3.3.22 \(\int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx\) [222]

3.3.22.1 Optimal result

Integrand size = 56, antiderivative size = 292 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=-\frac {1}{12} (1-3 x) (1-x)^{2/3} \sqrt [3]{1+x}+\frac {1}{4} \sqrt {1-x} x \sqrt {1+x}-\frac {1}{4} (1-x) (3+x)+\frac {1}{12} \sqrt [3]{1-x} (1+x)^{2/3} (1+3 x)+\frac {1}{12} \sqrt [6]{1-x} (1+x)^{5/6} (2+3 x)-\frac {1}{12} (1-x)^{5/6} \sqrt [6]{1+x} (10+3 x)+\frac {1}{6} \arctan \left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {4 \arctan \left (\frac {\sqrt [3]{1-x}-2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {5}{6} \arctan \left (\frac {\sqrt [3]{1-x}-\sqrt [3]{1+x}}{\sqrt [6]{1-x} \sqrt [6]{1+x}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{1-x} \sqrt [6]{1+x}}{\sqrt [3]{1-x}+\sqrt [3]{1+x}}\right )}{6 \sqrt {3}} \]

output

-1/12*(1-3*x)*(1-x)^(2/3)*(1+x)^(1/3)-1/4*(1-x)*(3+x)+1/12*(1-x)^(1/3)*(1+ 
x)^(2/3)*(1+3*x)+1/12*(1-x)^(1/6)*(1+x)^(5/6)*(2+3*x)-1/12*(1-x)^(5/6)*(1+ 
x)^(1/6)*(10+3*x)+1/6*arctan((1+x)^(1/6)/(1-x)^(1/6))-5/6*arctan(((1-x)^(1 
/3)-(1+x)^(1/3))/(1-x)^(1/6)/(1+x)^(1/6))-4/9*arctan(1/3*((1-x)^(1/3)-2*(1 
+x)^(1/3))/(1-x)^(1/3)*3^(1/2))*3^(1/2)+1/18*arctanh((1-x)^(1/6)*(1+x)^(1/ 
6)*3^(1/2)/((1-x)^(1/3)+(1+x)^(1/3)))*3^(1/2)+1/4*x*(1-x)^(1/2)*(1+x)^(1/2 
)
 

3.3.22.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 36.73 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=-\frac {1}{12} \sqrt [3]{1+x} \left ((1-3 x) (1-x)^{2/3}-\frac {3 \sqrt [3]{1-x} x (2+x)}{\sqrt [3]{1-x^2}}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-(1+3 x) \sqrt [3]{1-x^2}-\frac {(2+3 x) \sqrt {1-x^2}}{\sqrt [3]{1-x}}+\frac {(10+3 x) \left (1-x^2\right )^{5/6}}{1+x}-4\ 2^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},\frac {1+x}{2}\right )\right )+\frac {1}{36} \left (9 \arcsin (x)+14 \arctan \left (\frac {\sqrt [3]{1+x}}{\sqrt [6]{1-x^2}}\right )+7 \left (1+i \sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+7 \left (1-i \sqrt {3}\right ) \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+8 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{1-x^2}}{(1+x)^{2/3}}}{\sqrt {3}}\right )-\frac {15\ 2^{5/6} \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{6},\frac {7}{6},\frac {1-x}{2}\right )}{\sqrt [3]{1-x} \sqrt {1+x}}-8 \log \left ((1+x)^{2/3}+\sqrt [3]{1-x^2}\right )+4 \log \left (\sqrt [3]{1+x}+x \sqrt [3]{1+x}-(1+x)^{2/3} \sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \]

input

Integrate[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + 
(1 - x)^(2/3)*Sqrt[1 + x]),x]
 

