Integrand size = 61, antiderivative size = 110 \[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {x-x^2+x^6}}{x^2}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {x-x^2+x^6}}{x^2}\right ) \]
1/5*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*(x^6-x^2+x)^(1/2 )/x^2)-1/5*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*(x^6-x^2 +x)^(1/2)/x^2)
\[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx \]
Integrate[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10),x]
Integrate[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10), x]
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.
| \(\displaystyle \int \frac {\left (2 x^5+2 x-3\right ) \sqrt {x^6-x^2+x}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x^6-x^2+x} \int -\frac {\sqrt {x} \left (-2 x^5-2 x+3\right ) \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}dx}{\sqrt {x} \sqrt {x^5-x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x^6-x^2+x} \int \frac {\sqrt {x} \left (-2 x^5-2 x+3\right ) \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}dx}{\sqrt {x} \sqrt {x^5-x+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x^6-x^2+x} \int \frac {x \left (-2 x^5-2 x+3\right ) \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}d\sqrt {x}}{\sqrt {x} \sqrt {x^5-x+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x^6-x^2+x} \int \left (-\frac {2 \sqrt {x^5-x+1} x^6}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}-\frac {2 \sqrt {x^5-x+1} x^2}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}+\frac {3 \sqrt {x^5-x+1} x}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}\right )d\sqrt {x}}{\sqrt {x} \sqrt {x^5-x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x^6-x^2+x} \left (3 \int \frac {x \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}d\sqrt {x}-2 \int \frac {x^2 \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}d\sqrt {x}-2 \int \frac {x^6 \sqrt {x^5-x+1}}{x^{10}-x^8-3 x^6+2 x^5+x^4-x^3+x^2-2 x+1}d\sqrt {x}\right )}{\sqrt {x} \sqrt {x^5-x+1}}\) |
Int[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10),x]
3.17.24.3.1 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]
Time = 6.71 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {5}\, \left (\sqrt {-2+2 \sqrt {5}}\, \left (\sqrt {5}+1\right ) \operatorname {arctanh}\left (\frac {2 \sqrt {x \left (x^{5}-x +1\right )}}{x^{2} \sqrt {2+2 \sqrt {5}}}\right )-\arctan \left (\frac {2 \sqrt {x \left (x^{5}-x +1\right )}}{x^{2} \sqrt {-2+2 \sqrt {5}}}\right ) \sqrt {2+2 \sqrt {5}}\, \left (\sqrt {5}-1\right )\right )}{10}\) | \(92\) |
| trager | \(\text {Expression too large to display}\) | \(941\) |
int((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^2-2*x+ 1),x,method=_RETURNVERBOSE)
-1/10*5^(1/2)*((-2+2*5^(1/2))^(1/2)*(5^(1/2)+1)*arctanh(2*(x*(x^5-x+1))^(1 /2)/x^2/(2+2*5^(1/2))^(1/2))-arctan(2*(x*(x^5-x+1))^(1/2)/x^2/(-2+2*5^(1/2 ))^(1/2))*(2+2*5^(1/2))^(1/2)*(5^(1/2)-1))
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (77) = 154\).
Time = 0.49 (sec) , antiderivative size = 743, normalized size of antiderivative = 6.75 \[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} + \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} + {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} + \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} + \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} - {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} + \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} - \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} + {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} - \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} - \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} - {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} - \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) \]
integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^ 2-2*x+1),x, algorithm="fricas")
-1/20*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^6 + x^4 - 3*x^2 + sqrt(5)*( x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x^2 + x) + (x^10 + 5*x^8 - x^6 + 2* x^5 - 5*x^4 + 5*x^3 + x^2 + sqrt(5)*(x^10 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(2*sqrt(5) + 2))/(x^10 - x^8 - 3*x^6 + 2*x ^5 + x^4 - x^3 + x^2 - 2*x + 1)) + 1/20*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-( 4*(3*x^6 + x^4 - 3*x^2 + sqrt(5)*(x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x ^2 + x) - (x^10 + 5*x^8 - x^6 + 2*x^5 - 5*x^4 + 5*x^3 + x^2 + sqrt(5)*(x^1 0 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(2*sqrt( 5) + 2))/(x^10 - x^8 - 3*x^6 + 2*x^5 + x^4 - x^3 + x^2 - 2*x + 1)) + 1/20* sqrt(5)*sqrt(-2*sqrt(5) + 2)*log(-(4*(3*x^6 + x^4 - 3*x^2 - sqrt(5)*(x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x^2 + x) + (x^10 + 5*x^8 - x^6 + 2*x^5 - 5*x^4 + 5*x^3 + x^2 - sqrt(5)*(x^10 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(-2*sqrt(5) + 2))/(x^10 - x^8 - 3*x^6 + 2*x^5 + x^4 - x^3 + x^2 - 2*x + 1)) - 1/20*sqrt(5)*sqrt(-2*sqrt(5) + 2)*log(-(4*( 3*x^6 + x^4 - 3*x^2 - sqrt(5)*(x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x^2 + x) - (x^10 + 5*x^8 - x^6 + 2*x^5 - 5*x^4 + 5*x^3 + x^2 - sqrt(5)*(x^10 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(-2*sqrt(5) + 2))/(x^10 - x^8 - 3*x^6 + 2*x^5 + x^4 - x^3 + x^2 - 2*x + 1))
\[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\int \frac {\sqrt {x \left (x^{5} - x + 1\right )} \left (2 x^{5} + 2 x - 3\right )}{x^{10} - x^{8} - 3 x^{6} + 2 x^{5} + x^{4} - x^{3} + x^{2} - 2 x + 1}\, dx \]
integrate((2*x**5+2*x-3)*(x**6-x**2+x)**(1/2)/(x**10-x**8-3*x**6+2*x**5+x* *4-x**3+x**2-2*x+1),x)
Integral(sqrt(x*(x**5 - x + 1))*(2*x**5 + 2*x - 3)/(x**10 - x**8 - 3*x**6 + 2*x**5 + x**4 - x**3 + x**2 - 2*x + 1), x)
\[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\int { \frac {\sqrt {x^{6} - x^{2} + x} {\left (2 \, x^{5} + 2 \, x - 3\right )}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1} \,d x } \]
integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^ 2-2*x+1),x, algorithm="maxima")
integrate(sqrt(x^6 - x^2 + x)*(2*x^5 + 2*x - 3)/(x^10 - x^8 - 3*x^6 + 2*x^ 5 + x^4 - x^3 + x^2 - 2*x + 1), x)
\[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\int { \frac {\sqrt {x^{6} - x^{2} + x} {\left (2 \, x^{5} + 2 \, x - 3\right )}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1} \,d x } \]
integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^ 2-2*x+1),x, algorithm="giac")
integrate(sqrt(x^6 - x^2 + x)*(2*x^5 + 2*x - 3)/(x^10 - x^8 - 3*x^6 + 2*x^ 5 + x^4 - x^3 + x^2 - 2*x + 1), x)
Timed out. \[ \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx=\int \frac {\left (2\,x^5+2\,x-3\right )\,\sqrt {x^6-x^2+x}}{x^{10}-x^8-3\,x^6+2\,x^5+x^4-x^3+x^2-2\,x+1} \,d x \]
int(((2*x + 2*x^5 - 3)*(x - x^2 + x^6)^(1/2))/(x^2 - 2*x - x^3 + x^4 + 2*x ^5 - 3*x^6 - x^8 + x^10 + 1),x)