Integrand size = 25, antiderivative size = 255 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {1}{8} (1+4 x) \sqrt [4]{x^3+x^4}-\frac {29}{16} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {29}{16} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right ) \]
1/8*(1+4*x)*(x^4+x^3)^(1/4)-29/16*arctan(x/(x^4+x^3)^(1/4))+1/5*(10+10*5^( 1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))+1/5*(-10+10 *5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))+29/16*ar ctanh(x/(x^4+x^3)^(1/4))-1/5*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/ 2))^(1/2)*x/(x^4+x^3)^(1/4))-1/5*(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5 ^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))
Time = 0.70 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (10 x^{3/4} \sqrt [4]{1+x}+40 x^{7/4} \sqrt [4]{1+x}-145 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+16 \sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )+16 \sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )+145 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )-16 \sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )-16 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )\right )}{80 \left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(1 + x)^(3/4)*(10*x^(3/4)*(1 + x)^(1/4) + 40*x^(7/4)*(1 + x)^(1/4 ) - 145*ArcTan[(x/(1 + x))^(1/4)] + 16*Sqrt[10*(1 + Sqrt[5])]*ArcTan[Sqrt[ (-1 + Sqrt[5])/2]*(x/(1 + x))^(1/4)] + 16*Sqrt[10*(-1 + Sqrt[5])]*ArcTan[S qrt[(1 + Sqrt[5])/2]*(x/(1 + x))^(1/4)] + 145*ArcTanh[(x/(1 + x))^(1/4)] - 16*Sqrt[10*(1 + Sqrt[5])]*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]*(x/(1 + x))^(1/4 )] - 16*Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[Sqrt[(1 + Sqrt[5])/2]*(x/(1 + x))^ (1/4)]))/(80*(x^3*(1 + x))^(3/4))
Time = 0.78 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.37, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {2027, 2467, 25, 1201, 25, 60, 60, 73, 854, 827, 216, 219, 1202, 25, 73, 854, 827, 216, 219, 1205, 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.
| \(\displaystyle \int \frac {\left (x^2+x\right ) \sqrt [4]{x^4+x^3}}{x^2+x-1} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x (x+1) \sqrt [4]{x^4+x^3}}{x^2+x-1}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {x^{7/4} (x+1)^{5/4}}{-x^2-x+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{7/4} (x+1)^{5/4}}{-x^2-x+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 1201 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int -x^{3/4} \sqrt [4]{x+1}dx+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\int x^{3/4} \sqrt [4]{x+1}dx\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (-\frac {1}{8} \int \frac {x^{3/4}}{(x+1)^{3/4}}dx+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\sqrt [4]{x} (x+1)^{3/4}}dx-x^{3/4} \sqrt [4]{x+1}\right )+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (3 \int \frac {\sqrt {x}}{(x+1)^{3/4}}d\sqrt [4]{x}-x^{3/4} \sqrt [4]{x+1}\right )+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (3 \int \frac {\sqrt {x}}{1-x}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-x^{3/4} \sqrt [4]{x+1}\right )+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-x^{3/4} \sqrt [4]{x+1}\right )+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )+\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {x^{3/4} \sqrt [4]{x+1}}{-x^2-x+1}dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 1202 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (-\int -\frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx-\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4}}dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx-\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4}}dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx-4 \int \frac {\sqrt {x}}{(x+1)^{3/4}}d\sqrt [4]{x}+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx-4 \int \frac {\sqrt {x}}{1-x}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx-4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (-4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )+\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \frac {1}{\sqrt [4]{x} (x+1)^{3/4} \left (-x^2-x+1\right )}dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-4 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 1205 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\int \left (\frac {2}{\sqrt {5} \left (-2 x+\sqrt {5}-1\right ) (x+1)^{3/4} \sqrt [4]{x}}+\frac {2}{\sqrt {5} (x+1)^{3/4} \left (2 x+\sqrt {5}+1\right ) \sqrt [4]{x}}\right )dx+\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-4 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{8} \left (3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-x^{3/4} \sqrt [4]{x+1}\right )-4 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5}}-\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5}}+\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5}}+\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5}}-\frac {1}{2} \sqrt [4]{x+1} x^{7/4}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
-(((x^3 + x^4)^(1/4)*(-1/2*(x^(7/4)*(1 + x)^(1/4)) - (2^(3/4)*(3 + Sqrt[5] )^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/Sqrt[5] - (2^(3/4)*(3 - Sqrt[5])^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(1 + x)^(1/4)])/Sqrt[5] + (-(x^(3/4)*(1 + x)^(1/4)) + 3*(-1/2*ArcTan[x^(1/4)/ (1 + x)^(1/4)] + ArcTanh[x^(1/4)/(1 + x)^(1/4)]/2))/8 - 4*(-1/2*ArcTan[x^( 1/4)/(1 + x)^(1/4)] + ArcTanh[x^(1/4)/(1 + x)^(1/4)]/2) + (2^(3/4)*(3 + Sq rt[5])^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/Sqr t[5] + (2^(3/4)*(3 - Sqrt[5])^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x^(1/ 4))/(1 + x)^(1/4)])/Sqrt[5]))/(x^(3/4)*(1 + x)^(1/4)))
