\(\int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} (\sqrt {1-x}-\sqrt {1+x})} \, dx\) [221]

Optimal result

Integrand size = 52, antiderivative size = 304 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\frac {5}{16} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {7 \left (1-x^2\right )^{5/4}}{24 \sqrt {1-x}}+\frac {x \left (1-x^2\right )^{5/4}}{6 \sqrt {1-x}}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}} \] Output:

5/16*(1-x)^(3/4)*(1+x)^(1/4)-1/16*(1-x)^(1/4)*(1+x)^(3/4)+1/24*(1-x)^(5/4) 
*(1+x)^(3/4)+3/16*arctan(-1+(1-x)^(1/4)*2^(1/2)/(1+x)^(1/4))*2^(1/2)+3/16* 
arctan(1+(1-x)^(1/4)*2^(1/2)/(1+x)^(1/4))*2^(1/2)+1/16*ln(1-(1-x)^(1/4)*2^ 
(1/2)/(1+x)^(1/4)+(1-x)^(1/2)/(1+x)^(1/2))*2^(1/2)-1/16*ln(1+(1-x)^(1/4)*2 
^(1/2)/(1+x)^(1/4)+(1-x)^(1/2)/(1+x)^(1/2))*2^(1/2)+7/24*(-x^2+1)^(5/4)/(1 
-x)^(1/2)+1/6*x*(-x^2+1)^(5/4)/(1-x)^(1/2)+1/6*(-x^2+1)^(5/4)*(1+x)^(1/2)
 

Mathematica [A] (verified)

Time = 10.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\frac {1}{48} \sqrt {1+x} \sqrt [4]{1-x^2} \left (-7+2 x+8 x^2-\frac {\sqrt {1-x^2} \left (29+22 x+8 x^2\right )}{1+x}\right )+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt {1+x} \sqrt [4]{1-x^2}}{1+x-\sqrt {1-x^2}}\right )-2 \text {arctanh}\left (\frac {1+x+\sqrt {1-x^2}}{\sqrt {2} \sqrt {1+x} \sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}} \] Input:

Integrate[(x^2*Sqrt[1 + x]*(1 - x^2)^(1/4))/(Sqrt[1 - x]*(Sqrt[1 - x] - Sq 
rt[1 + x])),x]
 

Output:

-1/48*(Sqrt[1 + x]*(1 - x^2)^(1/4)*(-7 + 2*x + 8*x^2 - (Sqrt[1 - x^2]*(29 
+ 22*x + 8*x^2))/(1 + x))) + (3*ArcTan[(Sqrt[2]*Sqrt[1 + x]*(1 - x^2)^(1/4 
))/(1 + x - Sqrt[1 - x^2])] - 2*ArcTanh[(1 + x + Sqrt[1 - x^2])/(Sqrt[2]*S 
qrt[1 + x]*(1 - x^2)^(1/4))])/(8*Sqrt[2])
 

Rubi [A] (warning: unable to verify)

Time = 0.95 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.56, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.327, Rules used = {2003, 2528, 90, 60, 60, 73, 770, 755, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {x^2 \sqrt {x+1} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {x+1}\right )} \, dx\)

\(\Big \downarrow \) 2003

\(\displaystyle \int \frac {x^2 (x+1)^{3/4}}{\sqrt [4]{1-x} \left (\sqrt {1-x}-\sqrt {x+1}\right )}dx\)

\(\Big \downarrow \) 2528

\(\displaystyle -\frac {1}{2} \int \sqrt [4]{1-x} x (x+1)^{3/4}dx-\frac {1}{2} \int \frac {x (x+1)^{5/4}}{\sqrt [4]{1-x}}dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}-\frac {1}{6} \int \sqrt [4]{1-x} (x+1)^{3/4}dx\right )+\frac {1}{2} \left (\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}-\frac {1}{2} \int \frac {(x+1)^{5/4}}{\sqrt [4]{1-x}}dx\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \int \frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}dx\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \int \frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}dx\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-x} (x+1)^{3/4}}dx-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{(1-x)^{3/4} \sqrt [4]{x+1}}dx+\sqrt [4]{1-x} (x+1)^{3/4}\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \int \frac {1}{\sqrt [4]{x+1}}d\sqrt [4]{1-x}\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \int \frac {1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {\sqrt {1-x}+1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {\sqrt {1-x}+1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \int \frac {\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \int \frac {\sqrt {1-x}+1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {\sqrt {1-x}+1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )-(1-x)^{3/4} \sqrt [4]{x+1}\right )\right )+\frac {1}{3} (1-x)^{3/4} (x+1)^{9/4}\right )+\frac {1}{2} \left (\frac {1}{6} \left (\frac {1}{2} (1-x)^{5/4} (x+1)^{3/4}-\frac {3}{4} \left (\sqrt [4]{1-x} (x+1)^{3/4}-2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\right )\right )+\frac {1}{3} (1-x)^{5/4} (x+1)^{7/4}\right )\)

