Integrand size = 25, antiderivative size = 629 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-2 \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \text {arctanh}\left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+2 \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}+\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}-\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}+\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}} \]
e*arctanh((-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))/(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* arctanh((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))/(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)+1/2*e *ln((e*x+d)*c^(1/2)+(c*d^2-e*(I*b*d-a*e))^(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)-1/2*e*ln((e*x+d)*c^(1/2)+(c*d^2-e*( I*b*d-a*e))^(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)
Time = 0.67 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\frac {\sqrt {2} \left (\frac {\left (-2 i c d+\left (-b+\sqrt {b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d-i \left (-b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {-2 c d-i \left (-b+\sqrt {b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (b+\sqrt {b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+i \left (b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {-2 c d+i \left (b+\sqrt {b^2+4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2+4 a c}} \]
(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])
Time = 1.29 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.17, 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 |
\(\displaystyle 2 e \left (\frac {\frac {1}{2} \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )-\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d} \text {arctanh}\left (\frac {\sqrt {c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {c}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d} \text {arctanh}\left (\frac {\sqrt {c} \left (2 \sqrt {d+e x}+\frac {\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {c}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {1}{2} \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )\) |
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)]]))
3.24.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
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]
Time = 1.68 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(2 e \left (-\frac {\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}+\frac {\frac {\ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}-\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}\right )\) | \(691\) |
default | \(2 e \left (-\frac {\frac {\ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}+\frac {\frac {\ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}\, \arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}-\sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}\right )}{\sqrt {c}\, \sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}+i b e -2 c d}}}{2 \sqrt {2 \sqrt {-c \left (i b d e -a \,e^{2}-c \,d^{2}\right )}-i b e +2 c d}}\right )\) | \(691\) |
pseudoelliptic | \(-\frac {e \left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}+i b e -2 c d}}\right ) \sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}+i b e -2 c d}}\right ) \sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}-\frac {\ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right ) \sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}+i b e -2 c d}}{2}+\frac {\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}+i b e -2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2}\right )}{\sqrt {4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}+i b e -2 c d}\, \sqrt {c}\, \sqrt {2 \sqrt {c^{2} d^{2}+\left (-i b d e +a \,e^{2}\right ) c}-i b e +2 c d}}\) | \(698\) |
2*e*(-1/2/(2*(-c*(I*b*d*e-a*e^2-c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)*(1/2/c^(1 /2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(-c*(I*b*d*e-a*e^2-c*d^2))^(1/2)-I *b*e+2*c*d)^(1/2)+(-I*b*d*e+a*e^2+c*d^2)^(1/2))-1/c^(1/2)*(2*(-c*(I*b*d*e- a*e^2-c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)/(4*(-I*b*d*e+a*e^2+c*d^2)^(1/2)*c^( 1/2)-2*(-c*(I*b*d*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*b*d*e-a*e^2-c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2))/( 4*(-I*b*d*e+a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(-c*(I*b*d*e-a*e^2-c*d^2))^(1/2)+ I*b*e-2*c*d)^(1/2)))+1/2/(2*(-c*(I*b*d*e-a*e^2-c*d^2))^(1/2)-I*b*e+2*c*d)^ (1/2)*(1/2/c^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(-c*(I*b*d*e-a*e^2- c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)+(-I*b*d*e+a*e^2+c*d^2)^(1/2))+1/c^(1/2)*( 2*(-c*(I*b*d*e-a*e^2-c*d^2))^(1/2)-I*b*e+2*c*d)^(1/2)/(4*(-I*b*d*e+a*e^2+c *d^2)^(1/2)*c^(1/2)-2*(-c*(I*b*d*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*b*d*e-a*e^2-c*d^2))^(1/2)-I*b*e+ 2*c*d)^(1/2))/(4*(-I*b*d*e+a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(-c*(I*b*d*e-a*e^2 -c*d^2))^(1/2)+I*b*e-2*c*d)^(1/2))))
Time = 0.29 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.20 \[ \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 ) \]
-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)
\[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\int \frac {\sqrt {d + e x}}{a + i b x + c x^{2}}\, dx \]
\[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d}}{c x^{2} + i \, b x + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=\text {Timed out} \]
Time = 10.63 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )-\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )+\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}} \]
- 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)