\(\int \frac {A+B x}{\sqrt {x} (a+b x+c x^2)} \, dx\) [87]

Optimal result

Integrand size = 23, antiderivative size = 180 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\frac {\sqrt {2} \left (B-\frac {b B-2 A c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (B+\frac {b B-2 A c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\frac {\sqrt {2} \left (\frac {\left (-b B+2 A c+B \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b B-2 A c+B \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}} \] Input:

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

Output:

(Sqrt[2]*(((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]* 
Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((b*B 
 - 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + 
Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[c]*Sqrt[b^2 - 4*a 
*c])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1197, 1480, 218}

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.

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

\(\displaystyle 2 \left (\frac {\left (B-\frac {b B-2 A c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {b B-2 A c}{\sqrt {b^2-4 a c}}+B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\)

Input:

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

Output:

2*(((B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x]) 
/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c] 
]) + ((B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x 
])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a* 
c]]))
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(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] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1480

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

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(8 c \left (-\frac {\left (2 A c +B \sqrt {-4 a c +b^{2}}-B b \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (B b -2 A c +B \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\)	\(168\)
default	\(8 c \left (-\frac {\left (2 A c +B \sqrt {-4 a c +b^{2}}-B b \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (B b -2 A c +B \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\)	\(168\)

Input:

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

Output:

8*c*(-1/8*(2*A*c+B*(-4*a*c+b^2)^(1/2)-B*b)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(( 
-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2 
)^(1/2))*c)^(1/2))+1/8*(B*b-2*A*c+B*(-4*a*c+b^2)^(1/2))/c/(-4*a*c+b^2)^(1/ 
2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+( 
-4*a*c+b^2)^(1/2))*c)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1577 vs. \(2 (140) = 280\).

Time = 0.41 (sec) , antiderivative size = 1577, normalized size of antiderivative = 8.76 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*s 
qrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2 
*c - 4*a^2*c^2))*log(sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A 
^3*b^2)*c + (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt(( 
B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2* 
a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^ 
2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B 
^4*a^2 - A*B^3*a*b + A^3*B*b*c - A^4*c^2)*sqrt(x)) - 1/2*sqrt(2)*sqrt(-(B^ 
2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2* 
B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log( 
-sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c + (4*(2*B* 
a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*A^2*B^2* 
a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A^2 
*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^ 
2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b + 
 A^3*B*b*c - A^4*c^2)*sqrt(x)) + 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A 
^2*b)*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/( 
a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log(sqrt(2)*(A*B^2*a*b^2 
 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c - (4*(2*B*a^3 - A*a^2*b)*c^2 - 
(2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4488 vs. \(2 (165) = 330\).

Time = 9.14 (sec) , antiderivative size = 4488, normalized size of antiderivative = 24.93 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((A*log(sqrt(x) - sqrt(-a/b))/(b*sqrt(-a/b)) - A*log(sqrt(x) + sq 
rt(-a/b))/(b*sqrt(-a/b)) - B*a*log(sqrt(x) - sqrt(-a/b))/(b**2*sqrt(-a/b)) 
 + B*a*log(sqrt(x) + sqrt(-a/b))/(b**2*sqrt(-a/b)) + 2*B*sqrt(x)/b, Eq(c, 
0)), (-A*log(sqrt(x) - sqrt(-b/c))/(b*sqrt(-b/c)) + A*log(sqrt(x) + sqrt(- 
b/c))/(b*sqrt(-b/c)) - 2*A/(b*sqrt(x)) + B*log(sqrt(x) - sqrt(-b/c))/(c*sq 
rt(-b/c)) - B*log(sqrt(x) + sqrt(-b/c))/(c*sqrt(-b/c)), Eq(a, 0)), (2*sqrt 
(2)*A*b*c*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(2*b**2*c*sqrt(-b/c) + 4*b*c 
**2*x*sqrt(-b/c)) - 2*sqrt(2)*A*b*c*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(2 
*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)) + 8*A*c**2*sqrt(x)*sqrt(-b/c)/ 
(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)) + 4*sqrt(2)*A*c**2*x*log(sqr 
t(x) - sqrt(2)*sqrt(-b/c)/2)/(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)) 
 - 4*sqrt(2)*A*c**2*x*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(2*b**2*c*sqrt(- 
b/c) + 4*b*c**2*x*sqrt(-b/c)) + sqrt(2)*B*b**2*log(sqrt(x) - sqrt(2)*sqrt( 
-b/c)/2)/(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)) - sqrt(2)*B*b**2*lo 
g(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(- 
b/c)) - 4*B*b*c*sqrt(x)*sqrt(-b/c)/(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt( 
-b/c)) + 2*sqrt(2)*B*b*c*x*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(2*b**2*c*s 
qrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)) - 2*sqrt(2)*B*b*c*x*log(sqrt(x) + sqrt( 
2)*sqrt(-b/c)/2)/(2*b**2*c*sqrt(-b/c) + 4*b*c**2*x*sqrt(-b/c)), Eq(a, b**2 
/(4*c))), (-sqrt(2)*A*b*c*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x...
 

Maxima [F]

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

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

Output:

2*A*sqrt(x)/a - integrate((A*c*x^(3/2) - (B*a - A*b)*sqrt(x))/(a*c*x^2 + a 
*b*x + a^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1404 vs. \(2 (140) = 280\).

