\(\int \frac {A+B x}{\sqrt {x} (d+e x) \sqrt {a+b x+c x^2}} \, dx\) [26]

Optimal result

Integrand size = 32, antiderivative size = 386 \[ \int \frac {A+B x}{\sqrt {x} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{c e \sqrt {x} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} (B d-A e) \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{e \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.61 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{\sqrt {x} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {i \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {2+\frac {4 a}{b x-\sqrt {b^2-4 a c} x}} \left (A e \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+(B d-A e) \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2035, 2226, 27, 1416, 2220}

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.

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 2226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2220

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

Input:

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

Output:

2*(-1/2*((Sqrt[a]*B - A*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])/(a^(1/4)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*S 
qrt[a + b*x + c*x^2]) + ((B*d - A*e)*(-1/2*((Sqrt[c]*d - Sqrt[a]*e)*ArcTan 
[(Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[x])/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x + c*x 
^2])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) + ((Sqrt[c]*d + Sqrt[ 
a]*e)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2 
]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[c]*d)/Sqrt[a] - e)^2)/(Sqrt[c]*d*e), 2*A 
rcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4) 
*c^(1/4)*d*e*Sqrt[a + b*x + c*x^2])))/(Sqrt[c]*d - Sqrt[a]*e))
 

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 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 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2220

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

rule 2226

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

Maple [A] (verified)

Time = 3.90 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.63

method	result	size
elliptic	\(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {B \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{e c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (A e -B d \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, -\frac {b +\sqrt {-4 a c +b^{2}}}{2 c \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{e^{2} c \sqrt {c \,x^{3}+b \,x^{2}+a x}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\)	\(628\)
default	\(-\frac {2 \left (A \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c e +A \operatorname {EllipticPi}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b c e -B \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{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 e +B \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{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 d -B \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) c d +2 B \operatorname {EllipticF}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{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 e -B \operatorname {EllipticF}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{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} e +B \operatorname {EllipticF}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{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 c d -B \operatorname {EllipticPi}\left (\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}+2 c x}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b c d \right ) \sqrt {-\frac {c x}{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 {b +\sqrt {-4 a c +b^{2}}+2 c x}{b +\sqrt {-4 a c +b^{2}}}}}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, e c \left (e \sqrt {-4 a c +b^{2}}+b e -2 c d \right )}\)	\(938\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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