\(\int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx\) [550]

Optimal result

Integrand size = 25, antiderivative size = 479 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c d^2-e (i b d-a e)}+\sqrt {c} (d+e x)}\right )}{\sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}} \] Output:

e*arctanh(((2*c*d-I*b*e+2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)-2*c^( 
1/2)*(e*x+d)^(1/2))/(2*c*d-I*b*e-2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1 
/2))/c^(1/2)/(2*c*d-I*b*e-2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)-e*a 
rctanh(((2*c*d-I*b*e+2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)+2*c^(1/2 
)*(e*x+d)^(1/2))/(2*c*d-I*b*e-2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2) 
)/c^(1/2)/(2*c*d-I*b*e-2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)-e*arct 
anh((2*c*d-I*b*e+2*c^(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)*(e*x+d)^(1/2 
)/((c*d^2-e*(I*b*d-a*e))^(1/2)+c^(1/2)*(e*x+d)))/c^(1/2)/(2*c*d-I*b*e+2*c^ 
(1/2)*(c*d^2-e*(I*b*d-a*e))^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\frac {\sqrt {2} \left (\frac {\left (-2 i c d+\left (-b+\sqrt {b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d-i \left (-b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {-2 c d-i \left (-b+\sqrt {b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (b+\sqrt {b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+i \left (b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {-2 c d+i \left (b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2+4 a c}} \] Input:

Integrate[Sqrt[d + e*x]/(a + I*b*x + c*x^2),x]
 

Output:

(Sqrt[2]*((((-2*I)*c*d + (-b + Sqrt[b^2 + 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[ 
c]*Sqrt[d + e*x])/Sqrt[-2*c*d - I*(-b + Sqrt[b^2 + 4*a*c])*e]])/Sqrt[-2*c* 
d - I*(-b + Sqrt[b^2 + 4*a*c])*e] + (((2*I)*c*d + (b + Sqrt[b^2 + 4*a*c])* 
e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + I*(b + Sqrt[b^2 + 
4*a*c])*e]])/Sqrt[-2*c*d + I*(b + Sqrt[b^2 + 4*a*c])*e]))/(Sqrt[c]*Sqrt[b^ 
2 + 4*a*c])
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.53, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1148, 1449, 1142, 25, 27, 1083, 219, 1103}

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.

\(\displaystyle \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx\)

\(\Big \downarrow \) 1148

\(\displaystyle 2 e \int \frac {d+e x}{c d^2+c (d+e x)^2-e (i b d-a e)-(2 c d-i b e) (d+e x)}d\sqrt {d+e x}\)

\(\Big \downarrow \) 1449

\(\displaystyle 2 e \left (\frac {\int \frac {\sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {\int \frac {\sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 2 e \left (\frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}+\frac {1}{2} \int -\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{\sqrt {c} \left (d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}\right )}d\sqrt {d+e x}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\frac {1}{2} \int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{\sqrt {c} \left (d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e \left (\frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}-\frac {1}{2} \int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{\sqrt {c} \left (d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}\right )}d\sqrt {d+e x}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\frac {1}{2} \int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{\sqrt {c} \left (d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \left (\frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 2 e \left (\frac {-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d-\frac {i b e}{c}-e x-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \int \frac {1}{d-\frac {i b e}{c}-e x-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}}d\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{\sqrt {c}}+\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 e \left (\frac {-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \text {arctanh}\left (\frac {\sqrt {c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \text {arctanh}\left (\frac {\sqrt {c} \left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}+\frac {\int \frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}}{\sqrt {c}}}d\sqrt {d+e x}}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 e \left (\frac {\frac {1}{2} \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )-\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d} \text {arctanh}\left (\frac {\sqrt {c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {c}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d} \text {arctanh}\left (\frac {\sqrt {c} \left (2 \sqrt {d+e x}+\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {c}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {1}{2} \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )\)

Input:

Int[Sqrt[d + e*x]/(a + I*b*x + c*x^2),x]
 

Output:

2*e*((-((Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]*Arc 
Tanh[(Sqrt[c]*(-(Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a* 
e)]]/Sqrt[c]) + 2*Sqrt[d + e*x]))/Sqrt[2*c*d - I*b*e - 2*Sqrt[c]*Sqrt[c*d^ 
2 - e*(I*b*d - a*e)]]])/Sqrt[2*c*d - I*b*e - 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b 
*d - a*e)]]) + Log[Sqrt[c*d^2 - e*(I*b*d - a*e)] - Sqrt[2*c*d - I*b*e + 2* 
Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)]/ 
2)/(2*Sqrt[c]*Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)] 
]) - ((Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]*ArcTa 
nh[(Sqrt[c]*(Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]] 
/Sqrt[c] + 2*Sqrt[d + e*x]))/Sqrt[2*c*d - I*b*e - 2*Sqrt[c]*Sqrt[c*d^2 - e 
*(I*b*d - a*e)]]])/Sqrt[2*c*d - I*b*e - 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - 
a*e)]] + Log[Sqrt[c*d^2 - e*(I*b*d - a*e)] + Sqrt[2*c*d - I*b*e + 2*Sqrt[c 
]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)]/2)/(2* 
Sqrt[c]*Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1148

