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

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 
)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 34.79 (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
 

Rubi [A] (verified)

Time = 21.70 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {7292, 7296, 7293, 7239, 7293, 2009}

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

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

\(\Big \downarrow \) 2009

\(\displaystyle 6 \left (-\frac {1}{12} \arcsin \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{1-x}}{\sqrt [3]{x+1}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {1}{18} \arctan \left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )+\frac {1}{24} (1-x)^2-\frac {1}{24} \sqrt [3]{x+1} (1-x)^{5/3}-\frac {1}{24} \sqrt {x+1} (1-x)^{3/2}-\frac {1}{24} (x+1)^{2/3} (1-x)^{4/3}-\frac {1}{24} (x+1)^{5/6} (1-x)^{7/6}-\frac {1}{24} (x+1)^{7/6} (1-x)^{5/6}-\frac {7}{72} \sqrt [6]{x+1} (1-x)^{5/6}+\frac {1}{36} \sqrt [3]{x+1} (1-x)^{2/3}+\frac {1}{24} \sqrt {x+1} \sqrt {1-x}+\frac {1}{18} (x+1)^{2/3} \sqrt [3]{1-x}+\frac {5}{72} (x+1)^{5/6} \sqrt [6]{1-x}+\frac {x-1}{6}-\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{72 \sqrt {3}}+\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{72 \sqrt {3}}\right )\)

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:

6*((1 - x)^2/24 + (-1 + x)/6 - (7*(1 - x)^(5/6)*(1 + x)^(1/6))/72 + ((1 - 
x)^(2/3)*(1 + x)^(1/3))/36 - ((1 - x)^(5/3)*(1 + x)^(1/3))/24 + (Sqrt[1 - 
x]*Sqrt[1 + x])/24 - ((1 - x)^(3/2)*Sqrt[1 + x])/24 + ((1 - x)^(1/3)*(1 + 
x)^(2/3))/18 - ((1 - x)^(4/3)*(1 + x)^(2/3))/24 + (5*(1 - x)^(1/6)*(1 + x) 
^(5/6))/72 - ((1 - x)^(7/6)*(1 + x)^(5/6))/24 - ((1 - x)^(5/6)*(1 + x)^(7/ 
6))/24 - ArcSin[Sqrt[1 - x]/Sqrt[2]]/12 - ArcTan[(1 - x)^(1/6)/(1 + x)^(1/ 
6)]/9 + (2*ArcTan[(1 - (2*(1 - x)^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(9*Sqrt[ 
3]) + ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)]/18 - ArcTan[Sqrt[3 
] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)]/18 - Log[1 + (1 - x)^(1/3)/(1 + x)^(1 
/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(72*Sqrt[3]) + Log[1 + (1 - x 
)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(72*Sqrt[3] 
))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]]
 

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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (220) = 440\).

Time = 0.09 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Too large to display} \] 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) - 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)) - 5/72*sqrt(3)*log((sqrt(3)*(x 
+ 1)^(5/6)*(-x + 1)^(1/6) + x + (x + 1)^(2/3)*(-x + 1)^(1/3) + 1)/(x + 1)) 
 + 5/72*sqrt(3)*log(-(sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) - x - (x + 1)^( 
2/3)*(-x + 1)^(1/3) - 1)/(x + 1)) - 7/72*sqrt(3)*log((sqrt(3)*(x + 1)^(1/6 
)*(-x + 1)^(5/6) + x - (x + 1)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 7/72*s 
qrt(3)*log(-(sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) - x + (x + 1)^(1/3)*(-x 
+ 1)^(2/3) + 1)/(x - 1)) + 1/2*x - 5/36*arctan((sqrt(3)*(x + 1) + 2*(x + 1 
)^(5/6)*(-x + 1)^(1/6))/(x + 1)) - 5/36*arctan(-(sqrt(3)*(x + 1) - 2*(x + 
1)^(5/6)*(-x + 1)^(1/6))/(x + 1)) - 7/36*arctan((sqrt(3)*(x - 1) + 2*(x + 
1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) - 7/36*arctan(-(sqrt(3)*(x - 1) - 2*(x + 
 1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) - 7/18*arctan((-x + 1)^(1/6)/(x + 1)^(1 
/6)) - 5/18*arctan((x + 1)^(1/6)*(-x + 1)^(5/6)/(x - 1)) - 1/2*arctan((sqr 
t(x + 1)*sqrt(-x + 1) - 1)/x) - 2/9*log((x + (x + 1)^(2/3)*(-x + 1)^(1/3) 
+ 1)/(x + 1)) + 1/9*log((x - (x + 1)^(2/3)*(-x + 1)^(1/3) + (x + 1)^(1/3)* 
(-x + 1)^(2/3) + 1)/(x + 1)) - 1/9*log((x - (x + 1)^(2/3)*(-x + 1)^(1/3...
 

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
 

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)
 

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
 

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)
 

Reduce [F]

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