\(\int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx\) [200]

Optimal result

Integrand size = 27, antiderivative size = 401 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx=\frac {B \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {e}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} c^{3/2} \sqrt {e} \sqrt {a+x (b+c x)}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b B-2 A c+B \sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {e x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {e}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} c^{3/2} \sqrt {e} \sqrt {a+x (b+c x)}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.96 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx=-\frac {x^2 \left (-\frac {4 B \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))}{x^2}+\frac {i B \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {x}}-\frac {i \left (-b B+2 A c+B \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {x}}\right )}{2 c \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \sqrt {e x} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

-1/2*(x^2*((-4*B*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)))/x^2 + 
(I*B*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]* 
Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*Ellipt 
icE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqr 
t[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[x] - (I*(-(b*B) + 2*A*c + B 
*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a 
+ b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcS 
inh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4* 
a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[x]))/(c*Sqrt[a/(b + Sqrt[b^2 - 4*a*c] 
)]*Sqrt[e*x]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1241, 1240, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1241

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

\(\Big \downarrow \) 1240

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{\sqrt {e x}}\)

Input:

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

Output:

(2*Sqrt[x]*(-((B*(-((Sqrt[x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) 
 + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c 
]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqr 
t[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c]) + ((A + (Sqrt[a]*B)/ 
Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x 
)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c 
]))/4])/(2*a^(1/4)*c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[e*x]
 

Defintions of rubi rules used

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 1240

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), 
x_Symbol] :> Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, 
 Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 1241

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ 
)^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(f + g*x)/(Sqrt[x]*Sqrt[a 
+ b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
 

rule 1416

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

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (A \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c \sqrt {-4 a c +b^{2}}+A \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c b -B \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b -2 B \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) a c +4 B \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) a c -B \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b^{2}\right )}{\sqrt {c \,x^{2}+b x +a}\, \sqrt {e x}\, c^{2}}\)	\(538\)
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+b x +a \right ) e x}\, \left (\frac {A \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+a e x}}+\frac {B \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+b x +a}}\)	\(717\)

Input:

int((B*x+A)/(e*x)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/(c*x^2+b*x+a)^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)) 
)^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+ 
(-4*a*c+b^2)^(1/2)))^(1/2)*(A*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+( 
-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2) 
^(1/2))^(1/2))*c*(-4*a*c+b^2)^(1/2)+A*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2) 
+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4* 
a*c+b^2)^(1/2))^(1/2))*c*b-B*(-4*a*c+b^2)^(1/2)*EllipticE(((2*c*x+(-4*a*c+ 
b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^( 
1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b-2*B*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2 
)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4 
*a*c+b^2)^(1/2))^(1/2))*a*c+4*B*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b 
+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^ 
2)^(1/2))^(1/2))*a*c-B*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+ 
b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)) 
^(1/2))*b^2)/(e*x)^(1/2)/c^2
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.36 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} B c {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + {\left (B b - 3 \, A c\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right )}}{3 \, c^{2} e} \] Input:

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

Output:

-2/3*(3*sqrt(c*e)*B*c*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 
- 9*a*b*c)/c^3, weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 
9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) + (B*b - 3*A*c)*sqrt(c*e)*weierstrassPIn 
verse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/ 
c))/(c^2*e)
 

Sympy [F]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {e x} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

integrate((B*x+A)/(e*x)^(1/2)/(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)), x)
 

Giac [F]

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

integrate((B*x+A)/(e*x)^(1/2)/(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)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a +\left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{2}+b x +a}d x \right ) b \right )}{e} \] Input:

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

Output:

(sqrt(e)*(int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)* 
a + int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a + b*x + c*x**2),x)*b))/e