\(\int \frac {(-1+x^2) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx\) [3130]

Optimal result

Integrand size = 44, antiderivative size = 773 \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {c_3} c_4-\sqrt {c_1} c_5}\right ) (-c_1 c_2+c_0 c_3) c_5 \sqrt {\sqrt {c_3} \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}}{2 \sqrt {c_1} c_3{}^{3/2} \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}-\frac {\arctan \left (\frac {\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {c_3} c_4+\sqrt {c_1} c_5}\right ) (-c_1 c_2+c_0 c_3) c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}}{2 \sqrt {c_1} c_3{}^{3/2} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}-\frac {(c_1 c_2-c_0 c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3 \left (-c_1+\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right )}+\left (-c_1 c_2 c_5{}^2+c_0 c_3 c_5{}^2\right ) \text {RootSum}\left [c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-4 c_2{}^2 c_4{}^3 \text {$\#$1}^2-4 c_3{}^2 c_4{}^3 \text {$\#$1}^2+4 c_0 c_2 c_4 c_5{}^2 \text {$\#$1}^2+4 c_1 c_3 c_4 c_5{}^2 \text {$\#$1}^2+6 c_2{}^2 c_4{}^2 \text {$\#$1}^4+6 c_3{}^2 c_4{}^2 \text {$\#$1}^4-2 c_0 c_2 c_5{}^2 \text {$\#$1}^4-2 c_1 c_3 c_5{}^2 \text {$\#$1}^4-4 c_2{}^2 c_4 \text {$\#$1}^6-4 c_3{}^2 c_4 \text {$\#$1}^6+c_2{}^2 \text {$\#$1}^8+c_3{}^2 \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}-\text {$\#$1}\right ) \text {$\#$1}}{c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2-2 c_2{}^2 c_4 \text {$\#$1}^2-2 c_3{}^2 c_4 \text {$\#$1}^2+c_2{}^2 \text {$\#$1}^4+c_3{}^2 \text {$\#$1}^4}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 6.52 (sec) , antiderivative size = 656, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=4 (c_1 c_2-c_0 c_3) c_5{}^2 \left (\frac {\arctan \left (\frac {\sqrt {c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5}}\right )}{8 \sqrt {c_1} c_3 c_5 \sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5}}-\frac {\arctan \left (\frac {\sqrt {c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5}}\right )}{8 \sqrt {c_1} c_3 c_5 \sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5}}-\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{4 c_3 (-c_1 c_2+c_0 c_3) c_5{}^2}-\frac {1}{4} \text {RootSum}\left [c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-4 c_2{}^2 c_4{}^3 \text {$\#$1}^2-4 c_3{}^2 c_4{}^3 \text {$\#$1}^2+4 c_0 c_2 c_4 c_5{}^2 \text {$\#$1}^2+4 c_1 c_3 c_4 c_5{}^2 \text {$\#$1}^2+6 c_2{}^2 c_4{}^2 \text {$\#$1}^4+6 c_3{}^2 c_4{}^2 \text {$\#$1}^4-2 c_0 c_2 c_5{}^2 \text {$\#$1}^4-2 c_1 c_3 c_5{}^2 \text {$\#$1}^4-4 c_2{}^2 c_4 \text {$\#$1}^6-4 c_3{}^2 c_4 \text {$\#$1}^6+c_2{}^2 \text {$\#$1}^8+c_3{}^2 \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}-\text {$\#$1}\right ) \text {$\#$1}}{c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2-2 c_2{}^2 c_4 \text {$\#$1}^2-2 c_3{}^2 c_4 \text {$\#$1}^2+c_2{}^2 \text {$\#$1}^4+c_3{}^2 \text {$\#$1}^4}\&\right ]\right ) \] Input:

Integrate[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[ 
5]])/(1 + x^2),x]
 

Output:

4*(C[1]*C[2] - C[0]*C[3])*C[5]^2*(ArcTan[(Sqrt[C[3]]*Sqrt[C[4] + Sqrt[(C[0 
] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[ 
3]]*C[5]]]/(8*Sqrt[C[1]]*C[3]*C[5]*Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3 
]]*C[5]]) - ArcTan[(Sqrt[C[3]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x* 
C[3])]*C[5]])/Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]]/(8*Sqrt[C[1 
]]*C[3]*C[5]*Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]) - ((C[2] + x 
*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(4*C[3]*(- 
(C[1]*C[2]) + C[0]*C[3])*C[5]^2) - RootSum[C[2]^2*C[4]^4 + C[3]^2*C[4]^4 - 
 2*C[0]*C[2]*C[4]^2*C[5]^2 - 2*C[1]*C[3]*C[4]^2*C[5]^2 + C[0]^2*C[5]^4 + C 
[1]^2*C[5]^4 - 4*C[2]^2*C[4]^3*#1^2 - 4*C[3]^2*C[4]^3*#1^2 + 4*C[0]*C[2]*C 
[4]*C[5]^2*#1^2 + 4*C[1]*C[3]*C[4]*C[5]^2*#1^2 + 6*C[2]^2*C[4]^2*#1^4 + 6* 
C[3]^2*C[4]^2*#1^4 - 2*C[0]*C[2]*C[5]^2*#1^4 - 2*C[1]*C[3]*C[5]^2*#1^4 - 4 
*C[2]^2*C[4]*#1^6 - 4*C[3]^2*C[4]*#1^6 + C[2]^2*#1^8 + C[3]^2*#1^8 & , (Lo 
g[Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]] - #1]*#1)/(C[2]^ 
2*C[4]^2 + C[3]^2*C[4]^2 - C[0]*C[2]*C[5]^2 - C[1]*C[3]*C[5]^2 - 2*C[2]^2* 
C[4]*#1^2 - 2*C[3]^2*C[4]*#1^2 + C[2]^2*#1^4 + C[3]^2*#1^4) & ]/4)
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{x^2+1} \, dx\)

