Integrand size = 19, antiderivative size = 487 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=-\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {6 \sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+2 \sqrt [3]{2} \left (-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{5 b^3 \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]
-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)
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 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\frac {9 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {10}{3},-\frac {b \sqrt [3]{x}}{a}\right )}{7 \left (\left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}\right )^{2/3}} \]
(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))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1917, 864, 76, 74}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle \frac {x^{2/9} \left (a+b \sqrt [3]{x}\right )^{2/3} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}}dx}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\) |
\(\Big \downarrow \) 864 |
\(\displaystyle \frac {3 x^{2/9} \left (a+b \sqrt [3]{x}\right )^{2/3} \int \frac {x^{4/9}}{\left (a+b \sqrt [3]{x}\right )^{2/3}}d\sqrt [3]{x}}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\) |
\(\Big \downarrow \) 76 |
\(\displaystyle \frac {3 x^{2/9} \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{2/3} \int \frac {x^{4/9}}{\left (\frac {\sqrt [3]{x} b}{a}+1\right )^{2/3}}d\sqrt [3]{x}}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {9 x \left (\frac {b \sqrt [3]{x}}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {10}{3},-\frac {b \sqrt [3]{x}}{a}\right )}{7 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\) |
(9*(1 + (b*x^(1/3))/a)^(2/3)*x*Hypergeometric2F1[2/3, 7/3, 10/3, -((b*x^(1 /3))/a)])/(7*(a*x^(1/3) + b*x^(2/3))^(2/3))
3.5.38.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart [n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d* (x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !Integer Q[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0] && ((RationalQ[m] && !(EqQ[n, -2 ^(-1)] && EqQ[c^2 - d^2, 0])) || !RationalQ[n])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[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]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(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] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
\[\int \frac {1}{\left (a \,x^{\frac {1}{3}}+b \,x^{\frac {2}{3}}\right )^{\frac {2}{3}}}d x\]
Timed out. \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\int \frac {1}{\left (a \sqrt [3]{x} + b x^{\frac {2}{3}}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{\frac {2}{3}} + a x^{\frac {1}{3}}\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{\frac {2}{3}} + a x^{\frac {1}{3}}\right )}^{\frac {2}{3}}} \,d x } \]
Time = 9.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx=\frac {9\,x\,{\left (\frac {b\,x^{1/3}}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {2}{3},\frac {7}{3};\ \frac {10}{3};\ -\frac {b\,x^{1/3}}{a}\right )}{7\,{\left (a\,x^{1/3}+b\,x^{2/3}\right )}^{2/3}} \]