Integrand size = 25, antiderivative size = 226 \[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=2 \sqrt [4]{x^3+x^4}+\arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )-\text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right ) \]
2*(x^4+x^3)^(1/4)+arctan(x/(x^4+x^3)^(1/4))-(2+2*5^(1/2))^(1/2)*arctan(1/2 *(-2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))+(-2+2*5^(1/2))^(1/2)*arctan(1/2*( 2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))-arctanh(x/(x^4+x^3)^(1/4))+(2+2*5^(1 /2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))-(-2+2*5^(1/ 2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4+x^3)^(1/4))
Time = 0.78 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04 \[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (2 x^{3/4} \sqrt [4]{1+x}+\arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-\sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )+\sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )-\text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+\sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )-\sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )\right )}{\left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(1 + x)^(3/4)*(2*x^(3/4)*(1 + x)^(1/4) + ArcTan[(x/(1 + x))^(1/4) ] - Sqrt[2*(1 + Sqrt[5])]*ArcTan[Sqrt[(-1 + Sqrt[5])/2]*(x/(1 + x))^(1/4)] + Sqrt[2*(-1 + Sqrt[5])]*ArcTan[Sqrt[(1 + Sqrt[5])/2]*(x/(1 + x))^(1/4)] - ArcTanh[(x/(1 + x))^(1/4)] + Sqrt[2*(1 + Sqrt[5])]*ArcTanh[Sqrt[(-1 + Sq rt[5])/2]*(x/(1 + x))^(1/4)] - Sqrt[2*(-1 + Sqrt[5])]*ArcTanh[Sqrt[(1 + Sq rt[5])/2]*(x/(1 + x))^(1/4)]))/(x^3*(1 + x))^(3/4)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.75, 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, 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.
| \(\displaystyle \int \frac {(2 x+1) \sqrt [4]{x^4+x^3}}{x^2+x-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {x^{3/4} \sqrt [4]{x+1} (2 x+1)}{-x^2-x+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{3/4} \sqrt [4]{x+1} (2 x+1)}{-x^2-x+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \frac {x^{3/2} \sqrt [4]{x+1} (2 x+1)}{-x^2-x+1}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \left (\frac {(2-x) \sqrt {x} \sqrt [4]{x+1}}{-x^2-x+1}-2 \sqrt {x} \sqrt [4]{x+1}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \left (\frac {4 x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-x,-\frac {2 x}{1+\sqrt {5}}\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {4 x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {2 x}{1-\sqrt {5}},-x\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {1}{4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )+\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 \sqrt {5}}+\frac {1}{4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 \sqrt {5}}-\frac {1}{2} x^{3/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
(-4*(x^3 + x^4)^(1/4)*(-1/2*(x^(3/4)*(1 + x)^(1/4)) + (4*x^(3/4)*AppellF1[ 3/4, -1/4, 1, 7/4, -x, (-2*x)/(1 + Sqrt[5])])/(3*Sqrt[5]*(1 + Sqrt[5])) - (4*x^(3/4)*AppellF1[3/4, 1, -1/4, 7/4, (-2*x)/(1 - Sqrt[5]), -x])/(3*Sqrt[ 5]*(1 - Sqrt[5])) - ArcTan[x^(1/4)/(1 + x)^(1/4)]/4 + (((3 + Sqrt[5])/2)^( 1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(2*Sqrt[5]) + (((3 - Sqrt[5])/2)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4))/(1 + x )^(1/4)])/(2*Sqrt[5]) + ArcTanh[x^(1/4)/(1 + x)^(1/4)]/4 - (((3 + Sqrt[5]) /2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(2*Sqr t[5]) - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x^(1/4)) /(1 + x)^(1/4)])/(2*Sqrt[5])))/(x^(3/4)*(1 + x)^(1/4))
3.27.7.3.1 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_.) + (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 = 12.36 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(-\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right ) \sqrt {-2+2 \sqrt {5}}-\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right ) \sqrt {-2+2 \sqrt {5}}+\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right ) \sqrt {2+2 \sqrt {5}}+\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right ) \sqrt {2+2 \sqrt {5}}-\frac {\ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{2}-\arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\) | \(205\) |
| trager | \(\text {Expression too large to display}\) | \(2140\) |
| risch | \(\text {Expression too large to display}\) | \(3991\) |
-arctanh(2*(x^3*(1+x))^(1/4)/x/(2+2*5^(1/2))^(1/2))*(-2+2*5^(1/2))^(1/2)-a rctan(2*(x^3*(1+x))^(1/4)/x/(2+2*5^(1/2))^(1/2))*(-2+2*5^(1/2))^(1/2)+arct an(2*(x^3*(1+x))^(1/4)/x/(-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))*(2+2*5^(1/2))^(1/2)-1/2*ln((x +(x^3*(1+x))^(1/4))/x)+1/2*ln(((x^3*(1+x))^(1/4)-x)/x)-arctan((x^3*(1+x))^ (1/4)/x)+2*(x^3*(1+x))^(1/4)
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (168) = 336\).