\(\Big \downarrow \) 1103

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

Input:

Int[(x^2*Sqrt[1 + x]*(1 - x^2)^(1/4))/(Sqrt[1 - x]*(Sqrt[1 - x] - Sqrt[1 + 
 x])),x]
 

Output:

(((1 - x)^(3/4)*(1 + x)^(9/4))/3 + (((1 - x)^(3/4)*(1 + x)^(5/4))/2 - (5*( 
-((1 - x)^(3/4)*(1 + x)^(1/4)) - 2*((-(ArcTan[1 - (Sqrt[2]*(1 - x)^(1/4))/ 
(1 + x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4) 
]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4 
)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/ 
4)]/(2*Sqrt[2]))/2)))/4)/2)/2 + (((1 - x)^(5/4)*(1 + x)^(7/4))/3 + (((1 - 
x)^(5/4)*(1 + x)^(3/4))/2 - (3*((1 - x)^(1/4)*(1 + x)^(3/4) - 2*((-(ArcTan 
[1 - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2] 
*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[1 - x] - (S 
qrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - x] + (Sqrt 
[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt[2]))/2)))/4)/6)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 2003

Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : 
> Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} 
, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && 
  !IntegerQ[n]))
 

rule 2528

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), 
 x_Symbol] :> Simp[c/(e*(b*c - a*d))   Int[(u*Sqrt[a + b*x])/x, x], x] - Si 
mp[a/(f*(b*c - a*d))   Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c, d 
, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]
 

Maple [F]

\[\int \frac {x^{2} \left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {1+x}}{\sqrt {1-x}\, \left (-\sqrt {1+x}+\sqrt {1-x}\right )}d x\]

Input:

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/(-(1+x)^(1/2)+(1-x)^(1/2)), 
x)
 

Output:

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/(-(1+x)^(1/2)+(1-x)^(1/2)), 
x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.24 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\frac {1}{48} \, {\left (8 \, x^{2} + 2 \, x - 7\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + \frac {1}{48} \, {\left (8 \, x^{2} + 22 \, x + 29\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + \frac {1}{32} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + x + 1}{x + 1}\right ) + \frac {1}{32} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} - x - 1}{x + 1}\right ) + \frac {5}{32} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + x - 1}{x - 1}\right ) + \frac {5}{32} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} - x + 1}{x - 1}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + x + \sqrt {-x^{2} + 1} + 1}{x + 1}\right ) - \frac {1}{64} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} - x - \sqrt {-x^{2} + 1} - 1}{x + 1}\right ) + \frac {5}{64} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + x - \sqrt {-x^{2} + 1} - 1}{x - 1}\right ) - \frac {5}{64} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} - x + \sqrt {-x^{2} + 1} + 1}{x - 1}\right ) \] Input:

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1 
/2)),x, algorithm="fricas")
 

Output:

