\(\int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx\) [3137]

Optimal result

Integrand size = 41, antiderivative size = 884 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )}{\left (c_1 c_6-\frac {(c_0+x c_1) c_3 c_6}{c_2+x c_3}-c_0 c_7+\frac {(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right ) \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right )}+\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}-c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}-c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}+c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}} \] Output:

(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C1*_C2*_C4+_C0*_C3*_C4+ 
_C1*_C2*((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5-_C0*_C3*((_C1*x+_C0)/(_C3*x+_C 
2))^(1/2)*_C5)/(_C1*_C6-(_C1*x+_C0)*_C3*_C6/(_C3*x+_C2)-_C0*_C7+(_C1*x+_C0 
)*_C2*_C7/(_C3*x+_C2))/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4 
^2*_C6)+1/2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C2*_ 
C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*( 
-_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_ 
C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_C1*_C2*_C4*_C5*(-_C2*_C7+_C3*_C6)^( 
1/2)-_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1*_C6)^(1/2)/(- 
_C2*_C7+_C3*_C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4 
^2*_C6)/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_ 
C3*_C6)^(1/2))^(1/2)+1/2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^ 
(1/2)*(-_C2*_C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1 
*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1 
*_C6)^(1/2)+_C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)-_C1*_C2*_C4*_C5*(-_C2*_ 
C7+_C3*_C6)^(1/2)+_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1* 
_C6)^(1/2)/(-_C2*_C7+_C3*_C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^ 
2*_C7-_C3*_C4^2*_C6)/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1*_C6)^(1/2 
)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 821, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\frac {(c_1 c_2-c_0 c_3) \left (\frac {(c_2+x c_3) \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_1 c_6-c_0 c_7) (-c_3 c_6+c_2 c_7)}{(c_1 c_2-c_0 c_3) (c_6+x c_7)}+c_4 c_5 \sqrt {c_1 c_6-c_0 c_7} (c_3 c_6-c_2 c_7){}^{3/4} \left (-\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}-\frac {\text {arctanh}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )-\frac {1}{2} c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt [4]{c_3 c_6-c_2 c_7} \left (\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}-\text {arctanh}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}\right )\right )}{(-c_1 c_6+c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_5{}^2 (-c_1 c_6+c_0 c_7)+c_4{}^2 (c_3 c_6-c_2 c_7)\right )} \] Input:

Integrate[1/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] 
 + x*C[7])^2),x]
 

Output:

((C[1]*C[2] - C[0]*C[3])*(((C[2] + x*C[3])*(C[4] - Sqrt[(C[0] + x*C[1])/(C 
[2] + x*C[3])]*C[5])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5 
]]*(C[1]*C[6] - C[0]*C[7])*(-(C[3]*C[6]) + C[2]*C[7]))/((C[1]*C[2] - C[0]* 
C[3])*(C[6] + x*C[7])) + C[4]*C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] 
- C[2]*C[7])^(3/4)*(-(ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C 
[3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0] 
*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0 
]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - ArcTanh[(Sqrt[C[4] + Sqrt[( 
C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[ 
C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt 
[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - ( 
C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] - C[2]*C[7])^(1/4)*(ArcTan[(Sq 
rt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[ 
7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C 
[2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - 
C[2]*C[7]]] - ArcTanh[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C 
[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] 
+ C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] 
 + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]))/2))/((-(C[1]*C[6]) + C[0]*C[7])*(C[ 
3]*C[6] - C[2]*C[7])*(C[5]^2*(-(C[1]*C[6]) + C[0]*C[7]) + C[4]^2*(C[3]*...
 

Rubi [A] (warning: unable to verify)

Time = 1.55 (sec) , antiderivative size = 732, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {7268, 561, 27, 1492, 27, 27, 1480, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} (c_7 x+c_6){}^2} \, dx\)

\(\Big \downarrow \) 7268

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_1 c_6-c_0 c_7-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}\right ){}^2}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\)

\(\Big \downarrow \) 561

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-c_4}{c_5 \left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-c_4}{\left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (\frac {c_5{}^4 \int -\frac {2 (c_1 c_6-c_0 c_7) \left (\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}-2 c_4 (c_3 c_6-c_2 c_7)\right )}{c_5{}^2 \left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{8 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \int \frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-2 c_4\right ) (c_3 c_6-c_2 c_7)}{-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{4 (c_3 c_6-c_2 c_7) \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-2 c_4}{-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \left (\frac {1}{2} \left (1+\frac {\sqrt {c_3 c_6-c_2 c_7} c_4}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) \int \frac {1}{\frac {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}{c_5{}^2}-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{(c_2+x c_3) c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}+\frac {1}{2} \left (1-\frac {c_4 \sqrt {c_3 c_6-c_2 c_7}}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) \int \frac {1}{\frac {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}{c_5{}^2}-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{(c_2+x c_3) c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \left (\frac {\left (1+\frac {\sqrt {c_3 c_6-c_2 c_7} c_4}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) c_5{}^2 \text {arctanh}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{2 \sqrt {c_3 c_6-c_2 c_7} \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}+\frac {\left (1-\frac {c_4 \sqrt {c_3 c_6-c_2 c_7}}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) c_5{}^2 \text {arctanh}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{2 \sqrt {c_3 c_6-c_2 c_7} \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\)

Input:

Int[1/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C 
[7])^2),x]
 

Output:

(4*(C[1]*C[2] - C[0]*C[3])*(-1/4*(((C[0] + x*C[1])/(C[2] + x*C[3]) - 2*C[4 
])*C[5]^2*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/((C[3]* 
C[4]^2*C[6] - C[1]*C[5]^2*C[6] - (C[2]*C[4]^2 - C[0]*C[5]^2)*C[7])*(C[1]*C 
[6] - C[0]*C[7] - ((C[0] + x*C[1])^2*(C[3]*C[6] - C[2]*C[7]))/((C[2] + x*C 
[3])^2*C[5]^2) + (2*(C[0] + x*C[1])*C[4]*(C[3]*C[6] - C[2]*C[7]))/((C[2] + 
 x*C[3])*C[5]^2) - (C[4]^2*(C[3]*C[6] - C[2]*C[7]))/C[5]^2)) - (C[5]^2*((A 
rcTanh[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]* 
C[6] - C[2]*C[7]])/Sqrt[C[3]*C[4]*C[6] - C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C 
[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*C[5]^2*(1 + (C[4]*Sqrt[C[3] 
*C[6] - C[2]*C[7]])/(C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]])))/(2*Sqrt[C[3]*C[6] 
 - C[2]*C[7]]*Sqrt[C[3]*C[4]*C[6] - C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - 
 C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (ArcTanh[(Sqrt[C[4] + Sqrt[(C[ 
0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[C[3] 
*C[4]*C[6] - C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C 
[6] - C[2]*C[7]]]]*C[5]^2*(1 - (C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]])/(C[5]*Sq 
rt[C[1]*C[6] - C[0]*C[7]])))/(2*Sqrt[C[3]*C[6] - C[2]*C[7]]*Sqrt[C[3]*C[4] 
*C[6] - C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - 
 C[2]*C[7]]])))/(4*(C[3]*C[4]^2*C[6] - C[1]*C[5]^2*C[6] - (C[2]*C[4]^2 - C 
[0]*C[5]^2)*C[7]))))/C[5]^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

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]]
 

Maple [F]

\[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}d x\]

Input:

int(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2,x)
 

Output:

int(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56059 vs. \(2 (579) = 1158\).

Time = 43.59 (sec) , antiderivative size = 56059, normalized size of antiderivative = 63.42 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, 
x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Timed out} \] Input:

integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2)/(_C7*x+_C6)* 
*2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, 
x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Timed out} \] Input:

integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, 
x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\int \frac {1}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}\,{\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )}^2} \,d x \] Input:

int(1/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*(_C6 + _C7*x) 
^2),x)
 

Output:

int(1/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*(_C6 + _C7*x) 
^2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}d x \] Input:

int(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2,x)
 

Output:

int(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2,x)