Integrand size = 24, antiderivative size = 113 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\frac {1}{8} (1+4 x) \sqrt [4]{x^3+x^4}+\frac {3}{16} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{16} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (2 x^{3/4} \sqrt [4]{1+x}+8 x^{7/4} \sqrt [4]{1+x}+3 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-3 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+32 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt [4]{x}\right )+\log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{16 \left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(1 + x)^(3/4)*(2*x^(3/4)*(1 + x)^(1/4) + 8*x^(7/4)*(1 + x)^(1/4) + 3*ArcTan[(x/(1 + x))^(1/4)] - 3*ArcTanh[(x/(1 + x))^(1/4)] + 32*RootSum[ 3 - 3*#1^4 + #1^8 & , (-Log[x^(1/4)] + Log[(1 + x)^(1/4) - x^(1/4)*#1])/(- 3*#1^3 + 2*#1^7) & ]))/(16*(x^3*(1 + x))^(3/4))
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.73, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2467, 25, 2019, 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.
\(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt [4]{x^4+x^3}}{x^3+1} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {x^{3/4} \sqrt [4]{x+1} \left (1-x^3\right )}{x^3+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{3/4} \sqrt [4]{x+1} \left (1-x^3\right )}{x^3+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{3/4} \left (1-x^3\right )}{(x+1)^{3/4} \left (x^2-x+1\right )}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \frac {x^{3/2} \left (1-x^3\right )}{(x+1)^{3/4} \left (x^2-x+1\right )}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \left (-\frac {x^{5/2}}{(x+1)^{3/4}}-\frac {x^{3/2}}{(x+1)^{3/4}}+\frac {2 x^{3/2}}{(x+1)^{3/4} \left (x^2-x+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \left (-\frac {3}{64} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {i \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{\sqrt {3}}+\frac {\left (\sqrt {3}+i\right ) \sqrt [4]{-\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}+\frac {3}{64} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )+\frac {i \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{\sqrt {3}}-\frac {\left (\sqrt {3}+i\right ) \sqrt [4]{-\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}-\frac {1}{8} \sqrt [4]{x+1} x^{7/4}-\frac {1}{32} \sqrt [4]{x+1} x^{3/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
(-4*(x^3 + x^4)^(1/4)*(-1/32*(x^(3/4)*(1 + x)^(1/4)) - (x^(7/4)*(1 + x)^(1 /4))/8 - (3*ArcTan[x^(1/4)/(1 + x)^(1/4)])/64 - (I*((I - Sqrt[3])/(3*I - S qrt[3]))^(3/4)*ArcTan[x^(1/4)/(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/4)*(1 + x)^(1/4))])/Sqrt[3] + ((I + Sqrt[3])*(-((I + Sqrt[3])/(3*I + Sqrt[3])))^(1 /4)*ArcTan[x^(1/4)/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x)^(1/4))]) /(2*3^(3/4)) + (3*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/64 + (I*((I - Sqrt[3])/( 3*I - Sqrt[3]))^(3/4)*ArcTanh[x^(1/4)/(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/ 4)*(1 + x)^(1/4))])/Sqrt[3] - ((I + Sqrt[3])*(-((I + Sqrt[3])/(3*I + Sqrt[ 3])))^(1/4)*ArcTanh[x^(1/4)/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x) ^(1/4))])/(2*3^(3/4))))/(x^(3/4)*(1 + x)^(1/4))
3.17.86.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 = 23.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.55
method | result | size |
pseudoelliptic | \(-\frac {x^{6} \left (-16 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} x +3 \ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-64 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-3\right )}\right )-3 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+6 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}{32 {\left (x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}^{2} {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{2}}\) | \(175\) |
trager | \(\text {Expression too large to display}\) | \(1860\) |
risch | \(\text {Expression too large to display}\) | \(3868\) |
-1/32*x^6*(-16*(x^3*(1+x))^(1/4)*x+3*ln((x+(x^3*(1+x))^(1/4))/x)-64*sum(ln ((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^3/(2*_R^4-3),_R=RootOf(_Z^8-3*_Z^4+3))-3* ln(((x^3*(1+x))^(1/4)-x)/x)+6*arctan((x^3*(1+x))^(1/4)/x)-4*(x^3*(1+x))^(1 /4))/(x+(x^3*(1+x))^(1/4))^2/((x^3*(1+x))^(1/4)-x)^2/(x^2+(x^3*(1+x))^(1/2 ))^2