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

rule 1449

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = 
Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r)   Int[x^(m - 1)/(q 
 - r*x + x^2), x], x] - Simp[1/(2*c*r)   Int[x^(m - 1)/(q + r*x + x^2), x], 
 x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 
3] && NegQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 16.03 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.38

method	result	size
pseudoelliptic	\(-\frac {e \left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}+i b e -2 c d}}\right ) \sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}+i b e -2 c d}}\right ) \sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}-\frac {\ln \left (-\sqrt {e x +d}\, \sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}+\sqrt {c}\, \left (e x +d \right )\right ) \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}+i b e -2 c d}}{2}+\frac {\ln \left (\sqrt {e x +d}\, \sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}+\sqrt {c}\, \left (e x +d \right )\right ) \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}+i b e -2 c d}}{2}\right )}{\sqrt {2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}-i b e +2 c d}\, \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c \left (-i d b e +a \,e^{2}+c \,d^{2}\right )}+i b e -2 c d}\, \sqrt {c}}\)	\(662\)
derivativedivides	\(2 e \left (\frac {\frac {\ln \left (-\sqrt {c}\, \left (e x +d \right )+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}-\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}-\frac {\frac {\ln \left (\sqrt {c}\, \left (e x +d \right )+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}\right )\)	\(692\)
default	\(2 e \left (\frac {\frac {\ln \left (-\sqrt {c}\, \left (e x +d \right )+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}-\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}-\frac {\frac {\ln \left (\sqrt {c}\, \left (e x +d \right )+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i d b e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i d b e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}\right )\)	\(692\)

Input:

int((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x,method=_RETURNVERBOSE)
 

Output:

-1/(2*(c*(-I*d*b*e+a*e^2+c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)*e/(4*(-I*d*b*e+a 
*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(-I*d*b*e+a*e^2+c*d^2))^(1/2)+I*b*e-2*c*d)^ 
(1/2)/c^(1/2)*(arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(-I*d*b*e+a*e^2+c*d^ 
2))^(1/2)-I*b*e+2*c*d)^(1/2))/(4*(-I*d*b*e+a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c 
*(-I*d*b*e+a*e^2+c*d^2))^(1/2)+I*b*e-2*c*d)^(1/2))*(2*(c*(-I*d*b*e+a*e^2+c 
*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)-arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(-I* 
d*b*e+a*e^2+c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2))/(4*(-I*d*b*e+a*e^2+c*d^2)^(1 
/2)*c^(1/2)-2*(c*(-I*d*b*e+a*e^2+c*d^2))^(1/2)+I*b*e-2*c*d)^(1/2))*(2*(c*( 
-I*d*b*e+a*e^2+c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)-1/2*ln(-(e*x+d)^(1/2)*(2*( 
c*(-I*d*b*e+a*e^2+c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)+(-I*d*b*e+a*e^2+c*d^2)^ 
(1/2)+c^(1/2)*(e*x+d))*(4*(-I*d*b*e+a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(-I*d* 
b*e+a*e^2+c*d^2))^(1/2)+I*b*e-2*c*d)^(1/2)+1/2*ln((e*x+d)^(1/2)*(2*(c*(-I* 
d*b*e+a*e^2+c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)+(-I*d*b*e+a*e^2+c*d^2)^(1/2)+ 
c^(1/2)*(e*x+d))*(4*(-I*d*b*e+a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(-I*d*b*e+a* 
e^2+c*d^2))^(1/2)+I*b*e-2*c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e}{2 \, e}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt {e x + d} e}{2 \, e}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e}{2 \, e}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt {e x + d} e}{2 \, e}\right ) \] Input:

integrate((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x, algorithm="fricas")
 

Output:

