\(\int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx\) [49]

Optimal result

Integrand size = 36, antiderivative size = 434 \[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} (B f-A g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{g \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2*2^(1/2)*B*(-4*a*c+b^2)^(1/2)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e) 
)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/( 
-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a* 
c+b^2)^(1/2))*e))^(1/2))/c/g/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-4*2^(1/2)*( 
-4*a*c+b^2)^(1/2)*(-A*g+B*f)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^ 
(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticPi(1/2*(1+(2*c*x+b)/(- 
4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),-2*(-4*a*c+b^2)^(1/2)*g/(2*c*f-(b+(-4*a*c+ 
b^2)^(1/2))*g),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^ 
(1/2))/g/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2 
)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.78 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left ((-B d g+A e g) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+e (B f-A g) \operatorname {EllipticPi}\left (\frac {\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (e f-d g)}{2 \left (c d^2+e (-b d+a e)\right ) g},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} g (e f-d g) \sqrt {a+x (b+c x)}} \] Input:

Integrate[(A + B*x)/(Sqrt[d + e*x]*(f + g*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[ 
(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/(( 
-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((-(B*d*g) + A*e*g)*El 
lipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sq 
rt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a 
*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + e*(B*f - A*g)*Ellipt 
icPi[((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(e*f - d*g))/(2*(c*d^2 + e* 
(-(b*d) + a*e))*g), I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c* 
d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqr 
t[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e*Sqrt[ 
(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*g*(e* 
f - d*g)*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (warning: unable to verify)

Time = 2.38 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2154, 27, 1172, 321, 1279, 187, 25, 413, 413, 412}

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.

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

\(\Big \downarrow \) 2154

\(\displaystyle \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}dx+\int \frac {B}{g \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}dx+\frac {B \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{g}\)

\(\Big \downarrow \) 1172

\(\displaystyle \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}dx+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}dx+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {\sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {d+e x} (f+g x)}dx}{\sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 187

\(\displaystyle \frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \int -\frac {1}{\sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} (e f-d g+g (d+e x))}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \int \frac {1}{\sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} (e f-d g+g (d+e x))}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \int \frac {1}{\sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} (e f-d g+g (d+e x))}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2} \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} (e f-d g+g (d+e x))}d\sqrt {d+e x}}{\sqrt {a+b x+c x^2} \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}} \sqrt {\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\sqrt {2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {\sqrt {b^2-4 a c}+b+2 c x} \left (A-\frac {B f}{g}\right ) \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticPi}\left (-\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) g}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {c} \sqrt {a+b x+c x^2} (e f-d g) \sqrt {-\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}} \sqrt {\sqrt {b^2-4 a c}+b+\frac {2 c (d+e x)}{e}-\frac {2 c d}{e}}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

Input:

Int[(A + B*x)/(Sqrt[d + e*x]*(f + g*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(2*Sqrt[2]*B*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 
 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin 
[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqr 
t[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*g*Sqrt[d + e*x] 
*Sqrt[a + b*x + c*x^2]) + (Sqrt[2]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e] 
*(A - (B*f)/g)*Sqrt[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[b + Sqrt[b^2 - 4*a 
*c] + 2*c*x]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)] 
*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[ 
-1/2*((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*g)/(c*(e*f - d*g)), ArcSin[(Sqrt 
[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]], (2*c* 
d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt 
[c]*(e*f - d*g)*Sqrt[a + b*x + c*x^2]*Sqrt[b - Sqrt[b^2 - 4*a*c] - (2*c*d) 
/e + (2*c*(d + e*x))/e]*Sqrt[b + Sqrt[b^2 - 4*a*c] - (2*c*d)/e + (2*c*(d + 
 e*x))/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 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 15.94 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.55

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 B \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{g \sqrt {x^{3} e c +b e \,x^{2}+c d \,x^{2}+a e x +b d x +d a}}+\frac {2 \left (A g -B f \right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {f}{g}-\frac {d}{e}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{g^{2} \sqrt {x^{3} e c +b e \,x^{2}+c d \,x^{2}+a e x +b d x +d a}\, \left (\frac {f}{g}-\frac {d}{e}\right )}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(674\)
default	\(\text {Expression too large to display}\)	\(1542\)

Input:

int((B*x+A)/(e*x+d)^(1/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2*B/g*(d/ 
e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c 
))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^( 
1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^ 
(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF 
(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+ 
b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(A*g-B*f)/g^ 
2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/ 
2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b 
^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+ 
b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(f/g 
-d/e)*EllipticPi(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),(-d/e+ 
1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(f/g-d/e),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/ 
c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {d + e x} \left (f + g x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((B*x+A)/(e*x+d)**(1/2)/(g*x+f)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(sqrt(d + e*x)*(f + g*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*(g*x + f)), x)
 

Giac [F]

\[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*(g*x + f)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{\left (f+g\,x\right )\,\sqrt {d+e\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((A + B*x)/((f + g*x)*(d + e*x)^(1/2)*(a + b*x + c*x^2)^(1/2)),x)
 

Output:

int((A + B*x)/((f + g*x)*(d + e*x)^(1/2)*(a + b*x + c*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x}{\sqrt {d+e x} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {B x +A}{\sqrt {e x +d}\, \left (g x +f \right ) \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((B*x+A)/(e*x+d)^(1/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((B*x+A)/(e*x+d)^(1/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x)