Integrand size = 22, antiderivative size = 796 \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{4 \sqrt [3]{2}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {\sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}+\frac {3 \log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{8 \sqrt [3]{2}}-\frac {\log (x)}{4 \sqrt [3]{2}} \]
-1/2*(1-x)^(2/3)*(2-x)^(2/3)/x+3/16*ln(-(1-x)^(1/3)+1/2*(2-x)^(2/3)*2^(1/3 ))*2^(2/3)-1/8*ln(x)*2^(2/3)-1/8*arctan(1/3*3^(1/2)+1/3*2^(1/3)*(2-x)^(2/3 )/(1-x)^(1/3)*3^(1/2))*3^(1/2)*2^(2/3)-1/4*2^(2/3)*(x^2-3*x+2)^(1/3)*((3-2 *x)^2)^(1/2)*((-3+2*x)^2)^(1/2)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/(1+2^(2/3) *(x^2-3*x+2)^(1/3)+3^(1/2))-1/6*2^(1/6)*3^(3/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/ 3)*(x^2-3*x+2)^(1/3))*EllipticF((1+2^(2/3)*(x^2-3*x+2)^(1/3)-3^(1/2))/(1+2 ^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*((1-2^ (2/3)*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2 )^(1/3)+3^(1/2))^2)^(1/2)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^(1/2 )/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^ (1/2)+1/8*3^(1/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/3)*(x^2-3*x+2)^(1/3))*Elliptic E((1+2^(2/3)*(x^2-3*x+2)^(1/3)-3^(1/2))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/ 2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))*((1-2^(2/3 )*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2)^(1 /3)+3^(1/2))^2)^(1/2)*2^(2/3)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^ (1/2)/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)) ^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 21.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=-\frac {(1-x)^{2/3} \left (5 (2-x)^{2/3}+10 x \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-1+x,1-x\right )+(-1+x) x \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-1+x,1-x\right )\right )}{10 x} \]
-1/10*((1 - x)^(2/3)*(5*(2 - x)^(2/3) + 10*x*AppellF1[2/3, 1/3, 1, 5/3, -1 + x, 1 - x] + (-1 + x)*x*AppellF1[5/3, 1/3, 1, 8/3, -1 + x, 1 - x]))/x
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.20, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {134, 27, 175, 79, 133}
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx\) |
| \(\Big \downarrow \) 134 |
| \(\displaystyle \frac {1}{12} \int \frac {2 (x+3)}{\sqrt [3]{1-x} \sqrt [3]{2-x} x}dx-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {x+3}{\sqrt [3]{1-x} \sqrt [3]{2-x} x}dx-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}}dx+3 \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x}dx\right )-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {1}{6} \left (3 \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x}dx-\frac {3}{2} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x-1\right )\right )-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}\) |
| \(\Big \downarrow \) 133 |
| \(\displaystyle \frac {1}{6} \left (3 \left (-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{2 \sqrt [3]{2}}\right )-\frac {3}{2} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x-1\right )\right )-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}\) |
-1/2*((1 - x)^(2/3)*(2 - x)^(2/3))/x + ((-3*(1 - x)^(2/3)*Hypergeometric2F 1[1/3, 2/3, 5/3, -1 + x])/2 + 3*(-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3) *(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))])/2^(1/3) + (3*Log[-(1 - x)^(1/3) + (2 - x)^(2/3)/2^(2/3)])/(4*2^(1/3)) - Log[x]/(2*2^(1/3))))/6
3.9.81.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_)) ^(1/3)), x_] :> With[{q = Rt[b*((b*e - a*f)/(b*c - a*d)^2), 3]}, Simp[-Log[ a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[Sqrt[3]*(ArcTan[1/Sqrt[3] + 2*q*((c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3)))]/(2*q*(b*c - a*d))), x] + Simp[3*( Log[q*(c + d*x)^(2/3) - (e + f*x)^(1/3)]/(4*q*(b*c - a*d))), x])] /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]
Int[((a_.) + (b_.)*(x_))^(m_)/(((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x _))^(1/3)), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(2/3)*((e + f*x)^(2/3 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[f/(6*(m + 1)*(b*c - a*d)*(b *e - a*f)) Int[(a + b*x)^(m + 1)*((a*d*(3*m + 1) - 3*b*c*(3*m + 5) - 2*b* d*(3*m + 7)*x)/((c + d*x)^(1/3)*(e + f*x)^(1/3))), x], x] /; FreeQ[{a, b, c , d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0] && ILtQ[m, -1]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
\[\int \frac {1}{\left (1-x \right )^{\frac {1}{3}} \left (2-x \right )^{\frac {1}{3}} x^{2}}d x\]
\[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=\int { \frac {1}{x^{2} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=\int \frac {1}{x^{2} \sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \]
\[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=\int { \frac {1}{x^{2} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=\int { \frac {1}{x^{2} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx=\int \frac {1}{x^2\,{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]