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

3.32.30.1 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
 

3.32.30.2 Mathematica [A] (verified)

Time = 6.46 (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)
 

3.32.30.3 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
 

3.32.30.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7268

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears 
[u, x]}, Simp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls 
t[[2]])], x] /;  !FalseQ[lst]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.32.30.4 Maple [N/A] (verified)

Not integrable

Time = 0.06 (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)
 

3.32.30.5 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
 

3.32.30.6 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
 

3.32.30.7 Maxima [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.05 \[ \text {Unable to display latex} \]

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)
 

3.32.30.8 Giac [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Recursive assumption sageVAR_C0>=((sageVAR_C1*sageVAR_C 
2*sageVAR_C3-sageVAR_C1*sageVAR_C3*t_nostep^2)/sageVAR_C3^2) ignoredRecurs 
ive assumpti
 

3.32.30.9 Mupad [N/A]

Not integrable

Time = 8.03 (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)