\(\Big \downarrow \) 7268

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \int -\frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (1-\frac {\left (c_0-\frac {(c_0+x c_1) c_2}{c_2+x c_3}\right ){}^2}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2}\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (\frac {\left (c_0-\frac {(c_0+x c_1) c_2}{c_2+x c_3}\right ){}^2}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2}+1\right )}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 (c_1 c_2-c_0 c_3) \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (1-\frac {\left (c_0-\frac {(c_0+x c_1) c_2}{c_2+x c_3}\right ){}^2}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2}\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (\frac {\left (c_0-\frac {(c_0+x c_1) c_2}{c_2+x c_3}\right ){}^2}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2}+1\right )}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\)

\(\Big \downarrow \) 7267

\(\displaystyle -\frac {4 (c_1 c_2-c_0 c_3) \int \frac {(c_0+x c_1) \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) \left (1-\frac {\left (c_0-\frac {c_2 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right )}{(c_2+x c_3) \left (\frac {\left (c_0-\frac {c_2 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}+1\right ) \left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {4 (c_1 c_2-c_0 c_3) \int \frac {(c_0+x c_1) \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) \left (1-\frac {\left (c_0-\frac {c_2 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right )}{(c_2+x c_3) \left (\frac {\left (c_0-\frac {c_2 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2}+1\right ) \left (-\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_3 c_4 (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1-\frac {c_3 c_4{}^2}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 (c_1 c_2-c_0 c_3) \int \left (-\frac {\left (-c_3 c_4{}^2+\frac {(c_0+x c_1) c_3 c_4}{c_2+x c_3}+c_1 c_5{}^2\right ) c_5{}^4}{c_3 \left (\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}-\frac {c_5{}^4}{c_3 \left (\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {2 (c_0+x c_1) \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) c_5{}^4}{(c_2+x c_3) \left (\frac {c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) (c_0+x c_1){}^4}{(c_2+x c_3){}^4}-\frac {4 c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) c_4 (c_0+x c_1){}^3}{(c_2+x c_3){}^3}+\frac {6 c_2{}^2 c_4{}^2 \left (\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}+1\right ) (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {4 c_2{}^2 c_4{}^3 \left (\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}+1\right ) (c_0+x c_1)}{c_2+x c_3}+c_2{}^2 c_4{}^4 \left (\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (c_1 c_5{}^2-2 c_3 c_4{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}+1\right )\right )}\right )d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 (c_1 c_2-c_0 c_3) \left (2 c_4 \int \frac {c_0+x c_1}{(c_2+x c_3) \left (-\frac {c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) (c_0+x c_1){}^4}{(c_2+x c_3){}^4}+\frac {4 c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) c_4 (c_0+x c_1){}^3}{(c_2+x c_3){}^3}-\frac {6 c_2{}^2 c_4{}^2 \left (\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}+1\right ) (c_0+x c_1){}^2}{(c_2+x c_3){}^2}+\frac {4 c_2{}^2 c_4{}^3 \left (\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}+1\right ) (c_0+x c_1)}{c_2+x c_3}-c_2{}^2 c_4{}^4 \left (\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (c_1 c_5{}^2-2 c_3 c_4{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}+1\right )\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_5{}^4+2 \int \frac {(c_0+x c_1){}^2}{(c_2+x c_3){}^2 \left (\frac {c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) (c_0+x c_1){}^4}{(c_2+x c_3){}^4}-\frac {4 c_2{}^2 \left (\frac {c_3{}^2}{c_2{}^2}+1\right ) c_4 (c_0+x c_1){}^3}{(c_2+x c_3){}^3}+\frac {6 c_2{}^2 c_4{}^2 \left (\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}+1\right ) (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {4 c_2{}^2 c_4{}^3 \left (\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}+1\right ) (c_0+x c_1)}{c_2+x c_3}+c_2{}^2 c_4{}^4 \left (\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (c_1 c_5{}^2-2 c_3 c_4{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}+1\right )\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_5{}^4+\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_5{}^4}{4 c_3 \left (\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) c_5{}^3}{8 \sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) c_5{}^3}{8 \sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{c_5{}^2}\)

Input:

Int[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/( 
1 + x^2),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.04

\[\int \frac {\left (x^{2}-1\right ) \sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}{x^{2}+1}d x\]

Input:

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)
 

Output:

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=\text {Timed out} \] Input:

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1), 
x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=\text {Timed out} \] Input:

integrate((x**2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2)/(x**2 
+1),x)
 

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \text {Unable to generate Latex} \] Input:

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1), 
x, algorithm="maxima")
 

Output:

integrate((x^2 - 1)*sqrt(_C5*sqrt((_C1*x + _C0)/(_C3*x + _C2)) + _C4)/(x^2 
 + 1), x)
 

Giac [N/A]

Not integrable

Time = 47.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \text {Unable to generate Latex} \] Input:

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1), 
x, algorithm="giac")
 

Output:

integrate((x^2 - 1)*sqrt(_C5*sqrt((_C1*x + _C0)/(_C3*x + _C2)) + _C4)/(x^2 
 + 1), x)
 

Mupad [N/A]

Not integrable

Time = 10.99 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}}{x^2+1} \,d x \] Input:

int(((x^2 - 1)*(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2))/(x^2 
 + 1),x)
 

Output:

int(((x^2 - 1)*(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2))/(x^2 
 + 1), x)
 

Reduce [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx=\int \frac {\left (x^{2}-1\right ) \sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}{x^{2}+1}d x \] Input:

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)
 

Output:

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)