Integrand size = 27, antiderivative size = 147 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\sqrt [4]{-x^3+x^4}-\frac {7}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {7}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (4 \sqrt [4]{-1+x} x^{3/4}-14 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+14 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{4 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^(9/4)*(4*(-1 + x)^(1/4)*x^(3/4) - 14*ArcTan[((-1 + x)/x) ^(-1/4)] + 14*ArcTanh[((-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) & ]))/(4*((-1 + x)*x^3)^(3/ 4))
Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.58, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2467, 2035, 7279, 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.
| \(\displaystyle \int \frac {(x+1) \sqrt [4]{x^4-x^3}}{x^2-x+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/4} (x+1)}{x^2-x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/2} (x+1)}{x^2-x+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \int \left (\frac {\sqrt [4]{x-1} \sqrt {x} (2 x-1)}{x^2-x+1}+\sqrt [4]{x-1} \sqrt {x}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt [4]{x^4-x^3} \left (-\frac {2 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,\frac {2 x}{1-i \sqrt {3}}\right )}{3 \sqrt {3} \left (\sqrt {3}+i\right ) \sqrt [4]{1-x}}+\frac {2 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x}{1+i \sqrt {3}},x\right )}{3 \sqrt {3} \left (-\sqrt {3}+i\right ) \sqrt [4]{1-x}}-\frac {7}{8} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )+\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {3} \sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}+\frac {7}{8} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )-\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt {3} \sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}+\frac {1}{4} \sqrt [4]{x-1} x^{3/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
(4*(-x^3 + x^4)^(1/4)*(((-1 + x)^(1/4)*x^(3/4))/4 - (2*(-1 + x)^(1/4)*x^(3 /4)*AppellF1[3/4, -1/4, 1, 7/4, x, (2*x)/(1 - I*Sqrt[3])])/(3*Sqrt[3]*(I + Sqrt[3])*(1 - x)^(1/4)) + (2*(-1 + x)^(1/4)*x^(3/4)*AppellF1[3/4, 1, -1/4 , 7/4, (2*x)/(1 + I*Sqrt[3]), x])/(3*Sqrt[3]*(I - Sqrt[3])*(1 - x)^(1/4)) - (7*ArcTan[x^(1/4)/(-1 + x)^(1/4)])/8 + ((-((I - Sqrt[3])/(I + Sqrt[3]))) ^(1/4)*ArcTan[x^(1/4)/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(-1 + x)^(1/ 4))])/Sqrt[3] + ArcTan[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x^(1/4))/(- 1 + x)^(1/4)]/(Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)) + (7*ArcTan h[x^(1/4)/(-1 + x)^(1/4)])/8 - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*Arc Tanh[x^(1/4)/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(-1 + x)^(1/4))])/Sqr t[3] - ArcTanh[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x^(1/4))/(-1 + x)^( 1/4)]/(Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4))))/((-1 + x)^(1/4)*x ^(3/4))
3.21.51.3.1 Defintions of rubi rules used
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]
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]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 7.82 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+\frac {7 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {7 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )+\frac {7 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}\) | \(116\) |
| trager | \(\text {Expression too large to display}\) | \(1744\) |
| risch | \(\text {Expression too large to display}\) | \(3741\) |
(x^3*(-1+x))^(1/4)+7/4*ln((x+(x^3*(-1+x))^(1/4))/x)-7/4*ln(((x^3*(-1+x))^( 1/4)-x)/x)+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))+7/2*arctan((x^3*(-1+x))^(1/4)/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{4} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{4} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/2*sqrt(2)*sqrt(-sqrt(2*sqrt(-3) + 2))*log((sqrt(2)*x*sqrt(-sqrt(2*sqrt( -3) + 2)) + 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(2)*sqrt(-sqrt(2*sqrt(-3) + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(2*sqrt(-3) + 2)) - 2*(x^4 - x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(-sqrt(-2*sqrt(-3) + 2))*log((sqrt(2)*x*sqrt(-sqrt(-2*sq rt(-3) + 2)) + 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(2)*sqrt(-sqrt(-2*sqrt(-3 ) + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(-2*sqrt(-3) + 2)) - 2*(x^4 - x^3)^(1/4) )/x) - 1/2*sqrt(2)*(2*sqrt(-3) + 2)^(1/4)*log((sqrt(2)*x*(2*sqrt(-3) + 2)^ (1/4) + 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(2)*(2*sqrt(-3) + 2)^(1/4)*log(- (sqrt(2)*x*(2*sqrt(-3) + 2)^(1/4) - 2*(x^4 - x^3)^(1/4))/x) - 1/2*sqrt(2)* (-2*sqrt(-3) + 2)^(1/4)*log((sqrt(2)*x*(-2*sqrt(-3) + 2)^(1/4) + 2*(x^4 - x^3)^(1/4))/x) + 1/2*sqrt(2)*(-2*sqrt(-3) + 2)^(1/4)*log(-(sqrt(2)*x*(-2*s qrt(-3) + 2)^(1/4) - 2*(x^4 - x^3)^(1/4))/x) + (x^4 - x^3)^(1/4) + 7/2*arc tan((x^4 - x^3)^(1/4)/x) + 7/4*log((x + (x^4 - x^3)^(1/4))/x) - 7/4*log(-( x - (x^4 - x^3)^(1/4))/x)
Not integrable
Time = 1.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.14 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x + 1\right )}{x^{2} - x + 1}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{x^{2} - x + 1} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.61 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\frac {1}{2} \, {\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}{2} \, {\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}{2} \, {\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}{2} \, {\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}{4} \, {\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}{4} \, {\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}{4} \, {\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}{4} \, {\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 ) + x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{4} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{4} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-1/2*(sqrt(6) + sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/( sqrt(6) + sqrt(2))) - 1/2*(sqrt(6) + sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) - 1/2*(sqrt(6) - sqrt(2))*arctan ((sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) - 1/2*(sqrt (6) - sqrt(2))*arctan(-(sqrt(6) + sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) - 1/4*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/x + 1 )^(1/4) + sqrt(-1/x + 1) + 1) + 1/4*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1/4*(sqrt(6) - sqrt(2) )*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 1/4 *(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt( -1/x + 1) + 1) + x*(-1/x + 1)^(1/4) + 7/2*arctan((-1/x + 1)^(1/4)) + 7/4*l og((-1/x + 1)^(1/4) + 1) - 7/4*log(abs((-1/x + 1)^(1/4) - 1))
Not integrable
Time = 5.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x+1\right )}{x^2-x+1} \,d x \]