Integrand size = 57, antiderivative size = 466 \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\frac {\left (-1-a x+x^2+a x^3\right ) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{a \left (1+x^2\right )}+\frac {2 \sqrt {-(1+a)^2} \arctan \left (\frac {\frac {(1+a)^2}{\sqrt {-1-2 a-a^2} \sqrt {-1+a^2}}+\frac {(1+a)^2 x^2}{\sqrt {-1-2 a-a^2} \sqrt {-1+a^2}}}{(-1+x) (1+x) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right )}{\sqrt {-1+a^2}}-\frac {2 \sqrt {-(-1+a)^2} \arctan \left (\frac {\frac {(-1+a)^2}{\sqrt {-1+2 a-a^2} \sqrt {-1+a^2}}+\frac {(-1+a)^2 x^2}{\sqrt {-1+2 a-a^2} \sqrt {-1+a^2}}}{(-1+x) (1+x) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right )}{\sqrt {-1+a^2}}-\frac {2 \text {arctanh}\left (\frac {(-1+x) (1+x) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{1+x^2}\right )}{a} \]
(a*x^3-a*x+x^2-1)*((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x- 2*x^2+1))^(1/2)/a/(x^2+1)+2*(-(1+a)^2)^(1/2)*arctan(((1+a)^2/(-(1+a)^2)^(1 /2)/(a^2-1)^(1/2)+(1+a)^2*x^2/(-(1+a)^2)^(1/2)/(a^2-1)^(1/2))/(-1+x)/(1+x) /((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2))/ (a^2-1)^(1/2)-2*(-(a-1)^2)^(1/2)*arctan(((a-1)^2/(-(a-1)^2)^(1/2)/(a^2-1)^ (1/2)+(a-1)^2*x^2/(-(a-1)^2)^(1/2)/(a^2-1)^(1/2))/(-1+x)/(1+x)/((a*x^5+2*a *x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2))/(a^2-1)^(1/2 )-2*arctanh((-1+x)*(1+x)*((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x ^4+a*x-2*x^2+1))^(1/2)/(x^2+1))/a
Time = 0.42 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.52 \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\frac {\left (-1+x^2\right ) \sqrt {\frac {(-1+a x) \left (1+x^2\right )^2}{(1+a x) \left (-1+x^2\right )^2}} \left (-2 (-1+a) a \sqrt {-1+a^2} \sqrt {1+a x} \text {arctanh}\left (\frac {a-a x+\sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {-1+a^2}}\right )+(1+a) \left (-2 a \sqrt {-1+a^2} \sqrt {1+a x} \text {arctanh}\left (\frac {-a (1+x)+\sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {-1+a^2}}\right )+(-1+a) \left (\sqrt {-1+a x} (1+a x)+2 \sqrt {1+a x} \log \left (\sqrt {-1+a x}-\sqrt {1+a x}\right )\right )\right )\right )}{(-1+a) a (1+a) \sqrt {-1+a x} \left (1+x^2\right )} \]
Integrate[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)],x]
((-1 + x^2)*Sqrt[((-1 + a*x)*(1 + x^2)^2)/((1 + a*x)*(-1 + x^2)^2)]*(-2*(- 1 + a)*a*Sqrt[-1 + a^2]*Sqrt[1 + a*x]*ArcTanh[(a - a*x + Sqrt[-1 + a*x]*Sq rt[1 + a*x])/Sqrt[-1 + a^2]] + (1 + a)*(-2*a*Sqrt[-1 + a^2]*Sqrt[1 + a*x]* ArcTanh[(-(a*(1 + x)) + Sqrt[-1 + a*x]*Sqrt[1 + a*x])/Sqrt[-1 + a^2]] + (- 1 + a)*(Sqrt[-1 + a*x]*(1 + a*x) + 2*Sqrt[1 + a*x]*Log[Sqrt[-1 + a*x] - Sq rt[1 + a*x]]))))/((-1 + a)*a*(1 + a)*Sqrt[-1 + a*x]*(1 + x^2))
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.
