Integrand size = 25, antiderivative size = 148 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\frac {4}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Output:
Unintegrable
Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.09 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (16 \sqrt [4]{2} \left (\arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )-\text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left (x^3 (1+x)\right )^{3/4}} \] Input:
Integrate[((1 + x)*(x^3 + x^4)^(1/4))/(x*(-1 + x^3)),x]
Output:
(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 = 9.22 (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 |
\(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \left (\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1-i \sqrt {3}}\right )+\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1+i \sqrt {3}}\right )+\frac {1}{9} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},x,-x\right )+\frac {1}{72} \left (1-i \sqrt {3}\right ) \sqrt [4]{x+1} \left (3-\sqrt {x}\right )-\frac {1}{144} \left (\left (1+i \sqrt {3}\right )^2 \sqrt {x}+6 \left (1-i \sqrt {3}\right )\right ) \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
Input:
Int[((1 + x)*(x^3 + x^4)^(1/4))/(x*(-1 + x^3)),x]
Output:
(-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))
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 = 83.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{3}-\frac {2 \,2^{\frac {1}{4}} \ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\right )}{3}-\frac {4 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}\) | \(118\) |
trager | \(\text {Expression too large to display}\) | \(25163\) |
Input:
int((1+x)*(x^4+x^3)^(1/4)/x/(x^3-1),x,method=_RETURNVERBOSE)
Output:
-1/3*sum((_R^4-2)*ln((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootO f(_Z^8-_Z^4+1))-2/3*2^(1/4)*ln((2^(1/4)*x+(x^3*(1+x))^(1/4))/(-2^(1/4)*x+( x^3*(1+x))^(1/4)))-4/3*arctan(1/2*2^(3/4)/x*(x^3*(1+x))^(1/4))*2^(1/4)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.23 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx =\text {Too large to display} \] Input:
integrate((1+x)*(x^4+x^3)^(1/4)/x/(x^3-1),x, algorithm="fricas")
Output:
1/6*sqrt(2)*sqrt(-sqrt(6*sqrt(-1/3) + 2))*log((sqrt(2)*x*sqrt(-sqrt(6*sqrt (-1/3) + 2)) + 2*(x^4 + x^3)^(1/4))/x) - 1/6*sqrt(2)*sqrt(-sqrt(6*sqrt(-1/ 3) + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(6*sqrt(-1/3) + 2)) - 2*(x^4 + x^3)^(1/ 4))/x) + 1/6*sqrt(2)*sqrt(-sqrt(-6*sqrt(-1/3) + 2))*log((sqrt(2)*x*sqrt(-s qrt(-6*sqrt(-1/3) + 2)) + 2*(x^4 + x^3)^(1/4))/x) - 1/6*sqrt(2)*sqrt(-sqrt (-6*sqrt(-1/3) + 2))*log(-(sqrt(2)*x*sqrt(-sqrt(-6*sqrt(-1/3) + 2)) - 2*(x ^4 + x^3)^(1/4))/x) + 1/6*sqrt(2)*(6*sqrt(-1/3) + 2)^(1/4)*log((sqrt(2)*x* (6*sqrt(-1/3) + 2)^(1/4) + 2*(x^4 + x^3)^(1/4))/x) - 1/6*sqrt(2)*(6*sqrt(- 1/3) + 2)^(1/4)*log(-(sqrt(2)*x*(6*sqrt(-1/3) + 2)^(1/4) - 2*(x^4 + x^3)^( 1/4))/x) + 1/6*sqrt(2)*(-6*sqrt(-1/3) + 2)^(1/4)*log((sqrt(2)*x*(-6*sqrt(- 1/3) + 2)^(1/4) + 2*(x^4 + x^3)^(1/4))/x) - 1/6*sqrt(2)*(-6*sqrt(-1/3) + 2 )^(1/4)*log(-(sqrt(2)*x*(-6*sqrt(-1/3) + 2)^(1/4) - 2*(x^4 + x^3)^(1/4))/x ) + 4/3*2^(1/4)*arctan(2^(1/4)*(x^4 + x^3)^(3/4)/(x^3 + x^2)) - 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)
Not integrable
Time = 1.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \] 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)
Not integrable
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \] 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)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (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 {4}{3} \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 ) - \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))/(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) - 4/3*2^(1/4)*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 = 8.67 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\right )} \,d x \] Input:
int(((x^3 + x^4)^(1/4)*(x + 1))/(x*(x^3 - 1)),x)
Output:
int(((x^3 + x^4)^(1/4)*(x + 1))/(x*(x^3 - 1)), x)
Not integrable
Time = 1.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.26 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x \left (-1+x^3\right )} \, dx=\int \frac {x^{\frac {3}{4}} \left (x +1\right )^{\frac {1}{4}}}{x^{4}-x}d x +\int \frac {x^{\frac {3}{4}} \left (x +1\right )^{\frac {1}{4}}}{x^{3}-1}d x \] Input:
int((1+x)*(x^4+x^3)^(1/4)/x/(x^3-1),x)
Output:
int((x**(3/4)*(x + 1)**(1/4))/(x**4 - x),x) + int((x**(3/4)*(x + 1)**(1/4) )/(x**3 - 1),x)