\(\int \frac {\sqrt {x} (A+B x)}{(a+b x+c x^2)^2} \, dx\) [94]

Optimal result

Integrand size = 23, antiderivative size = 276 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {x} (A b-2 a B-(b B-2 A c) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b B-2 A c-\frac {b^2 B-4 A b c+4 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b B-2 A c+\frac {b^2 B-4 A b c+4 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\sqrt {x} (-A b+2 a B+b B x-2 A c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (-b^2 B+4 A b c-4 a B c+b B \sqrt {b^2-4 a c}-2 A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^2 B-4 A b c+4 a B c+b B \sqrt {b^2-4 a c}-2 A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Input:

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

Output:

(Sqrt[x]*(-(A*b) + 2*a*B + b*B*x - 2*A*c*x))/((b^2 - 4*a*c)*(a + b*x + c*x 
^2)) + ((-(b^2*B) + 4*A*b*c - 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 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]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (( 
b^2*B - 4*A*b*c + 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 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]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1234, 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.

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

\(\Big \downarrow \) 1234

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

-((Sqrt[x]*(A*b - 2*a*B - (b*B - 2*A*c)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^ 
2))) + (((b*B - 2*A*c - (b^2*B - 4*A*b*c + 4*a*B*c)/Sqrt[b^2 - 4*a*c])*Arc 
Tan[(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 - 2*A*c + (b^2*B - 4*A*b*c + 4*a*B 
*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^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 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (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 = 1.26 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {\frac {2 \left (2 A c -B b \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (A b -2 B a \right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (\frac {\left (2 A c \sqrt {-4 a c +b^{2}}+4 A b c -B b \sqrt {-4 a c +b^{2}}-4 a B c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (2 A c \sqrt {-4 a c +b^{2}}-4 A b c -B b \sqrt {-4 a c +b^{2}}+4 a B c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\)	\(295\)
default	\(\frac {\frac {2 \left (2 A c -B b \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (A b -2 B a \right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (\frac {\left (2 A c \sqrt {-4 a c +b^{2}}+4 A b c -B b \sqrt {-4 a c +b^{2}}-4 a B c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (2 A c \sqrt {-4 a c +b^{2}}-4 A b c -B b \sqrt {-4 a c +b^{2}}+4 a B c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\)	\(295\)

Input:

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

Output:

2*(1/2*(2*A*c-B*b)/(4*a*c-b^2)*x^(3/2)+1/2*(A*b-2*B*a)/(4*a*c-b^2)*x^(1/2) 
)/(c*x^2+b*x+a)+4/(4*a*c-b^2)*c*(1/8*(2*A*c*(-4*a*c+b^2)^(1/2)+4*A*b*c-B*b 
*(-4*a*c+b^2)^(1/2)-4*a*B*c-B*b^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))-1/8*(2*A*c*(-4*a*c+b^2)^(1/2)-4*A*b*c-B*b*(-4*a*c+b^2)^(1/2)+4*a*B 
*c+B*b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a 
rctanh(c*x^(1/2)*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. 3462 vs. \(2 (235) = 470\).

Time = 2.14 (sec) , antiderivative size = 3462, normalized size of antiderivative = 12.54 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3781 vs. \(2 (235) = 470\).

Time = 0.98 (sec) , antiderivative size = 3781, normalized size of antiderivative = 13.70 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(B*b*x^(3/2) - 2*A*c*x^(3/2) + 2*B*a*sqrt(x) - A*b*sqrt(x))/((c*x^2 + b*x 
+ a)*(b^2 - 4*a*c)) + 1/8*(2*(2*b^2*c^3 - 8*a*c^4 - sqrt(2)*sqrt(b^2 - 4*a 
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sq 
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( 
b^2 - 4*a*c)*c)*c^3 - 2*(b^2 - 4*a*c)*c^3)*(b^2 - 4*a*c)^2*A - (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*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*c^2)*(b^2 - 4*a*c)^2*B + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5* 
c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 2*sqrt(2)*sqrt(b 
*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 2*b^5*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*a^2*b*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b 
^2*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 16*a*b^3*c^3 - 
4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - 32*a^2*b*c^4 + 2*(b^2 
- 4*a*c)*b^3*c^2 - 8*(b^2 - 4*a*c)*a*b*c^3)*A*abs(b^2 - 4*a*c) - 4*(sqrt(2 
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a^2*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3* 
c^2 - 2*a*b^4*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 ...
 

