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

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...
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.95 (sec) , antiderivative size = 1337, normalized size of antiderivative = 0.78 \[ \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx =\text {Too large to display} \] 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]...
 

Rubi [A] (warning: unable to verify)

Time = 22.88 (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...
 

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]
 

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)
 

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
 

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)
 

Maxima [F]

\[ \text {Unable to generate Latex} \] 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)
 

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:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.Non regula 
r value [
 

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)
 

Reduce [F]

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