\(\int \frac {A+B x}{\sqrt {x} (a+b x+c x^2)^2} \, dx\) [95]

Optimal result

Integrand size = 23, antiderivative size = 292 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx=\frac {\sqrt {x} \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (A b-2 a B+\frac {4 a b B+A \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (A b-2 a B-\frac {A b^2+4 a b B-12 a A c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

x^(1/2)*(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+b*x+a) 
+1/2*c^(1/2)*(A*b-2*B*a+(4*a*b*B+A*(-12*a*c+b^2))/(-4*a*c+b^2)^(1/2))*arct 
an(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c 
+b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/2*c^(1/2)*(A*b-2*B*a-(-12*A*a*c+A*b^2 
+4*B*a*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))*2^(1/2)/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {x} \left (a B (b+2 c x)-A \left (b^2-2 a c+b c x\right )\right )}{a+x (b+c x)}-\frac {\sqrt {2} \sqrt {c} \left (-2 a B \left (-2 b+\sqrt {b^2-4 a c}\right )+A \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (2 a B \left (2 b+\sqrt {b^2-4 a c}\right )+A \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a \left (-b^2+4 a c\right )} \] Input:

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

Output:

((2*Sqrt[x]*(a*B*(b + 2*c*x) - A*(b^2 - 2*a*c + b*c*x)))/(a + x*(b + c*x)) 
 - (Sqrt[2]*Sqrt[c]*(-2*a*B*(-2*b + Sqrt[b^2 - 4*a*c]) + A*(b^2 - 12*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^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqr 
t[c]*(2*a*B*(2*b + Sqrt[b^2 - 4*a*c]) + A*(b^2 - 12*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^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(-b^2 + 4*a*c))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1235, 27, 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.

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {A b^2+a B b-6 a A c+(A b-2 a B) c x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a b B+A \left (b^2-6 a c\right )+(A b-2 a B) c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {\int \frac {a b B+A \left (b^2-6 a c\right )+(A b-2 a B) c x}{c x^2+b x+a}d\sqrt {x}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {1}{2} c \left (\frac {A \left (b^2-12 a c\right )+4 a b B}{\sqrt {b^2-4 a c}}-2 a B+A b\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} c \left (-\frac {-12 a A c+4 a b B+A b^2}{\sqrt {b^2-4 a c}}-2 a B+A b\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\sqrt {c} \left (\frac {A \left (b^2-12 a c\right )+4 a b B}{\sqrt {b^2-4 a c}}-2 a B+A b\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\frac {-12 a A c+4 a b B+A b^2}{\sqrt {b^2-4 a c}}-2 a B+A b\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

(Sqrt[x]*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*( 
a + b*x + c*x^2)) + ((Sqrt[c]*(A*b - 2*a*B + (4*a*b*B + A*(b^2 - 12*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[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(A*b - 2*a*B - (A 
*b^2 + 4*a*b*B - 12*a*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[b + Sqrt[b^2 - 4*a*c]]))/ 
(a*(b^2 - 4*a*c))
 

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 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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

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 = 1.18 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(32 c^{2} \left (-\frac {-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {-4 a c +b^{2}}\, \sqrt {x}}{16 a c \left (x +\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {\left (12 A a c \sqrt {-4 a c +b^{2}}-3 b^{2} A \sqrt {-4 a c +b^{2}}+28 A a b c -3 A \,b^{3}-8 B \,a^{2} c -6 B a \,b^{2}\right ) \left (-2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \left (4 a c -b^{2}\right ) c \sqrt {-4 a c +b^{2}}}-\frac {-\frac {\left (A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {-4 a c +b^{2}}\, \sqrt {x}}{16 a c \left (x +\frac {b}{2 c}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {\left (-28 A a b c +3 A \,b^{3}+12 A a c \sqrt {-4 a c +b^{2}}-3 b^{2} A \sqrt {-4 a c +b^{2}}+8 B \,a^{2} c +6 B a \,b^{2}\right ) \left (2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \left (4 a c -b^{2}\right ) c \sqrt {-4 a c +b^{2}}}\right )\)	\(468\)
default	\(32 c^{2} \left (-\frac {-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {-4 a c +b^{2}}\, \sqrt {x}}{16 a c \left (x +\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {\left (12 A a c \sqrt {-4 a c +b^{2}}-3 b^{2} A \sqrt {-4 a c +b^{2}}+28 A a b c -3 A \,b^{3}-8 B \,a^{2} c -6 B a \,b^{2}\right ) \left (-2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \left (4 a c -b^{2}\right ) c \sqrt {-4 a c +b^{2}}}-\frac {-\frac {\left (A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {-4 a c +b^{2}}\, \sqrt {x}}{16 a c \left (x +\frac {b}{2 c}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {\left (-28 A a b c +3 A \,b^{3}+12 A a c \sqrt {-4 a c +b^{2}}-3 b^{2} A \sqrt {-4 a c +b^{2}}+8 B \,a^{2} c +6 B a \,b^{2}\right ) \left (2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \left (4 a c -b^{2}\right ) c \sqrt {-4 a c +b^{2}}}\right )\)	\(468\)

