3.32.40 \(\int \frac {c_6+x c_7}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx\) [3140]

3.32.40.1 Optimal result

Integrand size = 39, antiderivative size = 1178 \[ \int \frac {c_6+x c_7}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx=-\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 ) \left (-8 c_1{}^2 c_2 c_3{}^{3/2} c_4 c_5 c_6+8 c_0 c_1 c_3{}^{5/2} c_4 c_5 c_6-8 c_1{}^{5/2} c_2 c_3 c_5{}^2 c_6+8 c_0 c_1{}^{3/2} c_3{}^2 c_5{}^2 c_6+6 c_1{}^2 c_2{}^2 \sqrt {c_3} c_4 c_5 c_7-4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5 c_7-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7+3 c_1{}^{5/2} c_2{}^2 c_5{}^2 c_7+2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2 c_7-5 c_0{}^2 \sqrt {c_1} c_3{}^2 c_5{}^2 c_7\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{5/2}}-\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 {\sqrt {c_3} c_4-\sqrt {c_1} c_5} \left (-8 c_1{}^2 c_2 c_3{}^{3/2} c_4 c_5 c_6+8 c_0 c_1 c_3{}^{5/2} c_4 c_5 c_6+8 c_1{}^{5/2} c_2 c_3 c_5{}^2 c_6-8 c_0 c_1{}^{3/2} c_3{}^2 c_5{}^2 c_6+6 c_1{}^2 c_2{}^2 \sqrt {c_3} c_4 c_5 c_7-4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5 c_7-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7-3 c_1{}^{5/2} c_2{}^2 c_5{}^2 c_7-2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2 c_7+5 c_0{}^2 \sqrt {c_1} c_3{}^2 c_5{}^2 c_7\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^3}+\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (4 c_2 c_3{}^2 c_4{}^3 c_6+4 x c_3{}^3 c_4{}^3 c_6-4 c_1 c_2 c_3 c_4 c_5{}^2 c_6-4 x c_1 c_3{}^2 c_4 c_5{}^2 c_6-2 c_2{}^2 c_3 c_4{}^3 c_7+2 x^2 c_3{}^3 c_4{}^3 c_7-c_1 c_2{}^2 c_4 c_5{}^2 c_7+3 c_0 c_2 c_3 c_4 c_5{}^2 c_7-3 x c_1 c_2 c_3 c_4 c_5{}^2 c_7+3 x c_0 c_3{}^2 c_4 c_5{}^2 c_7-2 x^2 c_1 c_3{}^2 c_4 c_5{}^2 c_7\right )}{4 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\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 (-8 c_1 c_2 c_3{}^2 c_4{}^2 c_5 c_6-8 x c_1 c_3{}^3 c_4{}^2 c_5 c_6+8 c_1{}^2 c_2 c_3 c_5{}^3 c_6+8 x c_1{}^2 c_3{}^2 c_5{}^3 c_6+5 c_1 c_2{}^2 c_3 c_4{}^2 c_5 c_7-c_0 c_2 c_3{}^2 c_4{}^2 c_5 c_7+x c_1 c_2 c_3{}^2 c_4{}^2 c_5 c_7-x c_0 c_3{}^3 c_4{}^2 c_5 c_7-4 x^2 c_1 c_3{}^3 c_4{}^2 c_5 c_7+c_1{}^2 c_2{}^2 c_5{}^3 c_7-5 c_0 c_1 c_2 c_3 c_5{}^3 c_7+5 x c_1{}^2 c_2 c_3 c_5{}^3 c_7-5 x c_0 c_1 c_3{}^2 c_5{}^3 c_7+4 x^2 c_1{}^2 c_3{}^2 c_5{}^3 c_7\right )}{8 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2} \]

