3.2308 \(\int \frac {1}{\sqrt {d+e x} (a+i b x+c x^2)} \, dx\)

Optimal. Leaf size=705 \[ -\frac {e \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c d^2-e (-a e+i b d)} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c d^2-e (-a e+i b d)} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \tanh ^{-1}\left (\frac {-2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c d^2-e (-a e+i b d)} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c d^2-e (-a e+i b d)} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}} \]

[Out]

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*d^2-e*(I*b*d-a*e))^(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*d^2-e*(I*b*d-a*e))^(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*d^2-e*(I*b*d-a*e))^(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*d^2-e*(I*b*d-a*e))^(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)

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Rubi [A]  time = 0.79, antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {707, 1094, 634, 618, 206, 628} \[ -\frac {e \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c d^2-e (-a e+i b d)} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c d^2-e (-a e+i b d)} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \tanh ^{-1}\left (\frac {-2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c d^2-e (-a e+i b d)} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c d^2-e (-a e+i b d)} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(a + I*b*x + c*x^2)),x]

[Out]

(e*ArcTanh[(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)]]])/(Sqrt[c*d^2 - e*(I*b*d - a*e)]*Sqrt[2*c*d - I*b*e - 2*S
qrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]]) - (e*ArcTanh[(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)]]])/(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)]]) - (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*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)]]) + (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*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)]])

Rule 206

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1094

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+i b x+c x^2\right )} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{c d^2-i b d e+a e^2-(2 c d-i b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}-x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\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)}}}+\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\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)}}}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}+\frac {e \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-\frac {e \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \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)}}}+\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \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)}}}\\ &=-\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 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)}}}+\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 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)}}}-\frac {e \operatorname {Subst}\left (\int \frac {1}{2 d-\frac {i b e}{c}-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-x^2} \, dx,x,-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{\sqrt {c} \sqrt {c d^2-e (i b d-a e)}}-\frac {e \operatorname {Subst}\left (\int \frac {1}{2 d-\frac {i b e}{c}-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{\sqrt {c} \sqrt {c d^2-e (i b d-a e)}}\\ &=\frac {e \tanh ^{-1}\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 {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)}}}-\frac {e \tanh ^{-1}\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 {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)}}}-\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 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)}}}+\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 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)}}}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 198, normalized size = 0.28 \[ \frac {2 \sqrt {2} \sqrt {c} \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {-4 a c-b^2}+i b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {-4 a c-b^2}+i b\right )}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {-4 a c-b^2}-i b e+2 c d}}\right )}{\sqrt {2 c d+e \left (\sqrt {-4 a c-b^2}-i b\right )}}\right )}{\sqrt {-4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a + I*b*x + c*x^2)),x]

[Out]

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

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fricas [B]  time = 1.03, size = 2725, normalized size = 3.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-(4*c*d - 2*I*b*e + (2*(b^2*c + 4*a*c^2)*d^2 - (2*I*b^3 + 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*
sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2
+ (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*
d*e + (a*b^2 + 4*a^2*c)*e^2))*log(1/4*(4*sqrt(e*x + d)*c*e - ((b^2 + 4*a*c)*e^2 + (2*(b^2*c^2 + 4*a*c^3)*d^3 -
(3*I*b^3*c + 12*I*a*b*c^2)*d^2*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (I*a*b^3 + 4*I*a^2*b*c)*e^3)*sqrt(-e
^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I
*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))*sqrt(-(4*c*d - 2*I*b*e + (2*(b^2*c + 4*a*c^2)*d^2 - (
2*I*b^3 + 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b
*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e
^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^2*c)*e^2)))/(c*e)) + 1/2*sqrt(-(4*c*d -
2*I*b*e + (2*(b^2*c + 4*a*c^2)*d^2 - (2*I*b^3 + 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2
+ 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I
*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^
2*c)*e^2))*log(1/4*(4*sqrt(e*x + d)*c*e + ((b^2 + 4*a*c)*e^2 + (2*(b^2*c^2 + 4*a*c^3)*d^3 + (-3*I*b^3*c - 12*I
*a*b*c^2)*d^2*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d*e^2 + (-I*a*b^3 - 4*I*a^2*b*c)*e^3)*sqrt(-e^2/((b^2*c^2 + 4*
a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*
b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))*sqrt(-(4*c*d - 2*I*b*e + (2*(b^2*c + 4*a*c^2)*d^2 - (2*I*b^3 + 8*I*a*b
*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^
4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4
*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^2*c)*e^2)))/(c*e)) + 1/2*sqrt(-(4*c*d - 2*I*b*e - (2*(b^
2*c + 4*a*c^2)*d^2 + (-2*I*b^3 - 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4
+ (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3
+ (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^2*c)*e^2))*log(1
/4*(4*sqrt(e*x + d)*c*e + ((b^2 + 4*a*c)*e^2 - (2*(b^2*c^2 + 4*a*c^3)*d^3 - (3*I*b^3*c + 12*I*a*b*c^2)*d^2*e -
(b^4 + 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (I*a*b^3 + 4*I*a^2*b*c)*e^3)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I
*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*
b^2 + 4*a^3*c)*e^4)))*sqrt(-(4*c*d - 2*I*b*e - (2*(b^2*c + 4*a*c^2)*d^2 + (-2*I*b^3 - 8*I*a*b*c)*d*e + 2*(a*b^
2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8
*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I
*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^2*c)*e^2)))/(c*e)) - 1/2*sqrt(-(4*c*d - 2*I*b*e - (2*(b^2*c + 4*a*c^2)*d^
2 + (-2*I*b^3 - 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8
*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^
3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*c)*d*e + (a*b^2 + 4*a^2*c)*e^2))*log(1/4*(4*sqrt(e*x +
d)*c*e - ((b^2 + 4*a*c)*e^2 - (2*(b^2*c^2 + 4*a*c^3)*d^3 + (-3*I*b^3*c - 12*I*a*b*c^2)*d^2*e - (b^4 + 2*a*b^2*
c - 8*a^2*c^2)*d*e^2 + (-I*a*b^3 - 4*I*a^2*b*c)*e^3)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*
b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*
e^4)))*sqrt(-(4*c*d - 2*I*b*e - (2*(b^2*c + 4*a*c^2)*d^2 + (-2*I*b^3 - 8*I*a*b*c)*d*e + 2*(a*b^2 + 4*a^2*c)*e^
2)*sqrt(-e^2/((b^2*c^2 + 4*a*c^3)*d^4 + (-2*I*b^3*c - 8*I*a*b*c^2)*d^3*e - (b^4 + 2*a*b^2*c - 8*a^2*c^2)*d^2*e
^2 + (-2*I*a*b^3 - 8*I*a^2*b*c)*d*e^3 + (a^2*b^2 + 4*a^3*c)*e^4)))/((b^2*c + 4*a*c^2)*d^2 + (-I*b^3 - 4*I*a*b*
c)*d*e + (a*b^2 + 4*a^2*c)*e^2)))/(c*e))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular val
ue [0,0,0,0,0] was discarded and replaced randomly by 0=[-40,-48,10,-58,31]index.cc index_m operator + Error:
Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might be
wrong.Non regular value [0,0,0,0,0] was discarded and replaced randomly by 0=[-64,83,-68,-60,2]index.cc index
_m operator + Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with para
meters. This might be wrong.Non regular value [0,0,0,0,0] was discarded and replaced randomly by 0=[84,-86,82,
76,-49]index.cc index_m operator + Error: Bad Argument ValueWarning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,0] was discarded and replaced rand
omly by 0=[17,-70,45,77,-80]index.cc index_m operator + Error: Bad Argument ValueDone

