3.22.38 \(\int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x (1+x^3)} \, dx\) [2138]

3.22.38.1 Optimal result

Integrand size = 27, antiderivative size = 156 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=-\frac {4}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

output

Unintegrable
 

3.22.38.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=-\frac {(-1+x)^{3/4} x^{9/4} \left (16 \sqrt [4]{2} \left (\arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left ((-1+x) x^3\right )^{3/4}} \]

input

Integrate[((-1 + x)*(-x^3 + x^4)^(1/4))/(x*(1 + x^3)),x]
 

output

-1/12*((-1 + x)^(3/4)*x^(9/4)*(16*2^(1/4)*(ArcTan[2^(1/4)/((-1 + x)/x)^(1/ 
4)] - ArcTanh[2^(1/4)/((-1 + x)/x)^(1/4)]) + RootSum[1 - #1^4 + #1^8 & , ( 
-2*Log[x] + 8*Log[(-1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(-1 + 
 x)^(1/4) - x^(1/4)*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]))/((-1 + x)*x^3)^(3/4)
 

3.22.38.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

input

Int[((-1 + x)*(-x^3 + x^4)^(1/4))/(x*(1 + x^3)),x]
 

output

$Aborted
 

3.22.38.3.1 Defintions of rubi rules used

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.22.38.4 Maple [N/A] (verified)

Time = 69.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(\frac {2 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}+\frac {4 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{3}\)	\(122\)
trager	\(\text {Expression too large to display}\)	\(25478\)

input

int((-1+x)*(x^4-x^3)^(1/4)/x/(x^3+1),x,method=_RETURNVERBOSE)
 

output

2/3*ln((-2^(1/4)*x-(x^3*(-1+x))^(1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4)))*2^( 
1/4)+4/3*arctan(1/2*2^(3/4)/x*(x^3*(-1+x))^(1/4))*2^(1/4)+1/3*sum((_R^4-2) 
*ln((-_R*x+(x^3*(-1+x))^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))
 

3.22.38.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.27 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.37 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {2}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

input

integrate((-1+x)*(x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="fricas")
 

output

-1/6*sqrt(2)*sqrt(-sqrt(2*I*sqrt(3) + 2))*log((sqrt(2)*x*sqrt(-sqrt(2*I*sq 
rt(3) + 2)) + 2*(x^4 - x^3)^(1/4))/x) + 1/6*sqrt(2)*sqrt(-sqrt(2*I*sqrt(3) 
 + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(2*I*sqrt(3) + 2)) - 2*(x^4 - x^3)^(1/4)) 
/x) - 1/6*sqrt(2)*sqrt(-sqrt(-2*I*sqrt(3) + 2))*log((sqrt(2)*x*sqrt(-sqrt( 
-2*I*sqrt(3) + 2)) + 2*(x^4 - x^3)^(1/4))/x) + 1/6*sqrt(2)*sqrt(-sqrt(-2*I 
*sqrt(3) + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(-2*I*sqrt(3) + 2)) - 2*(x^4 - x^ 
3)^(1/4))/x) - 1/6*sqrt(2)*(2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*x*(2*I*sqr 
t(3) + 2)^(1/4) + 2*(x^4 - x^3)^(1/4))/x) + 1/6*sqrt(2)*(2*I*sqrt(3) + 2)^ 
(1/4)*log(-(sqrt(2)*x*(2*I*sqrt(3) + 2)^(1/4) - 2*(x^4 - x^3)^(1/4))/x) - 
1/6*sqrt(2)*(-2*I*sqrt(3) + 2)^(1/4)*log((sqrt(2)*x*(-2*I*sqrt(3) + 2)^(1/ 
4) + 2*(x^4 - x^3)^(1/4))/x) + 1/6*sqrt(2)*(-2*I*sqrt(3) + 2)^(1/4)*log(-( 
sqrt(2)*x*(-2*I*sqrt(3) + 2)^(1/4) - 2*(x^4 - x^3)^(1/4))/x) + 2/3*2^(1/4) 
*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) - 2/3*2^(1/4)*log(-(2^(1/4)*x - (x 
^4 - x^3)^(1/4))/x) + 2/3*I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(1/4))/ 
x) - 2/3*I*2^(1/4)*log((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x)
 

3.22.38.6 Sympy [N/A]

Not integrable

Time = 2.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.17 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right )}{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

input

integrate((-1+x)*(x**4-x**3)**(1/4)/x/(x**3+1),x)
 

output

Integral((x**3*(x - 1))**(1/4)*(x - 1)/(x*(x + 1)*(x**2 - x + 1)), x)
 

3.22.38.7 Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.17 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )}}{{\left (x^{3} + 1\right )} x} \,d x } \]

input

integrate((-1+x)*(x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="maxima")
 

output

integrate((x^4 - x^3)^(1/4)*(x - 1)/((x^3 + 1)*x), x)
 

3.22.38.8 Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.36 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.51 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=-\frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]

input

integrate((-1+x)*(x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="giac")
 

output

-1/6*(sqrt(6) + sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/( 
sqrt(6) + sqrt(2))) - 1/6*(sqrt(6) + sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 
 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) - 1/6*(sqrt(6) - sqrt(2))*arctan 
((sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) - 1/6*(sqrt 
(6) - sqrt(2))*arctan(-(sqrt(6) + sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - 
 sqrt(2))) - 1/12*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/x + 
1)^(1/4) + sqrt(-1/x + 1) + 1) + 1/12*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6 
) + sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1/12*(sqrt(6) - sqrt 
(2))*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 
1/12*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + s 
qrt(-1/x + 1) + 1) + 1/3*8^(3/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) + 2/ 
3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) - 2/3*2^(1/4)*log(abs(-2^(1/4) + 
 (-1/x + 1)^(1/4)))
 

3.22.38.9 Mupad [N/A]

Not integrable

Time = 5.57 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.17 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x-1\right )}{x\,\left (x^3+1\right )} \,d x \]

input

int(((x^4 - x^3)^(1/4)*(x - 1))/(x*(x^3 + 1)),x)
 

output

int(((x^4 - x^3)^(1/4)*(x - 1))/(x*(x^3 + 1)), x)