output

-1/16*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))*(-8*_C1^2*_C2*_C3^(3/2)*_C4*_C5*_C6+8*_ 
C0*_C1*_C3^(5/2)*_C4*_C5*_C6-8*_C1^(5/2)*_C2*_C3*_C5^2*_C6+8*_C0*_C1^(3/2) 
*_C3^2*_C5^2*_C6+6*_C1^2*_C2^2*_C3^(1/2)*_C4*_C5*_C7-4*_C0*_C1*_C2*_C3^(3/ 
2)*_C4*_C5*_C7-2*_C0^2*_C3^(5/2)*_C4*_C5*_C7+3*_C1^(5/2)*_C2^2*_C5^2*_C7+2 
*_C0*_C1^(3/2)*_C2*_C3*_C5^2*_C7-5*_C0^2*_C1^(1/2)*_C3^2*_C5^2*_C7)/_C1^(3 
/2)/_C3^(7/4)/(_C3^(1/2)*_C4+_C1^(1/2)*_C5)^(5/2)-1/16*arctanh(_C3^(1/4)*( 
_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C3^(1/2)*_C4-_C1^(1/2)*_C 
5)^(1/2))*(_C3^(1/2)*_C4-_C1^(1/2)*_C5)^(1/2)*(-8*_C1^2*_C2*_C3^(3/2)*_C4* 
_C5*_C6+8*_C0*_C1*_C3^(5/2)*_C4*_C5*_C6+8*_C1^(5/2)*_C2*_C3*_C5^2*_C6-8*_C 
0*_C1^(3/2)*_C3^2*_C5^2*_C6+6*_C1^2*_C2^2*_C3^(1/2)*_C4*_C5*_C7-4*_C0*_C1* 
_C2*_C3^(3/2)*_C4*_C5*_C7-2*_C0^2*_C3^(5/2)*_C4*_C5*_C7-3*_C1^(5/2)*_C2^2* 
_C5^2*_C7-2*_C0*_C1^(3/2)*_C2*_C3*_C5^2*_C7+5*_C0^2*_C1^(1/2)*_C3^2*_C5^2* 
_C7)/_C1^(3/2)/_C3^(7/4)/(-_C3^(1/2)*_C4+_C1^(1/2)*_C5)^3+1/4*(_C4+((_C1*x 
+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-2*_C1*_C3^2*_C4*_C5^2*_C7*x^2+2*_C3^ 
3*_C4^3*_C7*x^2+3*_C0*_C3^2*_C4*_C5^2*_C7*x-3*_C1*_C2*_C3*_C4*_C5^2*_C7*x- 
4*_C1*_C3^2*_C4*_C5^2*_C6*x+4*_C3^3*_C4^3*_C6*x+3*_C0*_C2*_C3*_C4*_C5^2*_C 
7-_C1*_C2^2*_C4*_C5^2*_C7-4*_C1*_C2*_C3*_C4*_C5^2*_C6-2*_C2^2*_C3*_C4^3*_C 
7+4*_C2*_C3^2*_C4^3*_C6)/_C3/(-_C1*_C5^2+_C3*_C4^2)^2+1/8*((_C1*x+_C0)/(_C 
3*x+_C2))^(1/2)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(4*_C1^...
 

3.32.40.2 Mathematica [A] (verified)

Time = 6.26 (sec) , antiderivative size = 1140, normalized size of antiderivative = 0.97 \[ \int \frac {c_6+x c_7}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx=-\frac {1}{16} (c_1 c_2-c_0 c_3) \left (\frac {8 (c_2+x c_3){}^2 \left (c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ){}^{5/2} \left (c_3 c_4{}^2 \left (c_4-3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )+c_1 c_5{}^2 \left (3 c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7}{(c_1 c_2-c_0 c_3) \left (-c_3 c_4{}^2+c_1 c_5{}^2\right ){}^3}+\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 ) c_5 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ) \left (-2 c_0 c_3{}^{3/2} c_4 c_7-5 c_0 \sqrt {c_1} c_3 c_5 c_7+2 c_1 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7)+c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)\right )}{c_1{}^{3/2} c_3{}^{3/2} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 \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 ) c_5 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) \left (2 c_0 c_3{}^{3/2} c_4 c_7-5 c_0 \sqrt {c_1} c_3 c_5 c_7+c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)+2 c_1 \sqrt {c_3} c_4 (-4 c_3 c_6+3 c_2 c_7)\right )}{c_1{}^{3/2} c_3{}^{3/2} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 \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 {2 (c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_0 c_3{}^3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_4{}^4 c_5 c_7+c_1{}^3 c_5{}^4 \left (-8 c_3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6+c_2 \left (-2 c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_7\right )-c_1 c_3{}^2 c_4{}^2 \left (8 c_3 c_4{}^2 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6+\left (2 c_0 c_5{}^2 \left (13 c_4+4 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )+c_2 c_4{}^2 \left (-8 c_4+9 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7\right )+c_1{}^2 c_3 c_5{}^2 \left (16 c_3 c_4{}^2 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6+\left (c_0 c_5{}^2 \left (10 c_4-9 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )+2 c_2 c_4{}^2 \left (5 c_4+12 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7\right )\right )}{c_1 c_3 (c_1 c_2-c_0 c_3) \left (-c_3 c_4{}^2+c_1 c_5{}^2\right ){}^3}\right ) \]

input

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

output

-1/16*((C[1]*C[2] - C[0]*C[3])*((8*(C[2] + x*C[3])^2*(C[4] + Sqrt[(C[0] + 
x*C[1])/(C[2] + x*C[3])]*C[5])^(5/2)*(C[3]*C[4]^2*(C[4] - 3*Sqrt[(C[0] + x 
*C[1])/(C[2] + x*C[3])]*C[5]) + C[1]*C[5]^2*(3*C[4] - Sqrt[(C[0] + x*C[1]) 
/(C[2] + x*C[3])]*C[5]))*C[7])/((C[1]*C[2] - C[0]*C[3])*(-(C[3]*C[4]^2) + 
C[1]*C[5]^2)^3) + (ArcTan[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[(C[0] + x*C[1])/(C[2] + 
x*C[3])]*C[5]])]*C[5]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])*(-2*C[0]*C[3]^(3 
/2)*C[4]*C[7] - 5*C[0]*Sqrt[C[1]]*C[3]*C[5]*C[7] + 2*C[1]*Sqrt[C[3]]*C[4]* 
(4*C[3]*C[6] - 3*C[2]*C[7]) + C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C[2]*C[7])) 
)/(C[1]^(3/2)*C[3]^(3/2)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^2*Sqrt[-(C[3] 
*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[3]*C[4]^2 - C[1]*C[5]^2]) + (A 
rcTan[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[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])]*C[5 
]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])*(2*C[0]*C[3]^(3/2)*C[4]*C[7] - 5*C[0 
]*Sqrt[C[1]]*C[3]*C[5]*C[7] + C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C[2]*C[7]) 
+ 2*C[1]*Sqrt[C[3]]*C[4]*(-4*C[3]*C[6] + 3*C[2]*C[7])))/(C[1]^(3/2)*C[3]^( 
3/2)*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])^2*Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]* 
Sqrt[C[3]]*C[5]]*Sqrt[C[3]*C[4]^2 - C[1]*C[5]^2]) - (2*(C[2] + x*C[3])*Sqr 
t[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[0]*C[3]^3*Sqrt[(C[ 
0] + x*C[1])/(C[2] + x*C[3])]*C[4]^4*C[5]*C[7] + C[1]^3*C[5]^4*(-8*C[3]...
 

3.32.40.3 Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 856, normalized size of antiderivative = 0.73, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {7268, 2180, 27, 686, 27, 654, 25, 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.

