3.2296 \(\int \frac{\sqrt{d+e x}}{a+i b x+c x^2} \, dx\)

Optimal. Leaf size=629 \[ \frac{e \log \left (-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \log \left (\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left (\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left (\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}} \]

[Out]

(e*ArcTanh[(Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2 - e*(I*b*d - a*e)]] - 2*Sq
rt[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]*Sqrt[2*c*d - I*b*e - 2*Sqrt[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*Sqr
t[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]*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]*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]*Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2
- e*(I*b*d - a*e)]])

_______________________________________________________________________________________

Rubi [A]  time = 2.56405, antiderivative size = 629, 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 \[ \frac{e \log \left (-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \log \left (\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left (\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left (\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}} \]

Antiderivative was successfully verified.

[In]  Int[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*Sq
rt[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]*Sqrt[2*c*d - I*b*e - 2*Sqrt[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*Sqr
t[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]*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]*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]*Sqrt[2*c*d - I*b*e + 2*Sqrt[c]*Sqrt[c*d^2
- e*(I*b*d - a*e)]])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(a+I*b*x+c*x**2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.729125, size = 197, normalized size = 0.31 \[ \frac{\sqrt{2} \left (\sqrt{2 c d-e \left (\sqrt{-4 a c-b^2}+i b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{-4 a c-b^2}+i b\right )}}\right )-\sqrt{2 c d+e \left (\sqrt{-4 a c-b^2}-i b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{-4 a c-b^2}-i b e+2 c d}}\right )\right )}{\sqrt{c} \sqrt{-4 a c-b^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*(-(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*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[c]*Sqrt[-b^2 - 4*a*c])

_______________________________________________________________________________________

Maple [A]  time = 0.147, size = 610, normalized size = 1. \[{\frac{e}{2}\ln \left ( - \left ( ex+d \right ) \sqrt{c}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}\sqrt{ex+d}-\sqrt{c{d}^{2}-ibde+a{e}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{c}}}}-{e\arctan \left ({1 \left ( -2\,\sqrt{c}\sqrt{ex+d}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}}-{\frac{e}{2}\ln \left ( \left ( ex+d \right ) \sqrt{c}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}\sqrt{ex+d}+\sqrt{c{d}^{2}-ibde+a{e}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{c}}}}+{e\arctan \left ({1 \left ( 2\,\sqrt{c}\sqrt{ex+d}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)/c^(1/2)*ln(-(e*x+d)
*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)*(e*x+d)^(1/2)-(c
*d^2-I*b*d*e+a*e^2)^(1/2))-e/c^(1/2)/(4*c^(1/2)*(c*d^2-I*b*d*e+a*e^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)*(c*d^2-I*b
*d*e+a*e^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)/c^(1/2)*ln((e*x+d)*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)*(e*x+d)^(1/2)+(c*d^2-I*b
*d*e+a*e^2)^(1/2))+e/c^(1/2)/(4*c^(1/2)*(c*d^2-I*b*d*e+a*e^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)*(c*d^2-I*b*d*e+a*e^
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))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{c x^{2} + i \, b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.247233, size = 990, normalized size = 1.57 \[ -\frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e}{2 \, e}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt{e x + d} e}{2 \, e}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e}{2 \, e}\right ) - \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt{e x + d} e}{2 \, e}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-(4*c*d - 2*I*b*e + 2*(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*((b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3))*s
qrt(-(4*c*d - 2*I*b*e + 2*(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(-(4*c*d - 2*I*b*e + 2*(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*((b^2*c +
4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3))*sqrt(-(4*c*d - 2*I*b*e + 2*(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(-(4*c*d - 2*I*b*e - 2*(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*((b^2*c + 4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3))*
sqrt(-(4*c*d - 2*I*b*e - 2*(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(-(4*c*d - 2*I*b*e - 2*(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*((b^2*c +
4*a*c^2)*sqrt(-e^2/(b^2*c^2 + 4*a*c^3))*sqrt(-(4*c*d - 2*I*b*e - 2*(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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{a + i b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError