3.32.50 \(\int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx\) [3150]

3.32.50.1 Optimal result

Integrand size = 48, antiderivative size = 1716 \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx =\text {Too large to display} \]

output

2*arctanh(_C3^(1/4)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C3^( 
1/2)*_C4-_C1^(1/2)*_C5)^(1/2))*_C3^(1/4)*_C8/(_C3^(1/2)*_C4-_C1^(1/2)*_C5) 
^(1/2)/_C7+2*arctanh(_C3^(1/4)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^( 
1/2)/(_C3^(1/2)*_C4+_C1^(1/2)*_C5)^(1/2))*_C3^(1/4)*_C8/(_C3^(1/2)*_C4+_C1 
^(1/2)*_C5)^(1/2)/_C7+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))*(-_C2*_C7+_C3*_C6)^(1/2)*_C8/_ 
C7/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C 
6)^(1/2))^(1/2)+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))*(-_C2*_C7+_C3*_C6)^(1/2)*_C8/_C7/(-_ 
C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/ 
2))^(1/2)-2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(_C2*_C 
7-_C3*_C6)^(1/2)/(_C3*_C4*_C6-_C2*_C4*_C7-I*_C5*(-_C0*_C7+_C1*_C6)^(1/2)*( 
_C2*_C7-_C3*_C6)^(1/2))^(1/2))*_C6*(_C2*_C7-_C3*_C6)^(1/2)*_C9/_C7^2/(_C3* 
_C4*_C6-_C2*_C4*_C7-I*_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(_C2*_C7-_C3*_C6)^(1/2) 
)^(1/2)-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+I*_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(_C 
2*_C7-_C3*_C6)^(1/2))^(1/2))*_C6*(_C2*_C7-_C3*_C6)^(1/2)*_C9/_C7^2/(_C3*_C 
4*_C6-_C2*_C4*_C7+I*_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(_C2*_C7-_C3*_C6)^(1/2...
 

3.32.50.2 Mathematica [A] (verified)

Time = 10.87 (sec) , antiderivative size = 1337, normalized size of antiderivative = 0.78 \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx=-\frac {(c_1 c_2-c_0 c_3) \left (\frac {2 (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_7 c_9}{(c_1 c_2-c_0 c_3) \left (-c_3 c_4{}^2+c_1 c_5{}^2\right )}+\frac {4 \arctan \left (\frac {\sqrt {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}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \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 ) \sqrt {c_3 c_6-c_2 c_7} \left (c_1 c_5 c_6-c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right ) (-c_7 c_8+c_6 c_9)}{(c_1 c_2-c_0 c_3) \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6+\left (-c_2 c_4{}^2+c_0 c_5{}^2\right ) 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}}}+\frac {4 \arctan \left (\frac {\sqrt {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}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \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 ) \sqrt {c_3 c_6-c_2 c_7} \left (-c_1 c_5 c_6+c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right ) (-c_7 c_8+c_6 c_9)}{(c_1 c_2-c_0 c_3) \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6+\left (-c_2 c_4{}^2+c_0 c_5{}^2\right ) 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}}}-\frac {\arctan \left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\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}}\right ) \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) \left (c_0 c_3 c_5 c_7 c_9+4 \sqrt {c_1} c_3{}^{3/2} c_4 (c_7 c_8-c_6 c_9)-c_1 c_5 (c_2 c_7 c_9+4 c_3 (c_7 c_8-c_6 c_9))\right )}{\sqrt {c_1} \sqrt {c_3} (c_1 c_2-c_0 c_3) \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) \sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}-\frac {\arctan \left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\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}}\right ) \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ) \left (-c_0 c_3 c_5 c_7 c_9+4 \sqrt {c_1} c_3{}^{3/2} c_4 (c_7 c_8-c_6 c_9)+c_1 c_5 (c_2 c_7 c_9+4 c_3 (c_7 c_8-c_6 c_9))\right )}{\sqrt {c_1} \sqrt {c_3} (-c_1 c_2+c_0 c_3) \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) \sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}\right )}{2 c_7{}^2} \]

input

Integrate[(C[8] + x*C[9])/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3] 
)]*C[5]]*(C[6] + x*C[7])),x]
 

output

-1/2*((C[1]*C[2] - C[0]*C[3])*((2*(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[7]*C[9])/((C[1]*C[2] - C[0]*C[3])*(-(C[3]*C[4]^2) + C[1]*C[5]^ 
2)) + (4*ArcTan[Sqrt[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]]/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C 
[5]]*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]]])]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(C[1]*C[5 
]*C[6] - C[0]*C[5]*C[7] + C[4]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] 
- C[2]*C[7]])*(-(C[7]*C[8]) + C[6]*C[9]))/((C[1]*C[2] - C[0]*C[3])*Sqrt[C[ 
1]*C[6] - C[0]*C[7]]*Sqrt[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]]*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*ArcTan[Sqrt 
[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] 
]/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*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]]])]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[1]*C[5]*C[6]) + C[0]*C[5 
]*C[7] + C[4]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]])*(-( 
C[7]*C[8]) + C[6]*C[9]))/((C[1]*C[2] - C[0]*C[3])*Sqrt[C[1]*C[6] - C[0]*C[ 
7]]*Sqrt[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]]*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6]...
 

3.32.50.3 Rubi [A] (warning: unable to verify)

Time = 21.37 (sec) , antiderivative size = 1077, normalized size of antiderivative = 0.63, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {7268, 7267, 7292, 7279, 2009}

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.

\(\displaystyle \int \frac {c_9 x+c_8}{\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} (c_7 x+c_6)} \, dx\)

\(\Big \downarrow \) 7268

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (c_1 c_8-c_0 c_9-\frac {(c_0+x c_1) (c_3 c_8-c_2 c_9)}{c_2+x c_3}\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_1 c_6-c_0 c_7-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}\right )}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) \left (-\frac {(c_3 c_8-c_2 c_9) \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}+c_1 c_8-c_0 c_9\right )}{\left (c_1-\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}\right ){}^2 \left (-\frac {(c_3 c_6-c_2 c_7) \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ){}^2}{c_5{}^2}+c_1 c_6-c_0 c_7\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) \left (-\frac {(c_3 c_8-c_2 c_9) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_8-c_2 c_9) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_8-\frac {c_3 c_4{}^2 c_8}{c_5{}^2}-c_0 c_9+\frac {c_2 c_4{}^2 c_9}{c_5{}^2}\right )}{\left (-\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_3 c_4 (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1-\frac {c_3 c_4{}^2}{c_5{}^2}\right ){}^2 \left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-\frac {c_3 c_4{}^2 c_6}{c_5{}^2}-c_0 c_7+\frac {c_2 c_4{}^2 c_7}{c_5{}^2}\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 7279

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \left (\frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) c_9 c_5{}^4}{\left (\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2 c_7}+\frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) (c_3 c_6-c_2 c_7) (c_7 c_8-c_6 c_9) c_5{}^2}{(c_1 c_2-c_0 c_3) c_7{}^2 \left (\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}+\frac {c_3 \left (\frac {c_0+x c_1}{c_2+x c_3}-c_4\right ) (c_7 c_8-c_6 c_9) c_5{}^2}{(c_1 c_2-c_0 c_3) \left (-\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}+\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}-c_3 c_4{}^2+c_1 c_5{}^2\right ) c_7{}^2}\right )d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (\frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-2 c_4\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_9 c_5{}^4}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \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 ) c_7}-\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_9 c_5{}^3}{8 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{3/2} c_7}+\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_9 c_5{}^3}{8 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{3/2} 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 {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 ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9) c_5{}^2}{2 (c_1 c_2-c_0 c_3) c_7{}^2 \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 {\text {arctanh}\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 {\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 ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9) c_5{}^2}{2 (c_1 c_2-c_0 c_3) c_7{}^2 \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 {\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 ) \sqrt [4]{c_3} (c_7 c_8-c_6 c_9) c_5{}^2}{2 (c_1 c_2-c_0 c_3) \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7{}^2}+\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 ) \sqrt [4]{c_3} (c_7 c_8-c_6 c_9) c_5{}^2}{2 (c_1 c_2-c_0 c_3) \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7{}^2}\right )}{c_5{}^2}\)

input

Int[(C[8] + x*C[9])/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5 
]]*(C[6] + x*C[7])),x]
 

output

(4*(C[1]*C[2] - C[0]*C[3])*(-1/8*(ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[ 
0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5 
]]]*C[5]^3*C[9])/(Sqrt[C[1]]*C[3]^(3/4)*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5] 
)^(3/2)*C[7]) + (ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2 
] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]]]*C[5]^3*C[9])/ 
(8*Sqrt[C[1]]*C[3]^(3/4)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^(3/2)*C[7]) + 
 (((C[0] + x*C[1])/(C[2] + x*C[3]) - 2*C[4])*C[5]^4*Sqrt[C[4] + Sqrt[(C[0] 
 + x*C[1])/(C[2] + x*C[3])]*C[5]]*C[9])/(4*(C[3]*C[4]^2 - C[1]*C[5]^2)*((( 
C[0] + x*C[1])^2*C[3])/(C[2] + x*C[3])^2 - (2*(C[0] + x*C[1])*C[3]*C[4])/( 
C[2] + x*C[3]) + C[3]*C[4]^2 - C[1]*C[5]^2)*C[7]) + (ArcTanh[(C[3]^(1/4)*S 
qrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[ 
4] - Sqrt[C[1]]*C[5]]]*C[3]^(1/4)*C[5]^2*(C[7]*C[8] - C[6]*C[9]))/(2*(C[1] 
*C[2] - C[0]*C[3])*Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]]*C[7]^2) + (ArcT 
anh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/S 
qrt[Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]]]*C[3]^(1/4)*C[5]^2*(C[7]*C[8] - C[6 
]*C[9]))/(2*(C[1]*C[2] - C[0]*C[3])*Sqrt[Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5] 
]*C[7]^2) - (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*Sqrt[C 
[3]*C[6] - C[2]*C[7]]*(C[7]*C[8] - C[6]*C[9]))/(2*(C[1]*C[2] - C[0]*C[3...
 

3.32.50.3.1 Defintions of rubi rules used

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 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

rule 7292

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

3.32.50.4 Maple [F]

\[\int \frac {\textit {\_C9} x +\textit {\_C8}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )}d x\]

input

int((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6 
),x)
 

output

int((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6 
),x)
 

3.32.50.5 Fricas [F(-1)]

Timed out. \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx=\text {Timed out} \]

input

integrate((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7 
*x+_C6),x, algorithm="fricas")
 

output

Timed out
 

3.32.50.6 Sympy [F]

\[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx=\int \frac {_C8 + _C9 x}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}} \left (_C6 + _C7 x\right )}\, dx \]

input

integrate((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2)/(_ 
C7*x+_C6),x)
 

output

Integral((_C8 + _C9*x)/(sqrt(_C4 + _C5*sqrt(_C0/(_C2 + _C3*x) + _C1*x/(_C2 
 + _C3*x)))*(_C6 + _C7*x)), x)
 

3.32.50.7 Maxima [F]

\[ \text {Unable to display latex} \]

input

integrate((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7 
*x+_C6),x, algorithm="maxima")
 

output

integrate((_C9*x + _C8)/((_C7*x + _C6)*sqrt(_C5*sqrt((_C1*x + _C0)/(_C3*x 
+ _C2)) + _C4)), x)
 

3.32.50.8 Giac [F(-2)]

Exception generated. \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx=\text {Exception raised: TypeError} \]

input

integrate((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7 
*x+_C6),x, algorithm="giac")
 

output

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 Valuesym2poly/r2sym(con 
st gen &
 

3.32.50.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx=\int \frac {_{\mathrm {C8}}+_{\mathrm {C9}}\,x}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}\,\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )} \,d x \]

input

int((_C8 + _C9*x)/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*( 
_C6 + _C7*x)),x)
 

output

int((_C8 + _C9*x)/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*( 
_C6 + _C7*x)), x)