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[(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)]])

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Rubi [A]  time = 0.791481, antiderivative size = 705, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, 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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 0.627007, 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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Maple [A]  time = 0.38, size = 673, normalized size = 1. \begin{align*} -{\frac{e}{2}\ln \left ( \left ( ex+d \right ) \sqrt{c}-\sqrt{ex+d}\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}+\sqrt{-ibde+a{e}^{2}+c{d}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{-ibde+a{e}^{2}+c{d}^{2}}}}}+{e\arctan \left ({ \left ( 2\,\sqrt{c}\sqrt{ex+d}-\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{-ibde+a{e}^{2}+c{d}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{-ibde+a{e}^{2}+c{d}^{2}}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{-ibde+a{e}^{2}+c{d}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}}+{\frac{e}{2}\ln \left ( \left ( ex+d \right ) \sqrt{c}+\sqrt{ex+d}\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}+\sqrt{-ibde+a{e}^{2}+c{d}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{-ibde+a{e}^{2}+c{d}^{2}}}}}+{e\arctan \left ({ \left ( 2\,\sqrt{c}\sqrt{ex+d}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{-ibde+a{e}^{2}+c{d}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{-ibde+a{e}^{2}+c{d}^{2}}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{-ibde+a{e}^{2}+c{d}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + i \, b x + a\right )} \sqrt{e x + d}}\,{d x} \end{align*}

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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Fricas [B]  time = 3.26354, size = 5848, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d + e x} \left (a + i b x + c x^{2}\right )}\, dx \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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