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.99 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\left (\frac {1}{2187} i + \frac {1}{2187}\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i + 1\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} x - \left (81 i + 81\right ) \cdot 2187^{\frac {3}{8}} \sqrt {2} \left (-1\right )^{\frac {5}{8}} x + 2916 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{2187} i - \frac {1}{2187}\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i - 1\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + \left (81 i - 81\right ) \cdot 2187^{\frac {3}{8}} \sqrt {2} \left (-1\right )^{\frac {5}{8}} x + 2916 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{2187} i - \frac {1}{2187}\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i - 1\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} x - \left (81 i - 81\right ) \cdot 2187^{\frac {3}{8}} \sqrt {2} \left (-1\right )^{\frac {5}{8}} x + 2916 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{2187} i + \frac {1}{2187}\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i + 1\right ) \cdot 2187^{\frac {7}{8}} \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + \left (81 i + 81\right ) \cdot 2187^{\frac {3}{8}} \sqrt {2} \left (-1\right )^{\frac {5}{8}} x + 2916 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{2187} \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} x + 81 \cdot 2187^{\frac {3}{8}} \left (-1\right )^{\frac {5}{8}} x + 1458 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {2}{2187} \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} x + 81 \cdot 2187^{\frac {3}{8}} \left (-1\right )^{\frac {5}{8}} x - 1458 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{2187} i \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {i \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} x + 81 i \cdot 2187^{\frac {3}{8}} \left (-1\right )^{\frac {5}{8}} x + 1458 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {2}{2187} i \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-i \cdot 2187^{\frac {7}{8}} \left (-1\right )^{\frac {1}{8}} x - 81 i \cdot 2187^{\frac {3}{8}} \left (-1\right )^{\frac {5}{8}} x + 1458 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 1\right )} - \frac {3}{16} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{32} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {3}{32} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
(1/2187*I + 1/2187)*2187^(7/8)*sqrt(2)*(-1)^(1/8)*log(((I + 1)*2187^(7/8)* sqrt(2)*(-1)^(1/8)*x - (81*I + 81)*2187^(3/8)*sqrt(2)*(-1)^(5/8)*x + 2916* (x^4 + x^3)^(1/4))/x) - (1/2187*I - 1/2187)*2187^(7/8)*sqrt(2)*(-1)^(1/8)* log((-(I - 1)*2187^(7/8)*sqrt(2)*(-1)^(1/8)*x + (81*I - 81)*2187^(3/8)*sqr t(2)*(-1)^(5/8)*x + 2916*(x^4 + x^3)^(1/4))/x) + (1/2187*I - 1/2187)*2187^ (7/8)*sqrt(2)*(-1)^(1/8)*log(((I - 1)*2187^(7/8)*sqrt(2)*(-1)^(1/8)*x - (8 1*I - 81)*2187^(3/8)*sqrt(2)*(-1)^(5/8)*x + 2916*(x^4 + x^3)^(1/4))/x) - ( 1/2187*I + 1/2187)*2187^(7/8)*sqrt(2)*(-1)^(1/8)*log((-(I + 1)*2187^(7/8)* sqrt(2)*(-1)^(1/8)*x + (81*I + 81)*2187^(3/8)*sqrt(2)*(-1)^(5/8)*x + 2916* (x^4 + x^3)^(1/4))/x) + 2/2187*2187^(7/8)*(-1)^(1/8)*log((2187^(7/8)*(-1)^ (1/8)*x + 81*2187^(3/8)*(-1)^(5/8)*x + 1458*(x^4 + x^3)^(1/4))/x) - 2/2187 *2187^(7/8)*(-1)^(1/8)*log(-(2187^(7/8)*(-1)^(1/8)*x + 81*2187^(3/8)*(-1)^ (5/8)*x - 1458*(x^4 + x^3)^(1/4))/x) + 2/2187*I*2187^(7/8)*(-1)^(1/8)*log( (I*2187^(7/8)*(-1)^(1/8)*x + 81*I*2187^(3/8)*(-1)^(5/8)*x + 1458*(x^4 + x^ 3)^(1/4))/x) - 2/2187*I*2187^(7/8)*(-1)^(1/8)*log((-I*2187^(7/8)*(-1)^(1/8 )*x - 81*I*2187^(3/8)*(-1)^(5/8)*x + 1458*(x^4 + x^3)^(1/4))/x) + 1/8*(x^4 + x^3)^(1/4)*(4*x + 1) - 3/16*arctan((x^4 + x^3)^(1/4)/x) - 3/32*log((x + (x^4 + x^3)^(1/4))/x) + 3/32*log(-(x - (x^4 + x^3)^(1/4))/x)
Not integrable
Time = 0.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Not integrable
Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )}}{x^{3} + 1} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\frac {1}{8} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {3}{16} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
1/8*((1/x + 1)^(5/4) + 3*(1/x + 1)^(1/4))*x^2 - 3/16*arctan((1/x + 1)^(1/4 )) - 3/32*log((1/x + 1)^(1/4) + 1) + 3/32*log((1/x + 1)^(1/4) - 1)
Not integrable
Time = 5.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {\left (-1+x^3\right ) \sqrt [4]{x^3+x^4}}{1+x^3} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^3-1\right )}{x^3+1} \,d x \]