Input:

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

Output:

32*c^2*(-1/4/(4*a*c-b^2)/c/(-4*a*c+b^2)^(1/2)*(-1/16*(-A*(-4*a*c+b^2)^(1/2 
)-A*b+2*B*a)*(-4*a*c+b^2)^(1/2)/a/c*x^(1/2)/(x+1/2*b/c-1/2/c*(-4*a*c+b^2)^ 
(1/2))-1/16*(12*A*a*c*(-4*a*c+b^2)^(1/2)-3*b^2*A*(-4*a*c+b^2)^(1/2)+28*A*a 
*b*c-3*A*b^3-8*B*a^2*c-6*B*a*b^2)*(-2*b+(-4*a*c+b^2)^(1/2))/a/(4*a*c+3*b^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)))-1/4/(4*a*c-b^2)/c/(-4*a*c+b^2)^(1/2)*(-1/1 
6*(A*(-4*a*c+b^2)^(1/2)-A*b+2*B*a)*(-4*a*c+b^2)^(1/2)/a/c*x^(1/2)/(x+1/2*b 
/c+1/2/c*(-4*a*c+b^2)^(1/2))-1/16*(-28*A*a*b*c+3*A*b^3+12*A*a*c*(-4*a*c+b^ 
2)^(1/2)-3*b^2*A*(-4*a*c+b^2)^(1/2)+8*B*a^2*c+6*B*a*b^2)*(2*b+(-4*a*c+b^2) 
^(1/2))/a/(4*a*c+3*b^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. 4884 vs. \(2 (256) = 512\).

Time = 6.12 (sec) , antiderivative size = 4884, normalized size of antiderivative = 16.73 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

((B*a*b*c + (b^2*c - 6*a*c^2)*A)*x^(5/2) + 2*(a*b^2 - 4*a^2*c)*A*sqrt(x) + 
 ((b^3 - 5*a*b*c)*A + (a*b^2 - 2*a^2*c)*B)*x^(3/2))/(a^3*b^2 - 4*a^4*c + ( 
a^2*b^2*c - 4*a^3*c^2)*x^2 + (a^2*b^3 - 4*a^3*b*c)*x) - integrate(1/2*((B* 
a*b*c + (b^2*c - 6*a*c^2)*A)*x^(3/2) + ((b^3 - 7*a*b*c)*A + (a*b^2 + 2*a^2 
*c)*B)*sqrt(x))/(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^2 + (a^2*b^ 
3 - 4*a^3*b*c)*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4434 vs. \(2 (256) = 512\).

Time = 1.12 (sec) , antiderivative size = 4434, normalized size of antiderivative = 15.18 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-(2*B*a*c*x^(3/2) - A*b*c*x^(3/2) + B*a*b*sqrt(x) - A*b^2*sqrt(x) + 2*A*a* 
c*sqrt(x))/((a*b^2 - 4*a^2*c)*(c*x^2 + b*x + a)) + 1/8*((2*b^3*c^2 - 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*s 
qrt(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*c^2)* 
(a*b^2 - 4*a^2*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)*s 
qrt(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)*(a*b^2 - 4*a^2*c)^2*B + 2* 
(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 - 14*sqrt(2)*sqrt(b*c + sqr 
t(b^2 - 4*a*c)*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a* 
b^5*c - 2*a*b^6*c + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^2 
 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 + sqrt(2)*sqrt(b 
*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 28*a^2*b^4*c^2 - 96*sqrt(2)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a^4*c^3 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a^3*b*c^3 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 12 
8*a^3*b^2*c^3 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 192*a 
^4*c^4 + 2*(b^2 - 4*a*c)*a*b^4*c - 20*(b^2 - 4*a*c)*a^2*b^2*c^2 + 48*(b...
 