Time = 0.28 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.94 \[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/2*sqrt(2*sqrt(5) + 2)*log(((sqrt(5)*x - x)*sqrt(2*sqrt(5) + 2) + 4*(x^4 + x^3)^(1/4))/x) - 1/2*sqrt(2*sqrt(5) + 2)*log(-((sqrt(5)*x - x)*sqrt(2*sq rt(5) + 2) - 4*(x^4 + x^3)^(1/4))/x) - 1/2*sqrt(2*sqrt(5) - 2)*log(((sqrt( 5)*x + x)*sqrt(2*sqrt(5) - 2) + 4*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(2*sqrt( 5) - 2)*log(-((sqrt(5)*x + x)*sqrt(2*sqrt(5) - 2) - 4*(x^4 + x^3)^(1/4))/x ) - 1/2*sqrt(-2*sqrt(5) + 2)*log(((sqrt(5)*x + x)*sqrt(-2*sqrt(5) + 2) + 4 *(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-2*sqrt(5) + 2)*log(-((sqrt(5)*x + x)*sq rt(-2*sqrt(5) + 2) - 4*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-2*sqrt(5) - 2)*lo g(((sqrt(5)*x - x)*sqrt(-2*sqrt(5) - 2) + 4*(x^4 + x^3)^(1/4))/x) - 1/2*sq rt(-2*sqrt(5) - 2)*log(-((sqrt(5)*x - x)*sqrt(-2*sqrt(5) - 2) - 4*(x^4 + x ^3)^(1/4))/x) + 2*(x^4 + x^3)^(1/4) - arctan((x^4 + x^3)^(1/4)/x) - 1/2*lo g((x + (x^4 + x^3)^(1/4))/x) + 1/2*log(-(x - (x^4 + x^3)^(1/4))/x)
\[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (2 x + 1\right )}{x^{2} + x - 1}\, dx \]
\[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (2 \, x + 1\right )}}{x^{2} + x - 1} \,d x } \]
Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=-\sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 2 \, x {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-sqrt(2*sqrt(5) - 2)*arctan((1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) + 1/2)) + sqr t(2*sqrt(5) + 2)*arctan((1/x + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/2)) - 1/2*sqr t(2*sqrt(5) - 2)*log(sqrt(1/2*sqrt(5) + 1/2) + (1/x + 1)^(1/4)) + 1/2*sqrt (2*sqrt(5) + 2)*log(sqrt(1/2*sqrt(5) - 1/2) + (1/x + 1)^(1/4)) + 1/2*sqrt( 2*sqrt(5) - 2)*log(abs(-sqrt(1/2*sqrt(5) + 1/2) + (1/x + 1)^(1/4))) - 1/2* sqrt(2*sqrt(5) + 2)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (1/x + 1)^(1/4))) + 2*x*(1/x + 1)^(1/4) - arctan((1/x + 1)^(1/4)) - 1/2*log((1/x + 1)^(1/4) + 1) + 1/2*log(abs((1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {(1+2 x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (2\,x+1\right )}{x^2+x-1} \,d x \]