Time = 0.66 (sec) , antiderivative size = 1404, normalized size of antiderivative = 7.80 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 
3*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqr 
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 
 - 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^2 + 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqr 
t(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a* 
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( 
b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 - 2*(b^ 
2 - 4*a*c)*b*c^2)*A - 2*(2*a*b^2*c^2 - 8*a^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a* 
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr 
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a*b*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b 
^2 - 4*a*c)*c)*a*c^2 - 2*(b^2 - 4*a*c)*a*c^2)*B)*arctan(2*sqrt(1/2)*sqrt(x 
)/sqrt((b + sqrt(b^2 - 4*a*c))/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16* 
a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2*c^3)*abs(c)) + 1/2*((sqrt(2)*sqr 
t(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)* 
c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 1 
6*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c ...
 

Mupad [B] (verification not implemented)

Time = 11.91 (sec) , antiderivative size = 4141, normalized size of antiderivative = 23.01 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((((-(B^2*a*b^3 + B^2*a*(-(4*a*c - b^2)^3)^(1/2) + A^2*b^3*c - A^2*c 
*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2*c^2 - 4*A^2*a*b*c^2 - 4*B^2*a^2*b*c 
 - 4*A*B*a*b^2*c)/(2*(16*a^3*c^3 - 8*a^2*b^2*c^2 + a*b^4*c)))^(1/2)*(x^(1/ 
2)*(8*b^3*c^2 - 32*a*b*c^3)*(-(B^2*a*b^3 + B^2*a*(-(4*a*c - b^2)^3)^(1/2) 
+ A^2*b^3*c - A^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2*c^2 - 4*A^2*a*b* 
c^2 - 4*B^2*a^2*b*c - 4*A*B*a*b^2*c)/(2*(16*a^3*c^3 - 8*a^2*b^2*c^2 + a*b^ 
4*c)))^(1/2) - 8*A*b^2*c^2 + 32*A*a*c^3) + x^(1/2)*(16*A^2*c^3 - 16*B^2*a* 
c^2 + 8*B^2*b^2*c - 16*A*B*b*c^2))*(-(B^2*a*b^3 + B^2*a*(-(4*a*c - b^2)^3) 
^(1/2) + A^2*b^3*c - A^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2*c^2 - 4*A 
^2*a*b*c^2 - 4*B^2*a^2*b*c - 4*A*B*a*b^2*c)/(2*(16*a^3*c^3 - 8*a^2*b^2*c^2 
 + a*b^4*c)))^(1/2)*1i + ((-(B^2*a*b^3 + B^2*a*(-(4*a*c - b^2)^3)^(1/2) + 
A^2*b^3*c - A^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2*c^2 - 4*A^2*a*b*c^ 
2 - 4*B^2*a^2*b*c - 4*A*B*a*b^2*c)/(2*(16*a^3*c^3 - 8*a^2*b^2*c^2 + a*b^4* 
c)))^(1/2)*(x^(1/2)*(8*b^3*c^2 - 32*a*b*c^3)*(-(B^2*a*b^3 + B^2*a*(-(4*a*c 
 - b^2)^3)^(1/2) + A^2*b^3*c - A^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^2 
*c^2 - 4*A^2*a*b*c^2 - 4*B^2*a^2*b*c - 4*A*B*a*b^2*c)/(2*(16*a^3*c^3 - 8*a 
^2*b^2*c^2 + a*b^4*c)))^(1/2) + 8*A*b^2*c^2 - 32*A*a*c^3) + x^(1/2)*(16*A^ 
2*c^3 - 16*B^2*a*c^2 + 8*B^2*b^2*c - 16*A*B*b*c^2))*(-(B^2*a*b^3 + B^2*a*( 
-(4*a*c - b^2)^3)^(1/2) + A^2*b^3*c - A^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16* 
A*B*a^2*c^2 - 4*A^2*a*b*c^2 - 4*B^2*a^2*b*c - 4*A*B*a*b^2*c)/(2*(16*a^3...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.96 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b c -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a c +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b^{2}+2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b c +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a c -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {x}\, \sqrt {c}}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b^{2}+\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b c -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b c -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) a c +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b^{2}+2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) a c -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}+\sqrt {a}+\sqrt {c}\, x \right ) b^{2}}{2 c \left (4 a c -b^{2}\right )} \] Input:

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

Output:

( - 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b 
) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b*c - 4*sqrt(c)*sqrt(2 
*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c 
))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*c + 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b 
)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sq 
rt(a) + b))*b**2 + 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt 
(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b*c + 4 
*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2 
*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*c - 2*sqrt(c)*sqrt(2*sqrt 
(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sq 
rt(2*sqrt(c)*sqrt(a) + b))*b**2 + sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b)*log( 
 - sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*b*c - sqrt(a 
)*sqrt(2*sqrt(c)*sqrt(a) - b)*log(sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sq 
rt(a) + sqrt(c)*x)*b*c - 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - b)*log( - sqrt 
(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*a*c + sqrt(c)*sqrt( 
2*sqrt(c)*sqrt(a) - b)*log( - sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a 
) + sqrt(c)*x)*b**2 + 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - b)*log(sqrt(x)*sq 
rt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*a*c - sqrt(c)*sqrt(2*sqrt 
(c)*sqrt(a) - b)*log(sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt( 
c)*x)*b**2)/(2*c*(4*a*c - b**2))