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

\(\Big \downarrow \) 7268

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (c_1 c_6-c_0 c_7-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\)

\(\Big \downarrow \) 2180

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (\frac {\int -\frac {c_1 \left (\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_4 c_5 c_7-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (5 \left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 c_5{}^2+8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_6-\frac {c_2 c_7}{c_3}\right )\right )\right )}{2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{4 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\int \frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_4 c_5 c_7-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \left (5 \left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 c_5{}^2+8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_6-\frac {c_2 c_7}{c_3}\right )\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 686

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\frac {\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right )}-\frac {\int -\frac {c_5 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )+2 c_3 c_4 \left (c_0 c_3 c_7 c_4{}^2+4 c_1{}^2 c_5{}^2 c_6-c_1 \left (4 c_3 c_4{}^2 c_6-\left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\frac {c_5 \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )+2 c_3 c_4 \left (c_0 c_3 c_7 c_4{}^2+4 c_1{}^2 c_5{}^2 c_6-c_1 \left (4 c_3 c_4{}^2 c_6-\left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_7\right )\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{4 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 654

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\frac {c_5 \int -\frac {c_5 \left (c_4 \left (c_0 c_3{}^2 c_7 c_4{}^2+c_1{}^2 c_5{}^2 (16 c_3 c_6-3 c_2 c_7)-c_1 c_3 \left (16 c_3 c_4{}^2 c_6-\left (15 c_2 c_4{}^2-13 c_0 c_5{}^2\right ) c_7\right )\right )+\frac {(c_0+x c_1) \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )}{c_2+x c_3}\right )}{\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\frac {\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right )}-\frac {c_5 \int \frac {c_5 \left (c_4 \left (c_0 c_3{}^2 c_7 c_4{}^2+c_1{}^2 c_5{}^2 (16 c_3 c_6-3 c_2 c_7)-c_1 c_3 \left (16 c_3 c_4{}^2 c_6-\left (15 c_2 c_4{}^2-13 c_0 c_5{}^2\right ) c_7\right )\right )+\frac {(c_0+x c_1) \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )}{c_2+x c_3}\right )}{\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\frac {\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right )}-\frac {c_5{}^2 \int \frac {c_4 \left (c_0 c_3{}^2 c_7 c_4{}^2+c_1{}^2 c_5{}^2 (16 c_3 c_6-3 c_2 c_7)-c_1 c_3 \left (16 c_3 c_4{}^2 c_6-\left (15 c_2 c_4{}^2-13 c_0 c_5{}^2\right ) c_7\right )\right )+\frac {(c_0+x c_1) \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )}{c_2+x c_3}}{\frac {c_3 (c_0+x c_1){}^2}{(c_2+x c_3){}^2}-\frac {2 c_3 c_4 (c_0+x c_1)}{c_2+x c_3}+c_3 c_4{}^2-c_1 c_5{}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) c_7 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7}{4 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {c_5{}^2 \left (\frac {\left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 \left (-c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1+5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right ) \int \frac {1}{\frac {(c_0+x c_1) c_3}{c_2+x c_3}-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{2 \sqrt {c_1} c_5}-\frac {\left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 \left (c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1-5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right ) \int \frac {1}{\frac {(c_0+x c_1) c_3}{c_2+x c_3}-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{2 \sqrt {c_1} c_5}\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle 2 (c_1 c_2-c_0 c_3) \left (-\frac {\left (c_0-\frac {c_1 c_2}{c_3}\right ) \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7}{4 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_7 c_4{}^2-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{2 c_1 c_3 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {c_5{}^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 \left (c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1-5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right )}{2 \sqrt {c_1} c_3{}^{3/4} c_5 \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 \left (-c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1+5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right )}{2 \sqrt {c_1} c_3{}^{3/4} c_5 \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{2 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}}{8 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\right )\)

input

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

output

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

3.32.40.3.1 Defintions of rubi rules used

rule 25

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

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 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 2180

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) 
 + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 
 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* 
(p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p 
+ 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e 
, m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] &&  !(IGtQ[ 
m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

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]]
 

3.32.40.4 Maple [F]

\[\int \frac {\textit {\_C7} x +\textit {\_C6}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}d x\]

input

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

output

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

3.32.40.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.32.40.6 Sympy [F]

\[ \int \frac {c_6+x c_7}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx=\int \frac {_C6 + _C7 x}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}}}\, dx \]

input

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

output

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

3.32.40.7 Maxima [F]

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

input

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

output

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

3.32.40.8 Giac [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const ind 
ex_m & i,con
 

3.32.40.9 Mupad [F(-1)]

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

input

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

output

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