Mupad [B] (verification not implemented)

Time = 15.91 (sec) , antiderivative size = 12364, normalized size of antiderivative = 42.34 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((1536*A*a^5*c^6 + 4*A*a*b^8*c^2 - 256*B*a^5*b*c^5 - 72*A*a^2*b^6*c 
^3 + 480*A*a^3*b^4*c^4 - 1408*A*a^4*b^2*c^5 + 4*B*a^2*b^7*c^2 - 48*B*a^3*b 
^5*c^3 + 192*B*a^4*b^3*c^4)/(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4* 
b^2*c^2) - (2*x^(1/2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^2*(-(4*a*c - b^2)^ 
9)^(1/2) + B^2*a^2*(-(4*a*c - b^2)^9)^(1/2) + 2*A*B*a*b^10 + 288*A^2*a^2*b 
^7*c^2 - 1504*A^2*a^3*b^5*c^3 + 3840*A^2*a^4*b^3*c^4 - 96*B^2*a^4*b^5*c^2 
+ 512*B^2*a^5*b^3*c^3 + 3072*A*B*a^6*c^5 - 27*A^2*a*b^9*c - 9*A^2*a*c*(-(4 
*a*c - b^2)^9)^(1/2) - 3840*A^2*a^5*b*c^5 - 768*B^2*a^6*b*c^4 + 192*A*B*a^ 
3*b^6*c^2 - 128*A*B*a^4*b^4*c^3 - 1536*A*B*a^5*b^2*c^4 + 2*A*B*a*b*(-(4*a* 
c - b^2)^9)^(1/2) - 36*A*B*a^2*b^8*c)/(8*(a^3*b^12 + 4096*a^9*c^6 - 24*a^4 
*b^10*c + 240*a^5*b^8*c^2 - 1280*a^6*b^6*c^3 + 3840*a^7*b^4*c^4 - 6144*a^8 
*b^2*c^5)))^(1/2)*(256*a^5*b*c^5 - 4*a^2*b^7*c^2 + 48*a^3*b^5*c^3 - 192*a^ 
4*b^3*c^4))/(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c))*(-(A^2*b^11 + B^2*a^2*b^ 
9 + A^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*(-(4*a*c - b^2)^9)^(1/2) + 
2*A*B*a*b^10 + 288*A^2*a^2*b^7*c^2 - 1504*A^2*a^3*b^5*c^3 + 3840*A^2*a^4*b 
^3*c^4 - 96*B^2*a^4*b^5*c^2 + 512*B^2*a^5*b^3*c^3 + 3072*A*B*a^6*c^5 - 27* 
A^2*a*b^9*c - 9*A^2*a*c*(-(4*a*c - b^2)^9)^(1/2) - 3840*A^2*a^5*b*c^5 - 76 
8*B^2*a^6*b*c^4 + 192*A*B*a^3*b^6*c^2 - 128*A*B*a^4*b^4*c^3 - 1536*A*B*a^5 
*b^2*c^4 + 2*A*B*a*b*(-(4*a*c - b^2)^9)^(1/2) - 36*A*B*a^2*b^8*c)/(8*(a^3* 
b^12 + 4096*a^9*c^6 - 24*a^4*b^10*c + 240*a^5*b^8*c^2 - 1280*a^6*b^6*c^...
 

Reduce [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 2372, normalized size of antiderivative = 8.12 \[ \int \frac {A+B x}{\sqrt {x} \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*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))*a**2*b*c - 4*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))*a*b**3 + 8*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))*a*b**2*c*x + 8*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))*a*b*c**2*x**2 - 4*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqr 
t(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b**4*x 
 - 4*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**3*c*x**2 - 24*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**3*c + 10*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**2*b**2 - 24*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)*sqr 
t(a) + b))*a**2*b*c*x - 24*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 
**2*c**2*x**2 + 10*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*b**3...