-1/48*(8*x^2 + 2*x - 7)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + 1/48*(8*x^2 + 22*x 
+ 29)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + 1/32*sqrt(2)*arctan((sqrt(2)*(-x^2 + 
 1)^(1/4)*sqrt(x + 1) + x + 1)/(x + 1)) + 1/32*sqrt(2)*arctan((sqrt(2)*(-x 
^2 + 1)^(1/4)*sqrt(x + 1) - x - 1)/(x + 1)) + 5/32*sqrt(2)*arctan((sqrt(2) 
*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + x - 1)/(x - 1)) + 5/32*sqrt(2)*arctan((sq 
rt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) - x + 1)/(x - 1)) + 1/64*sqrt(2)*log(( 
sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + sqrt(-x^2 + 1) + 1)/(x + 1)) - 
1/64*sqrt(2)*log(-(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - sqrt(-x^2 + 
1) - 1)/(x + 1)) + 5/64*sqrt(2)*log((sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) 
 + x - sqrt(-x^2 + 1) - 1)/(x - 1)) - 5/64*sqrt(2)*log(-(sqrt(2)*(-x^2 + 1 
)^(1/4)*sqrt(-x + 1) - x + sqrt(-x^2 + 1) + 1)/(x - 1))
 

Sympy [F]

\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int \frac {x^{2} \sqrt [4]{- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x + 1}}{\sqrt {1 - x} \left (\sqrt {1 - x} - \sqrt {x + 1}\right )}\, dx \] Input:

integrate(x**2*(-x**2+1)**(1/4)*(1+x)**(1/2)/(1-x)**(1/2)/((1-x)**(1/2)-(1 
+x)**(1/2)),x)
 

Output:

Integral(x**2*(-(x - 1)*(x + 1))**(1/4)*sqrt(x + 1)/(sqrt(1 - x)*(sqrt(1 - 
 x) - sqrt(x + 1))), x)
 

Maxima [F]

\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int { -\frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}} \,d x } \] Input:

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1 
/2)),x, algorithm="maxima")
 

Output:

-integrate((-x^2 + 1)^(1/4)*sqrt(x + 1)*x^2/(sqrt(-x + 1)*(sqrt(x + 1) - s 
qrt(-x + 1))), x)
 

Giac [F]

\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int { -\frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}} \,d x } \] Input:

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1 
/2)),x, algorithm="giac")
 

Output:

integrate(-(-x^2 + 1)^(1/4)*sqrt(x + 1)*x^2/(sqrt(-x + 1)*(sqrt(x + 1) - s 
qrt(-x + 1))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\int \frac {x^2\,{\left (1-x^2\right )}^{1/4}\,\sqrt {x+1}}{\left (\sqrt {x+1}-\sqrt {1-x}\right )\,\sqrt {1-x}} \,d x \] Input:

int(-(x^2*(1 - x^2)^(1/4)*(x + 1)^(1/2))/(((x + 1)^(1/2) - (1 - x)^(1/2))* 
(1 - x)^(1/2)),x)
 

Output:

-int((x^2*(1 - x^2)^(1/4)*(x + 1)^(1/2))/(((x + 1)^(1/2) - (1 - x)^(1/2))* 
(1 - x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\frac {\sqrt {1-x}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x^{2}}{6}+\frac {11 \sqrt {1-x}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x}{24}+\frac {11 \sqrt {1-x}\, \left (-x^{2}+1\right )^{\frac {1}{4}}}{12}-\frac {\sqrt {x +1}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x^{2}}{6}-\frac {\sqrt {x +1}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x}{24}+\frac {\sqrt {x +1}\, \left (-x^{2}+1\right )^{\frac {1}{4}}}{12}-\frac {5 \left (\int \frac {\sqrt {1-x}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x}{x^{2}-1}d x \right )}{16}+\frac {\left (\int \frac {\sqrt {x +1}\, \left (-x^{2}+1\right )^{\frac {1}{4}} x}{x^{2}-1}d x \right )}{16} \] Input:

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x 
)
 

Output:

(8*sqrt( - x + 1)*( - x**2 + 1)**(1/4)*x**2 + 22*sqrt( - x + 1)*( - x**2 + 
 1)**(1/4)*x + 44*sqrt( - x + 1)*( - x**2 + 1)**(1/4) - 8*sqrt(x + 1)*( - 
x**2 + 1)**(1/4)*x**2 - 2*sqrt(x + 1)*( - x**2 + 1)**(1/4)*x + 4*sqrt(x + 
1)*( - x**2 + 1)**(1/4) - 15*int((sqrt( - x + 1)*( - x**2 + 1)**(1/4)*x)/( 
x**2 - 1),x) + 3*int((sqrt(x + 1)*( - x**2 + 1)**(1/4)*x)/(x**2 - 1),x))/4 
8