Mupad [B] (verification not implemented)

Time = 15.45 (sec) , antiderivative size = 9434, normalized size of antiderivative = 34.18 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^(1/2)*(A*b - 2*B*a))/(4*a*c - b^2) + (x^(3/2)*(2*A*c - B*b))/(4*a*c - 
b^2))/(a + b*x + c*x^2) - atan(((((4*A*b^7*c^2 + 512*B*a^4*c^5 - 48*A*a*b^ 
5*c^3 - 256*A*a^3*b*c^5 - 8*B*a*b^6*c^2 + 192*A*a^2*b^3*c^4 + 96*B*a^2*b^4 
*c^3 - 384*B*a^3*b^2*c^4)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) 
 - (2*x^(1/2)*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9*c + 
A^2*c*(-(4*a*c - b^2)^9)^(1/2) - 96*A^2*a^2*b^5*c^3 + 512*A^2*a^3*b^3*c^4 
- 96*B^2*a^3*b^5*c^2 + 512*B^2*a^4*b^3*c^3 + 1024*A*B*a^5*c^5 - 768*A^2*a^ 
4*b*c^5 - 768*B^2*a^5*b*c^4 + 128*A*B*a^2*b^6*c^2 - 384*A*B*a^3*b^4*c^3 - 
12*A*B*a*b^8*c)/(8*(4096*a^7*c^7 - 24*a^2*b^10*c^2 + 240*a^3*b^8*c^3 - 128 
0*a^4*b^6*c^4 + 3840*a^5*b^4*c^5 - 6144*a^6*b^2*c^6 + a*b^12*c)))^(1/2)*(4 
*b^7*c^2 - 48*a*b^5*c^3 - 256*a^3*b*c^5 + 192*a^2*b^3*c^4))/(b^4 + 16*a^2* 
c^2 - 8*a*b^2*c))*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9* 
c + A^2*c*(-(4*a*c - b^2)^9)^(1/2) - 96*A^2*a^2*b^5*c^3 + 512*A^2*a^3*b^3* 
c^4 - 96*B^2*a^3*b^5*c^2 + 512*B^2*a^4*b^3*c^3 + 1024*A*B*a^5*c^5 - 768*A^ 
2*a^4*b*c^5 - 768*B^2*a^5*b*c^4 + 128*A*B*a^2*b^6*c^2 - 384*A*B*a^3*b^4*c^ 
3 - 12*A*B*a*b^8*c)/(8*(4096*a^7*c^7 - 24*a^2*b^10*c^2 + 240*a^3*b^8*c^3 - 
 1280*a^4*b^6*c^4 + 3840*a^5*b^4*c^5 - 6144*a^6*b^2*c^6 + a*b^12*c)))^(1/2 
) - (2*x^(1/2)*(B^2*b^4*c - 8*A^2*a*c^4 + 10*A^2*b^2*c^3 + 8*B^2*a^2*c^3 - 
 6*A*B*b^3*c^2 + 2*B^2*a*b^2*c^2 - 8*A*B*a*b*c^3))/(b^4 + 16*a^2*c^2 - 8*a 
*b^2*c))*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9*c + A^...
 

Reduce [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 1813, normalized size of antiderivative = 6.57 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^(1/2)*(B*x+A)/(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*c**2 + 6*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 - 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 - 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*c**3*x**2 + 6*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 + 6*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**2*c**2*x**2 - 
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)*sqrt(a) + b))*a*b**3 - 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)*sqrt(a) + b))*b**4*x - 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)* 
sqrt(a) + b))*b**3*c*x**2 + 8*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sq 
rt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b) 
)*a**2*c**2 - 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s 
qrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**2*c ...