 )}{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 (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 ) \sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \]

[Out]

-45/28*a*(a+b*x^(1/3))*x^(1/3)/b^2/(a*x^(1/3)+b*x^(2/3))^(1/3)+9/7*(a+b*x^(1/3))*x^(2/3)/b/(a*x^(1/3)+b*x^(2/3
))^(1/3)-45/28*a^2*(a+2*b*x^(1/3))*(-b*(a*x^(1/3)+b*x^(2/3))/a^2)^(1/3)*2^(2/3)/b^3/(a*x^(1/3)+b*x^(2/3))^(1/3
)/(1-2^(2/3)*(-b*(a+b*x^(1/3))*x^(1/3)/a^2)^(1/3)-3^(1/2))+15/14*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)^(1/3)*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^(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)*2^(1/6)/b^3/(a+2*b*x^(1/3))/(a*x^(1/3)+b*x^(2/3))^(1/3)/((-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)-45/56*3^(1/4)*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)^(1/3)*EllipticE((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^(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)*(1/2*6^(1/2)+1/2*2^(1/2))*2^(2/3)/b^3/(a+2*b*x^(1/3))/(a
*x^(1/3)+b*x^(2/3))^(1/3)/((-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 = 1.57 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2036, 348, 52, 63, 636, 633, 241, 310, 225, 1893} \[ \int \frac {1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx=-\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 (\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 ) 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}} \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 ) 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}}} \]

[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 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 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 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 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

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 (\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \left (1+\sqrt {3}\right ) 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 ) \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 )}{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^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] (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 = 0.06 \[ \int \frac {1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx=\frac {9 \sqrt [3]{1+\frac {b \sqrt [3]{x}}{a}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {8}{3},\frac {11}{3},-\frac {b \sqrt [3]{x}}{a}\right )}{8 \sqrt [3]{\left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}} \]

[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))

Maple [F]

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

[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)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 9.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.04 \[ \int \frac {1}{\sqrt [3]{a \sqrt [3]{x}+b x^{2/3}}} \, dx=\frac {9\,x\,{\left (\frac {b\,x^{1/3}}{a}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {8}{3};\ \frac {11}{3};\ -\frac {b\,x^{1/3}}{a}\right )}{8\,{\left (a\,x^{1/3}+b\,x^{2/3}\right )}^{1/3}} \]

[In]

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

[Out]

(9*x*((b*x^(1/3))/a + 1)^(1/3)*hypergeom([1/3, 8/3], 11/3, -(b*x^(1/3))/a))/(8*(a*x^(1/3) + b*x^(2/3))^(1/3))