Integrand size = 22, antiderivative size = 93 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \]
-2*arctan(x/(x^4-x^3)^(1/4))+2*2^(1/4)*arctan(2^(1/4)*x/(x^4-x^3)^(1/4))+2 *arctanh(x/(x^4-x^3)^(1/4))-2*2^(1/4)*arctanh(2^(1/4)*x/(x^4-x^3)^(1/4))
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=-\frac {2 (-1+x)^{3/4} x^{9/4} \left (\arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \]
(-2*(-1 + x)^(3/4)*x^(9/4)*(ArcTan[((-1 + x)/x)^(-1/4)] - 2^(1/4)*ArcTan[2 ^(1/4)/((-1 + x)/x)^(1/4)] - ArcTanh[((-1 + x)/x)^(-1/4)] + 2^(1/4)*ArcTan h[2^(1/4)/((-1 + x)/x)^(1/4)]))/((-1 + x)*x^3)^(3/4)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1948, 140, 27, 73, 104, 770, 756, 216, 219, 827, 216, 219}
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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-x^3}}{x (x+1)} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1}}{\sqrt [4]{x} (x+1)}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx+\int -\frac {2}{(x-1)^{3/4} \sqrt [4]{x} (x+1)}dx\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx-2 \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+1)}dx\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}-2 \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+1)}dx\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}-8 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \int \frac {1}{2-x}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}-8 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \int \frac {1}{\sqrt {x-1}+1}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )-8 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )-8 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )-8 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (1-\frac {2 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )-8 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}\right )\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )-8 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}\right )\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )-8 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2\ 2^{3/4}}\right )\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
((-x^3 + x^4)^(1/4)*(4*(ArcTan[(-1 + x)^(1/4)/x^(1/4)]/2 + ArcTanh[(-1 + x )^(1/4)/x^(1/4)]/2) - 8*(-1/2*ArcTan[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)]/2^( 3/4) + ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)]/(2*2^(3/4)))))/((-1 + x)^ (1/4)*x^(3/4))
3.13.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 4.86 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )+\ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x}{x}\right )-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}\) | \(121\) |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x -4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{2}\) | \(449\) |
-ln(((x^3*(x-1))^(1/4)-x)/x)+2*arctan((x^3*(x-1))^(1/4)/x)+ln(((x^3*(x-1)) ^(1/4)+x)/x)-ln((-2^(1/4)*x-(x^3*(x-1))^(1/4))/(2^(1/4)*x-(x^3*(x-1))^(1/4 )))*2^(1/4)-2*arctan(1/2*2^(3/4)/x*(x^3*(x-1))^(1/4))*2^(1/4)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=-2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
-2^(1/4)*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x) - I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(1/4))/ x) + I*2^(1/4)*log((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2*arctan((x^4 - x^3)^(1/4)/x) + log((x + (x^4 - x^3)^(1/4))/x) - log(-(x - (x^4 - x^3)^(1 /4))/x)
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (x + 1\right )}\, dx \]
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x + 1\right )} x} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-2*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) - 2^(1/4)*log(2^(1/4) + (- 1/x + 1)^(1/4)) + 2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 2*arctan ((-1/x + 1)^(1/4)) + log((-1/x + 1)^(1/4) + 1) - log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (1+x)} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (x+1\right )} \,d x \]