output

-1/12*((1 + x)^(1/3)*((1 - 3*x)*(1 - x)^(2/3) - (3*(1 - x)^(1/3)*x*(2 + x) 
)/(1 - x^2)^(1/3) - 3*(1 - x)^(1/3)*x*(1 - x^2)^(1/6) - (1 + 3*x)*(1 - x^2 
)^(1/3) - ((2 + 3*x)*Sqrt[1 - x^2])/(1 - x)^(1/3) + ((10 + 3*x)*(1 - x^2)^ 
(5/6))/(1 + x) - 4*2^(2/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 + x)/2])) + 
 (9*ArcSin[x] + 14*ArcTan[(1 + x)^(1/3)/(1 - x^2)^(1/6)] + 7*(1 + I*Sqrt[3 
])*ArcTan[((1 - I*Sqrt[3])*(1 + x)^(1/3))/(2*(1 - x^2)^(1/6))] + 7*(1 - I* 
Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*(1 + x)^(1/3))/(2*(1 - x^2)^(1/6))] + 8*S 
qrt[3]*ArcTan[(1 - (2*(1 - x^2)^(1/3))/(1 + x)^(2/3))/Sqrt[3]] - (15*2^(5/ 
6)*Sqrt[1 - x^2]*Hypergeometric2F1[1/6, 1/6, 7/6, (1 - x)/2])/((1 - x)^(1/ 
3)*Sqrt[1 + x]) - 8*Log[(1 + x)^(2/3) + (1 - x^2)^(1/3)] + 4*Log[(1 + x)^( 
1/3) + x*(1 + x)^(1/3) - (1 + x)^(2/3)*(1 - x^2)^(1/3) + (1 - x^2)^(2/3)]) 
/36
 

3.3.22.3 Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt {1-x} x (x+1)^{2/3}}{(1-x)^{2/3} \sqrt {x+1}-(1-x)^{5/6} \sqrt [3]{x+1}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x \sqrt [3]{x+1}}{\sqrt [6]{1-x} \left (\sqrt [6]{x+1}-\sqrt [6]{1-x}\right )}dx\)

\(\Big \downarrow \) 7296

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle 6 \int \left (\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}-\frac {(1-x)^{5/3} \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}\right )d\sqrt [6]{1-x}\)

\(\Big \downarrow \) 7239

\(\displaystyle 6 \int \frac {(1-x)^{2/3} x \sqrt [3]{x+1}}{\sqrt [6]{1-x}-\sqrt [6]{x+1}}d\sqrt [6]{1-x}\)

input

Int[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + (1 - x 
)^(2/3)*Sqrt[1 + x]),x]
 

output

$Aborted
 

3.3.22.3.1 Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7296

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst]]
 

3.3.22.4 Maple [F]

\[\int \frac {x \left (1+x \right )^{\frac {2}{3}} \sqrt {1-x}}{-\left (1-x \right )^{\frac {5}{6}} \left (1+x \right )^{\frac {1}{3}}+\left (1-x \right )^{\frac {2}{3}} \sqrt {1+x}}d x\]

input

int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^ 
(1/2)),x)
 

output

int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^ 
(1/2)),x)
 

3.3.22.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Too large to display} \]

input

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)* 
(1+x)^(1/2)),x, algorithm="fricas")
 