3.28.43.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_)^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[(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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Simp[g/c^2 Int[Simp[2*c*e*f + c*d*g - b* e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Simp[1/c^2 Int[Simp[c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2 *c^2*d*f*g - 2*b*c*e*f*g - b*c*d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]*(d + e* x)^(m - 1)*((f + g*x)^(n - 2)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1 ]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*(g/c) Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Simp[1/c Int[Simp[c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*((f + g*x)^(n - 1)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^ n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]
Time = 15.51 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\frac {x^{6} \left (\frac {5 \left (-\frac {1}{4}-x \right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2}+\left (\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \sqrt {5}\, \sqrt {-2+2 \sqrt {5}}+\sqrt {2+2 \sqrt {5}}\, \left (\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )\right ) \sqrt {5}+\frac {145 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{32}-\frac {145 \ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{32}-\frac {145 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{16}\right )}{5 {\left (x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}^{2} {\left (-\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x \right )}^{2} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{2}}\) | \(244\) |
| trager | \(\text {Expression too large to display}\) | \(2153\) |
| risch | \(\text {Expression too large to display}\) | \(4381\) |
-1/5*x^6*(5/2*(-1/4-x)*(x^3*(1+x))^(1/4)+(arctanh(2*(x^3*(1+x))^(1/4)/x/(2 +2*5^(1/2))^(1/2))+arctan(2*(x^3*(1+x))^(1/4)/x/(2+2*5^(1/2))^(1/2)))*5^(1 /2)*(-2+2*5^(1/2))^(1/2)+(2+2*5^(1/2))^(1/2)*(arctanh(2*(x^3*(1+x))^(1/4)/ x/(-2+2*5^(1/2))^(1/2))+arctan(2*(x^3*(1+x))^(1/4)/x/(-2+2*5^(1/2))^(1/2)) )*5^(1/2)+145/32*ln(((x^3*(1+x))^(1/4)-x)/x)-145/32*ln((x+(x^3*(1+x))^(1/4 ))/x)-145/16*arctan((x^3*(1+x))^(1/4)/x))/(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
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (177) = 354\).
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.84 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=-\frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} - 4 \, {\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 {29}{16} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {29}{32} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {29}{32} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/10*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(((sqrt(5)*x - x)*sqrt(2*sqrt(5) + 2) + 4*(x^4 + x^3)^(1/4))/x) + 1/10*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-((sqrt( 5)*x - x)*sqrt(2*sqrt(5) + 2) - 4*(x^4 + x^3)^(1/4))/x) - 1/10*sqrt(5)*sqr t(2*sqrt(5) - 2)*log(((sqrt(5)*x + x)*sqrt(2*sqrt(5) - 2) + 4*(x^4 + x^3)^ (1/4))/x) + 1/10*sqrt(5)*sqrt(2*sqrt(5) - 2)*log(-((sqrt(5)*x + x)*sqrt(2* sqrt(5) - 2) - 4*(x^4 + x^3)^(1/4))/x) - 1/10*sqrt(5)*sqrt(-2*sqrt(5) + 2) *log(((sqrt(5)*x + x)*sqrt(-2*sqrt(5) + 2) + 4*(x^4 + x^3)^(1/4))/x) + 1/1 0*sqrt(5)*sqrt(-2*sqrt(5) + 2)*log(-((sqrt(5)*x + x)*sqrt(-2*sqrt(5) + 2) - 4*(x^4 + x^3)^(1/4))/x) - 1/10*sqrt(5)*sqrt(-2*sqrt(5) - 2)*log(((sqrt(5 )*x - x)*sqrt(-2*sqrt(5) - 2) + 4*(x^4 + x^3)^(1/4))/x) + 1/10*sqrt(5)*sqr t(-2*sqrt(5) - 2)*log(-((sqrt(5)*x - x)*sqrt(-2*sqrt(5) - 2) - 4*(x^4 + x^ 3)^(1/4))/x) + 1/8*(x^4 + x^3)^(1/4)*(4*x + 1) + 29/16*arctan((x^4 + x^3)^ (1/4)/x) + 29/32*log((x + (x^4 + x^3)^(1/4))/x) - 29/32*log(-(x - (x^4 + x ^3)^(1/4))/x)
\[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {x \sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x^{2} + x - 1}\, dx \]
\[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + x\right )}}{x^{2} + x - 1} \,d x } \]
Time = 0.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.93 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {1}{8} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {29}{16} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {29}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {29}{32} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/8*((1/x + 1)^(5/4) + 3*(1/x + 1)^(1/4))*x^2 - 1/5*sqrt(10*sqrt(5) - 10)* arctan((1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) + 1/2)) - 1/5*sqrt(10*sqrt(5) + 10 )*arctan((1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/2)) - 1/10*sqrt(10*sqrt(5) - 10)*log(sqrt(1/2*sqrt(5) + 1/2) + (1/x + 1)^(1/4)) - 1/10*sqrt(10*sqrt(5) + 10)*log(sqrt(1/2*sqrt(5) - 1/2) + (1/x + 1)^(1/4)) + 1/10*sqrt(10*sqrt( 5) - 10)*log(abs(-sqrt(1/2*sqrt(5) + 1/2) + (1/x + 1)^(1/4))) + 1/10*sqrt( 10*sqrt(5) + 10)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (1/x + 1)^(1/4))) + 29 /16*arctan((1/x + 1)^(1/4)) + 29/32*log((1/x + 1)^(1/4) + 1) - 29/32*log(a bs((1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2+x\right )}{x^2+x-1} \,d x \]