| \(\displaystyle \int \sqrt {\frac {a x^5+2 a x^3+a x-x^4-2 x^2-1}{a x^5-2 a x^3+a x+x^4-2 x^2+1}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \sqrt {\frac {\left (x^2+1\right )^2 (a x-1)}{\left (x^2-1\right )^2 (a x+1)}}dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sqrt {\frac {\left (x^2+1\right )^2 (a x-1)}{\left (x^2-1\right )^2 (a x+1)}}dx\) |
Int[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a *x^3 + x^4 + a*x^5)],x]
3.31.55.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.14 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {\left (x^{2}+1\right )^{2} \left (a x -1\right )}{\left (x^{2}-1\right )^{2} \left (a x +1\right )}}\, \left (x^{2}-1\right )}{a \left (x^{2}+1\right )}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {\left (1-a \right ) \ln \left (\frac {2 a^{2}-2+2 a^{2} \left (-1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (-1+x \right )^{2}+2 a^{2} \left (-1+x \right )+a^{2}-1}}{-1+x}\right )}{\sqrt {a^{2}-1}}+\frac {\left (-a -1\right ) \ln \left (\frac {2 a^{2}-2-2 a^{2} \left (1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (1+x \right )^{2}-2 a^{2} \left (1+x \right )+a^{2}-1}}{1+x}\right )}{\sqrt {a^{2}-1}}\right ) \sqrt {\frac {\left (x^{2}+1\right )^{2} \left (a x -1\right )}{\left (x^{2}-1\right )^{2} \left (a x +1\right )}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (x^{2}-1\right )}{\left (x^{2}+1\right ) \left (a x -1\right )}\) | \(281\) |
| default | \(-\frac {\sqrt {\frac {a \,x^{5}+2 a \,x^{3}-x^{4}+a x -2 x^{2}-1}{a \,x^{5}-2 a \,x^{3}+x^{4}+a x -2 x^{2}+1}}\, \left (x^{2}-1\right ) \left (a x +1\right ) \left (\sqrt {a^{2}}\, \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{-1+x}\right ) a^{2}+\sqrt {a^{2}}\, \sqrt {a^{2}-1}\, \ln \left (\frac {-2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{1+x}\right ) a^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2}-2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2}-\sqrt {a^{2}}\, \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{-1+x}\right ) a +\sqrt {a^{2}}\, \sqrt {a^{2}-1}\, \ln \left (\frac {-2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{1+x}\right ) a +2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3}-\ln \left (\frac {a^{2} x +\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{\left (x^{2}+1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (1+a \right ) \left (a -1\right ) a \sqrt {a^{2}}}\) | \(475\) |
int(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2 ),x,method=_RETURNVERBOSE)
1/a*(a*x+1)*((x^2+1)^2*(a*x-1)/(x^2-1)^2/(a*x+1))^(1/2)/(x^2+1)*(x^2-1)+(- ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)+(1-a)/(a^2-1)^(1/2)*ln ((2*a^2-2+2*a^2*(-1+x)+2*(a^2-1)^(1/2)*(a^2*(-1+x)^2+2*a^2*(-1+x)+a^2-1)^( 1/2))/(-1+x))+(-a-1)/(a^2-1)^(1/2)*ln((2*a^2-2-2*a^2*(1+x)+2*(a^2-1)^(1/2) *(a^2*(1+x)^2-2*a^2*(1+x)+a^2-1)^(1/2))/(1+x)))*((x^2+1)^2*(a*x-1)/(x^2-1) ^2/(a*x+1))^(1/2)*((a*x+1)*(a*x-1))^(1/2)/(x^2+1)/(a*x-1)*(x^2-1)