output

1/4*x^2 + 1/12*(3*x + 2)*(x + 1)^(5/6)*(-x + 1)^(1/6) + 1/12*(3*x + 1)*(x 
+ 1)^(2/3)*(-x + 1)^(1/3) + 1/4*sqrt(x + 1)*x*sqrt(-x + 1) + 1/12*(3*x - 1 
)*(x + 1)^(1/3)*(-x + 1)^(2/3) - 1/12*(3*x + 10)*(x + 1)^(1/6)*(-x + 1)^(5 
/6) - 1/72*sqrt(2)*sqrt(25*I*sqrt(3) + 25)*log((sqrt(2)*(x + 1)*sqrt(25*I* 
sqrt(3) + 25) + 10*(x + 1)^(5/6)*(-x + 1)^(1/6))/(x + 1)) + 1/72*sqrt(2)*s 
qrt(25*I*sqrt(3) + 25)*log(-(sqrt(2)*(x + 1)*sqrt(25*I*sqrt(3) + 25) - 10* 
(x + 1)^(5/6)*(-x + 1)^(1/6))/(x + 1)) - 1/72*sqrt(2)*sqrt(-25*I*sqrt(3) + 
 25)*log((sqrt(2)*(x + 1)*sqrt(-25*I*sqrt(3) + 25) + 10*(x + 1)^(5/6)*(-x 
+ 1)^(1/6))/(x + 1)) + 1/72*sqrt(2)*sqrt(-25*I*sqrt(3) + 25)*log(-(sqrt(2) 
*(x + 1)*sqrt(-25*I*sqrt(3) + 25) - 10*(x + 1)^(5/6)*(-x + 1)^(1/6))/(x + 
1)) - 1/72*sqrt(2)*sqrt(49*I*sqrt(3) + 49)*log((sqrt(2)*(x - 1)*sqrt(49*I* 
sqrt(3) + 49) + 14*(x + 1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) + 1/72*sqrt(2)*s 
qrt(49*I*sqrt(3) + 49)*log(-(sqrt(2)*(x - 1)*sqrt(49*I*sqrt(3) + 49) - 14* 
(x + 1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) - 1/72*sqrt(2)*sqrt(-49*I*sqrt(3) + 
 49)*log((sqrt(2)*(x - 1)*sqrt(-49*I*sqrt(3) + 49) + 14*(x + 1)^(1/6)*(-x 
+ 1)^(5/6))/(x - 1)) + 1/72*sqrt(2)*sqrt(-49*I*sqrt(3) + 49)*log(-(sqrt(2) 
*(x - 1)*sqrt(-49*I*sqrt(3) + 49) - 14*(x + 1)^(1/6)*(-x + 1)^(5/6))/(x - 
1)) - 2/9*sqrt(3)*arctan(-1/3*(sqrt(3)*(x + 1) - 2*sqrt(3)*(x + 1)^(2/3)*( 
-x + 1)^(1/3))/(x + 1)) - 2/9*sqrt(3)*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt 
(3)*(x + 1)^(1/3)*(-x + 1)^(2/3))/(x - 1)) + 1/2*x - 5/18*arctan((-x + ...
 

3.3.22.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Timed out} \]

input

integrate(x*(1+x)**(2/3)*(1-x)**(1/2)/(-(1-x)**(5/6)*(1+x)**(1/3)+(1-x)**( 
2/3)*(1+x)**(1/2)),x)
 

output

Timed out
 

3.3.22.7 Maxima [F]

\[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {2}{3}} x \sqrt {-x + 1}}{\sqrt {x + 1} {\left (-x + 1\right )}^{\frac {2}{3}} - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {5}{6}}} \,d x } \]

input

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)* 
(1+x)^(1/2)),x, algorithm="maxima")
 

output

integrate((x + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 
1)^(1/3)*(-x + 1)^(5/6)), x)
 

3.3.22.8 Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Timed out} \]

input

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)* 
(1+x)^(1/2)),x, algorithm="giac")
 

output

Timed out
 

3.3.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\int \frac {x\,\sqrt {1-x}\,{\left (x+1\right )}^{2/3}}{{\left (1-x\right )}^{2/3}\,\sqrt {x+1}-{\left (1-x\right )}^{5/6}\,{\left (x+1\right )}^{1/3}} \,d x \]

input

int((x*(1 - x)^(1/2)*(x + 1)^(2/3))/((1 - x)^(2/3)*(x + 1)^(1/2) - (1 - x) 
^(5/6)*(x + 1)^(1/3)),x)
 

output

int((x*(1 - x)^(1/2)*(x + 1)^(2/3))/((1 - x)^(2/3)*(x + 1)^(1/2) - (1 - x) 
^(5/6)*(x + 1)^(1/3)), x)
 

3.3.22.10 Reduce [F]

\[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\int -\frac {\left (x +1\right )^{\frac {2}{3}} \sqrt {1-x}\, x}{\left (x +1\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {5}{6}}-\sqrt {x +1}\, \left (1-x \right )^{\frac {2}{3}}}d x \]

input

int(( - (x + 1)**(2/3)*sqrt( - x + 1)*x)/((x + 1)**(1/3)*( - x + 1)**(5/6) 
 - sqrt(x + 1)*( - x + 1)**(2/3)),x)
 

output

int(( - (x + 1)**(2/3)*sqrt( - x + 1)*x)/((x + 1)**(1/3)*( - x + 1)**(5/6) 
 - sqrt(x + 1)*( - x + 1)**(2/3)),x)