-1/2*sqrt(2)*sqrt(-(2*c*d - I*b*e + (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 
 4*a*c^3)))/(b^2*c + 4*a*c^2))*log(1/2*(sqrt(2)*(b^2*c + 4*a*c^2)*sqrt(-e^ 
2/(b^2*c^2 + 4*a*c^3))*sqrt(-(2*c*d - I*b*e + (b^2*c + 4*a*c^2)*sqrt(-e^2/ 
(b^2*c^2 + 4*a*c^3)))/(b^2*c + 4*a*c^2)) + 2*sqrt(e*x + d)*e)/e) + 1/2*sqr 
t(2)*sqrt(-(2*c*d - I*b*e + (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3 
)))/(b^2*c + 4*a*c^2))*log(-1/2*(sqrt(2)*(b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2* 
c^2 + 4*a*c^3))*sqrt(-(2*c*d - I*b*e + (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^ 
2 + 4*a*c^3)))/(b^2*c + 4*a*c^2)) - 2*sqrt(e*x + d)*e)/e) + 1/2*sqrt(2)*sq 
rt(-(2*c*d - I*b*e - (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3)))/(b^ 
2*c + 4*a*c^2))*log(1/2*(sqrt(2)*(b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4* 
a*c^3))*sqrt(-(2*c*d - I*b*e - (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a* 
c^3)))/(b^2*c + 4*a*c^2)) + 2*sqrt(e*x + d)*e)/e) - 1/2*sqrt(2)*sqrt(-(2*c 
*d - I*b*e - (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3)))/(b^2*c + 4* 
a*c^2))*log(-1/2*(sqrt(2)*(b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3)) 
*sqrt(-(2*c*d - I*b*e - (b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3)))/ 
(b^2*c + 4*a*c^2)) - 2*sqrt(e*x + d)*e)/e)
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\int \frac {\sqrt {d + e x}}{a + i b x + c x^{2}}\, dx \] Input:

integrate((e*x+d)**(1/2)/(a+I*b*x+c*x**2),x)
 

Output:

Integral(sqrt(d + e*x)/(a + I*b*x + c*x**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d}}{c x^{2} + i \, b x + a} \,d x } \] Input:

integrate((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)/(c*x^2 + I*b*x + a), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 6.45 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )-\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )+\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}} \] Input:

int((d + e*x)^(1/2)/(a + b*x*1i + c*x^2),x)
 

Output:

- 2*atanh(((8*c^2*(d + e*x)^(1/2)*(b^2*e^4 - 2*c^2*d^2*e^2 + 2*a*c*e^4 + b 
*c*d*e^3*2i) - (4*c^2*(d + e*x)^(1/2)*(b^3*c*e^3*1i - 2*b^2*c^2*d*e^2 + a* 
b*c^2*e^3*4i - 8*a*c^3*d*e^2)*(e*(-(4*a*c + b^2)^3)^(1/2) - b^3*e*1i + 8*a 
*c^2*d + 2*b^2*c*d - a*b*c*e*4i))/(b^4*c + 16*a^2*c^3 + 8*a*b^2*c^2))*(-(e 
*(-(4*a*c + b^2)^3)^(1/2) - b^3*e*1i + 8*a*c^2*d + 2*b^2*c*d - a*b*c*e*4i) 
/(2*(b^4*c + 16*a^2*c^3 + 8*a*b^2*c^2)))^(1/2))/(8*c^2*(a*e^5 + c*d^2*e^3 
- b*d*e^4*1i)))*(-(e*(-(4*a*c + b^2)^3)^(1/2) - b^3*e*1i + 8*a*c^2*d + 2*b 
^2*c*d - a*b*c*e*4i)/(2*(b^4*c + 16*a^2*c^3 + 8*a*b^2*c^2)))^(1/2) - 2*ata 
nh(((8*c^2*(d + e*x)^(1/2)*(b^2*e^4 - 2*c^2*d^2*e^2 + 2*a*c*e^4 + b*c*d*e^ 
3*2i) + (4*c^2*(d + e*x)^(1/2)*(b^3*c*e^3*1i - 2*b^2*c^2*d*e^2 + a*b*c^2*e 
^3*4i - 8*a*c^3*d*e^2)*(e*(-(4*a*c + b^2)^3)^(1/2) + b^3*e*1i - 8*a*c^2*d 
- 2*b^2*c*d + a*b*c*e*4i))/(b^4*c + 16*a^2*c^3 + 8*a*b^2*c^2))*((e*(-(4*a* 
c + b^2)^3)^(1/2) + b^3*e*1i - 8*a*c^2*d - 2*b^2*c*d + a*b*c*e*4i)/(2*(b^4 
*c + 16*a^2*c^3 + 8*a*b^2*c^2)))^(1/2))/(8*c^2*(a*e^5 + c*d^2*e^3 - b*d*e^ 
4*1i)))*((e*(-(4*a*c + b^2)^3)^(1/2) + b^3*e*1i - 8*a*c^2*d - 2*b^2*c*d + 
a*b*c*e*4i)/(2*(b^4*c + 16*a^2*c^3 + 8*a*b^2*c^2)))^(1/2)
 

Reduce [F]

\[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\int \frac {\sqrt {e x +d}}{i b x +c \,x^{2}+a}d x \] Input:

int((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x)
 

Output:

int((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x)