Integrand size = 25, antiderivative size = 148 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\frac {4}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.09 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (16 \sqrt [4]{2} \left (\arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )-\text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(1 + x)^(3/4)*(16*2^(1/4)*(ArcTan[2^(1/4)*(x/(1 + x))^(1/4)] - Ar cTanh[2^(1/4)*(x/(1 + 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) & ]))/(12*(x^3*(1 + x))^(3/4))
Result contains complex when optimal does not.
Time = 9.02 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2467, 25, 2035, 7276, 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.
| \(\displaystyle \int \frac {(x+1) \sqrt [4]{x^4+x^3}}{x \left (x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {(x+1)^{5/4}}{\sqrt [4]{x} \left (1-x^3\right )}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {(x+1)^{5/4}}{\sqrt [4]{x} \left (1-x^3\right )}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \frac {\sqrt {x} (x+1)^{5/4}}{1-x^3}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \left (\frac {\sqrt {x} (x+1)^{5/4}}{2 \left (x^{3/2}+1\right )}-\frac {\sqrt {x} (x+1)^{5/4}}{2 \left (x^{3/2}-1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \left (\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1-i \sqrt {3}}\right )+\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1+i \sqrt {3}}\right )+\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},x,-x\right )+\frac {1}{72} \left (1-i \sqrt {3}\right ) \sqrt [4]{x+1} \left (3-\sqrt {x}\right )-\frac {1}{144} \left (\left (1+i \sqrt {3}\right )^2 \sqrt {x}+6 \left (1-i \sqrt {3}\right )\right ) \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
(-4*(x^3 + x^4)^(1/4)*(((1 - I*Sqrt[3])*(3 - Sqrt[x])*(1 + x)^(1/4))/72 - ((6*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])^2*Sqrt[x])*(1 + x)^(1/4))/144 + (x^( 3/4)*AppellF1[3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 - I*Sqrt[3])])/9 + (x^(3/4) *AppellF1[3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 + I*Sqrt[3])])/9 + (x^(3/4)*App ellF1[3/4, 1, -5/4, 7/4, x, -x])/9))/(x^(3/4)*(1 + x)^(1/4))
3.21.60.3.1 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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 8.76 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82
| 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 | \(\text {Expression too large to display}\) | \(3025\) |
-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)*l n((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.39 \[ \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 ) \]
1/6*sqrt(2)*sqrt(-sqrt(2*I*sqrt(3) + 2))*log((sqrt(2)*x*sqrt(-sqrt(2*I*sqr t(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*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) - 1/6*sqrt(2)*(-2*I*sqrt(3) + 2)^(1/4)*log(-(s qrt(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)
Not integrable
Time = 1.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.38 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.45 \[ \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 ) \]
1/6*(sqrt(6) + sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(1/x + 1)^(1/4))/(sq rt(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((s qrt(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) + sqr t(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) - 1/3*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 )))
Not integrable
Time = 6.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\right )} \,d x \]