Time = 0.28 (sec) , antiderivative size = 1023, normalized size of antiderivative = 2.20 \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\left [\frac {{\left (a x^{2} + a\right )} \sqrt {\frac {a + 1}{a - 1}} \log \left (-\frac {a^{2} x^{3} + a^{2} x + x^{2} + {\left ({\left (a^{2} - a\right )} x^{3} + {\left (a - 1\right )} x^{2} - {\left (a^{2} - a\right )} x - a + 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {\frac {a + 1}{a - 1}} + 1}{x^{3} + x^{2} + x + 1}\right ) + {\left (a x^{2} + a\right )} \sqrt {\frac {a - 1}{a + 1}} \log \left (\frac {a^{2} x^{3} + a^{2} x - x^{2} - {\left ({\left (a^{2} + a\right )} x^{3} + {\left (a + 1\right )} x^{2} - {\left (a^{2} + a\right )} x - a - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {\frac {a - 1}{a + 1}} - 1}{x^{3} - x^{2} + x - 1}\right ) - {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) + {\left (x^{2} + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) + {\left (a x^{3} - a x + x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}}}{a x^{2} + a}, -\frac {2 \, {\left (a x^{2} + a\right )} \sqrt {-\frac {a + 1}{a - 1}} \arctan \left (\frac {{\left ({\left (a - 1\right )} x^{2} - a + 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {-\frac {a + 1}{a - 1}}}{{\left (a + 1\right )} x^{2} + a + 1}\right ) - 2 \, {\left (a x^{2} + a\right )} \sqrt {-\frac {a - 1}{a + 1}} \arctan \left (\frac {{\left ({\left (a + 1\right )} x^{2} - a - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {-\frac {a - 1}{a + 1}}}{{\left (a - 1\right )} x^{2} + a - 1}\right ) + {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) - {\left (a x^{3} - a x + x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}}}{a x^{2} + a}\right ] \]
integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1) )^(1/2),x, algorithm="fricas")
[((a*x^2 + a)*sqrt((a + 1)/(a - 1))*log(-(a^2*x^3 + a^2*x + x^2 + ((a^2 - a)*x^3 + (a - 1)*x^2 - (a^2 - a)*x - a + 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt((a + 1)/( a - 1)) + 1)/(x^3 + x^2 + x + 1)) + (a*x^2 + a)*sqrt((a - 1)/(a + 1))*log( (a^2*x^3 + a^2*x - x^2 - ((a^2 + a)*x^3 + (a + 1)*x^2 - (a^2 + a)*x - a - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt((a - 1)/(a + 1)) - 1)/(x^3 - x^2 + x - 1)) - (x^2 + 1)*log((x^2 + (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/( a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) + (x^2 + 1)*log( -(x^2 - (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) + (a*x^3 - a*x + x^2 - 1 )*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)))/(a*x^2 + a), -(2*(a*x^2 + a)*sqrt(-(a + 1)/(a - 1))*arc tan(((a - 1)*x^2 - a + 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/( a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt(-(a + 1)/(a - 1))/((a + 1)* x^2 + a + 1)) - 2*(a*x^2 + a)*sqrt(-(a - 1)/(a + 1))*arctan(((a + 1)*x^2 - a - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt(-(a - 1)/(a + 1))/((a - 1)*x^2 + a - 1)) + (x ^2 + 1)*log((x^2 + (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1 )/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) - (x^2 + 1...
Timed out. \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\text {Timed out} \]
integrate(((a*x**5+2*a*x**3-x**4+a*x-2*x**2-1)/(a*x**5-2*a*x**3+x**4+a*x-2 *x**2+1))**(1/2),x)
\[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\int { \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \,d x } \]
integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1) )^(1/2),x, algorithm="maxima")
integrate(sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)), x)
Exception generated. \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\text {Exception raised: TypeError} \]
integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1) )^(1/2),x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx=\int \sqrt {\frac {a\,x^5-x^4+2\,a\,x^3-2\,x^2+a\,x-1}{a\,x^5+x^4-2\,a\,x^3-2\,x^2+a\,x+1}} \,d x \]
int(((a*x + 2*a*x^3 + a*x^5 - 2*x^2 - x^4 - 1)/(a*x - 2*a*x^3 + a*x^5 - 2* x^2 + x^4 + 1))^(1/2),x)