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maple [A]  time = 0.39, size = 673, normalized size = 0.95 \[ \frac {e \arctan \left (\frac {2 \sqrt {e x +d}\, \sqrt {c}-\sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}\right )}{\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}+\frac {e \arctan \left (\frac {2 \sqrt {e x +d}\, \sqrt {c}+\sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}\right )}{\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}-\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}}+\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + i \, b x + a\right )} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.91, size = 4530, normalized size = 6.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan((((8*c^2*(b^2*e^3 + 4*a*c*e^3) - 8*c^2*(d + e*x)^(1/2)*(-(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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*
b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*c*e^3*4i - 8*a*c
^2*d*e^2 - 2*b^2*c*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)/(2*(
a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2
*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*(-(e*(-(4*a*c + b^2)^3)^(1/2) - b^3*e*1
i + 8*a*c^2*d + 2*b^2*c*d - a*b*c*e*4i)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*
1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*1i - ((8*c^2*(b^2*e^3 + 4
*a*c*e^3) + 8*c^2*(d + e*x)^(1/2)*(-(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*4
i)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^
2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a
^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e
*8i)))^(1/2) - 16*c^3*e^2*(d + e*x)^(1/2))*(-(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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2
*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*1i)/(((8*c^2*(b^2*e^3 + 4*a*c*e^3) - 8*c^2*(d + e*x)^
(1/2)*(-(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*(a*b^4*e^2 + b^4*c*d^2
+ 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^
3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2
- b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) + 16*c^3*e^2*(d
+ e*x)^(1/2))*(-(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*(a*b^4*e^2 +
b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*1
6i - a*b^3*c*d*e*8i)))^(1/2) + ((8*c^2*(b^2*e^3 + 4*a*c*e^3) + 8*c^2*(d + e*x)^(1/2)*(-(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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*
e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i
+ a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2
+ 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) - 16*c^3*e^2*(d + e*x)^(1/2))*(-(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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16
*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)))*
(-(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*(a*b^4*e^2 + b^4*c*d^2 + 16*
a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*
e*8i)))^(1/2)*2i - atan((((8*c^2*(b^2*e^3 + 4*a*c*e^3) - 8*c^2*(d + e*x)^(1/2)*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b
^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*
c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*
b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) + 16*c^3*e^2*(d + e*x)^(1/2))*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*
e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*1i - ((8*c^
2*(b^2*e^3 + 4*a*c*e^3) + 8*c^2*(d + e*x)^(1/2)*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8
*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^
2*c*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)/(2*(a*b^4*e^2 + b^4*
c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i -
a*b^3*c*d*e*8i)))^(1/2) - 16*c^3*e^2*(d + e*x)^(1/2))*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*
d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*1i)/(((8*c^2*(b^2*e^3 + 4*a*c*e^3) - 8*c^2
*(d + e*x)^(1/2)*((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*(a*b^4*e^2 +
b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*
16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3*e^3*1i + a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3
*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) + 16*c
^3*e^2*(d + e*x)^(1/2))*((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*(a*b^
4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c
^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) + ((8*c^2*(b^2*e^3 + 4*a*c*e^3) + 8*c^2*(d + e*x)^(1/2)*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a
^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2)*(b^3
*e^3*1i + a*b*c*e^3*4i - 8*a*c^2*d*e^2 - 2*b^2*c*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)/(2*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^
2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/2) - 16*c^3*e^2*(d + e*x)^(1/2))*((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*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2
+ 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*c*d*e*8i)))^(1/
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*(a*b^4*e^2 + b^4*c*d^2 +
16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e*1i + 8*a*b^2*c^2*d^2 + 8*a^2*b^2*c*e^2 - a^2*b*c^2*d*e*16i - a*b^3*
c*d*e*8i)))^(1/2)*2i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d + e x} \left (a + i b x + c x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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