\(\int \frac {A+B x}{x^2 (a+b x+c x^2)^{3/2}} \, dx\) [159]

Optimal result

Integrand size = 23, antiderivative size = 149 \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {A}{a x \sqrt {a+b x+c x^2}}+\frac {2 a B \left (b^2-2 a c\right )-A \left (3 b^3-10 a b c\right )-c \left (3 A b^2-2 a b B-8 a A c\right ) x}{a^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2}} \] Output:

-A/a/x/(c*x^2+b*x+a)^(1/2)+(2*a*B*(-2*a*c+b^2)-A*(-10*a*b*c+3*b^3)-c*(-8*A 
*a*c+3*A*b^2-2*B*a*b)*x)/a^2/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+1/2*(3*A*b-2 
*B*a)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \left (-4 a^2 c (A-B x)+3 A b^2 x (b+c x)-2 a b B x (b+c x)+a A \left (b^2-10 b c x-8 c^2 x^2\right )\right )}{x \sqrt {a+x (b+c x)}}+(3 A b-2 a B) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2} \left (-b^2+4 a c\right )} \] Input:

Integrate[(A + B*x)/(x^2*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

((Sqrt[a]*(-4*a^2*c*(A - B*x) + 3*A*b^2*x*(b + c*x) - 2*a*b*B*x*(b + c*x) 
+ a*A*(b^2 - 10*b*c*x - 8*c^2*x^2)))/(x*Sqrt[a + x*(b + c*x)]) + (3*A*b - 
2*a*B)*(b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]]) 
/(a^(5/2)*(-b^2 + 4*a*c))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1235, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {3 A b^2-2 a B b-8 a A c+2 (A b-2 a B) c x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 A b^2-2 a B b-8 a A c+2 (A b-2 a B) c x}{x^2 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {\left (b^2-4 a c\right ) (3 A b-2 a B) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a x}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) (3 A b-2 a B) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a x}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) (3 A b-2 a B) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a x}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

Int[(A + B*x)/(x^2*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x*Sqrt[ 
a + b*x + c*x^2]) + (-(((3*A*b^2 - 2*a*b*B - 8*a*A*c)*Sqrt[a + b*x + c*x^2 
])/(a*x)) + ((3*A*b - 2*a*B)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]* 
Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

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] 
)
 

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.63

method	result	size
default	\(A \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+B \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )\)	\(243\)
risch	\(-\frac {A \sqrt {c \,x^{2}+b x +a}}{a^{2} x}-\frac {\frac {2 b^{2} A \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+a \left (3 A b -2 B a \right ) \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )+A b c \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {4 A a c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{2 a^{2}}\)	\(256\)

Input:

int((B*x+A)/x^2/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

A*(-1/a/x/(c*x^2+b*x+a)^(1/2)-3/2*b/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+ 
b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+ 
b*x+a)^(1/2))/x))-4*c/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+B*(1/a/ 
(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2 
)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (134) = 268\).

Time = 0.30 (sec) , antiderivative size = 657, normalized size of antiderivative = 4.41 \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} - {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} - {\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {a} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (A a^{2} b^{2} - 4 \, A a^{3} c - {\left (8 \, A a^{2} c^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}, -\frac {{\left ({\left (4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} - {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} - {\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (A a^{2} b^{2} - 4 \, A a^{3} c - {\left (8 \, A a^{2} c^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}\right ] \] Input:

integrate((B*x+A)/x^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(((4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*x^3 - (2*B*a* 
b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3*A*a*b^ 
3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*x)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^ 
2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(A*a^2*b 
^2 - 4*A*a^3*c - (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2* 
b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a 
^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c 
)*x), -1/2*(((4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*x^3 - ( 
2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3* 
A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*x)*sqrt(-a)*arctan(1/2*sqrt(c*x^2 + b 
*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) + 2*(A*a^2*b^2 - 4*A 
*a^3*c - (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3* 
A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^3*b^2*c 
 - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)]
 

Sympy [F]

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

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

Output:

Integral((A + B*x)/(x**2*(a + b*x + c*x**2)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/x^2/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (B a^{3} b c - A a^{2} b^{2} c + 2 \, A a^{3} c^{2}\right )} x}{a^{4} b^{2} - 4 \, a^{5} c} + \frac {B a^{3} b^{2} - A a^{2} b^{3} - 2 \, B a^{4} c + 3 \, A a^{3} b c}{a^{4} b^{2} - 4 \, a^{5} c}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )} a^{2}} \] Input:

integrate((B*x+A)/x^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x)/(x^2*(a + b*x + c*x^2)^(3/2)),x)
 

Output:

int((A + B*x)/(x^2*(a + b*x + c*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.03 \[ \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-8 \sqrt {c \,x^{2}+b x +a}\, a^{3} c +2 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2}-12 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c x -16 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2} x^{2}+2 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x +2 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c \,x^{2}+4 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b c x -\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{3} x +4 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{2} c \,x^{2}+4 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b \,c^{2} x^{3}-\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{4} x^{2}-\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{3} c \,x^{3}-4 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b c x +\sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{3} x -4 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c \,x^{2}-4 \sqrt {a}\, \mathrm {log}\left (x \right ) a b \,c^{2} x^{3}+\sqrt {a}\, \mathrm {log}\left (x \right ) b^{4} x^{2}+\sqrt {a}\, \mathrm {log}\left (x \right ) b^{3} c \,x^{3}}{2 a^{2} x \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:

int((B*x+A)/x^2/(c*x^2+b*x+a)^(3/2),x)
 

Output:

( - 8*sqrt(a + b*x + c*x**2)*a**3*c + 2*sqrt(a + b*x + c*x**2)*a**2*b**2 - 
 12*sqrt(a + b*x + c*x**2)*a**2*b*c*x - 16*sqrt(a + b*x + c*x**2)*a**2*c** 
2*x**2 + 2*sqrt(a + b*x + c*x**2)*a*b**3*x + 2*sqrt(a + b*x + c*x**2)*a*b* 
*2*c*x**2 + 4*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x) 
*a**2*b*c*x - sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x) 
*a*b**3*x + 4*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x) 
*a*b**2*c*x**2 + 4*sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - 
 b*x)*a*b*c**2*x**3 - sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2* 
a - b*x)*b**4*x**2 - sqrt(a)*log( - 2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
 - b*x)*b**3*c*x**3 - 4*sqrt(a)*log(x)*a**2*b*c*x + sqrt(a)*log(x)*a*b**3* 
x - 4*sqrt(a)*log(x)*a*b**2*c*x**2 - 4*sqrt(a)*log(x)*a*b*c**2*x**3 + sqrt 
(a)*log(x)*b**4*x**2 + sqrt(a)*log(x)*b**3*c*x**3)/(2*a**2*x*(4*a**2*c - a 
*b**2 + 4*a*b*c*x + 4*a*c**2*x**2 - b**3*x - b**2*c*x**2))