\(\int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx\) [151]

Optimal result

Integrand size = 23, antiderivative size = 167 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{24 a^3 x}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{7/2}} \] Output:

-1/3*A*(c*x^2+b*x+a)^(1/2)/a/x^3+1/12*(5*A*b-6*B*a)*(c*x^2+b*x+a)^(1/2)/a^ 
2/x^2-1/24*(-16*A*a*c+15*A*b^2-18*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^3/x+1/16*(- 
12*A*a*b*c+5*A*b^3+8*B*a^2*c-6*B*a*b^2)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x 
^2+b*x+a)^(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (-15 A b^2 x^2-4 a^2 (2 A+3 B x)+2 a x (5 A b+9 b B x+8 A c x)\right )}{x^3}-3 \left (5 A b^3+8 a^2 B c\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-18 a b (b B+2 A c) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{24 a^{7/2}} \] Input:

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

Output:

((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-15*A*b^2*x^2 - 4*a^2*(2*A + 3*B*x) + 2*a 
*x*(5*A*b + 9*b*B*x + 8*A*c*x)))/x^3 - 3*(5*A*b^3 + 8*a^2*B*c)*ArcTanh[(Sq 
rt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - 18*a*b*(b*B + 2*A*c)*ArcTanh[( 
-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(24*a^(7/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1237, 27, 1237, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {5 A b-6 a B+4 A c x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {5 A b-6 a B+4 A c x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {15 A b^2-18 a B b-16 a A c+2 (5 A b-6 a B) c x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{2 a x^2}}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {15 A b^2-18 a B b-16 a A c+2 (5 A b-6 a B) c x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{2 a x^2}}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {3 \left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{a x}}{4 a}-\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{2 a x^2}}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {\frac {3 \left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{a x}}{4 a}-\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{2 a x^2}}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {3 \left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{a x}}{4 a}-\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{2 a x^2}}{6 a}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}\)

Input:

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

Output:

-1/3*(A*Sqrt[a + b*x + c*x^2])/(a*x^3) - (-1/2*((5*A*b - 6*a*B)*Sqrt[a + b 
*x + c*x^2])/(a*x^2) - (-(((15*A*b^2 - 18*a*b*B - 16*a*A*c)*Sqrt[a + b*x + 
 c*x^2])/(a*x)) + (3*(5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTan 
h[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(4*a))/(6*a 
)
 

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 1237

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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-16 A a c \,x^{2}+15 x^{2} b^{2} A -18 B a \,x^{2} b -10 a b A x +12 a^{2} B x +8 a^{2} A \right )}{24 a^{3} x^{3}}-\frac {\left (12 A a b c -5 A \,b^{3}-8 B \,a^{2} c +6 B a \,b^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{16 a^{\frac {7}{2}}}\)	\(128\)
default	\(A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )+B \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )\)	\(331\)

Input:

int((B*x+A)/x^4/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(c*x^2+b*x+a)^(1/2)*(-16*A*a*c*x^2+15*A*b^2*x^2-18*B*a*b*x^2-10*A*a* 
b*x+12*B*a^2*x+8*A*a^2)/a^3/x^3-1/16*(12*A*a*b*c-5*A*b^3-8*B*a^2*c+6*B*a*b 
^2)/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.92 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\left [\frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {a} x^{3} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (8 \, A a^{3} - {\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{4} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (8 \, A a^{3} - {\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{4} x^{3}}\right ] \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/96*(3*(6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*sqrt(a)*x^3*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*(8*A*a^3 - (18*B*a^2*b - 15*A*a*b^2 + 16*A*a^2*c)*x^2 + 
 2*(6*B*a^3 - 5*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a))/(a^4*x^3), 1/48*(3*(6*B 
*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*sqrt(-a)*x^3*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*(8*A*a^3 
 - (18*B*a^2*b - 15*A*a*b^2 + 16*A*a^2*c)*x^2 + 2*(6*B*a^3 - 5*A*a^2*b)*x) 
*sqrt(c*x^2 + b*x + a))/(a^4*x^3)]
 

Sympy [F]

\[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{x^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((B*x+A)/x**4/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(x**4*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(1/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (145) = 290\).

Time = 0.25 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\frac {{\left (6 \, B a b^{2} - 5 \, A b^{3} - 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} - 48 \, A a^{3} b^{2} \sqrt {c} + 32 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{3}} \] Input:

integrate((B*x+A)/x^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/8*(6*B*a*b^2 - 5*A*b^3 - 8*B*a^2*c + 12*A*a*b*c)*arctan(-(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) - 1/24*(18*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^5*B*a*b^2 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A* 
b^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*c + 36*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^5*A*a*b*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 
*B*a^2*b^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3 - 96*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^2*B*a^3*b*sqrt(c) - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^3 
*c^(3/2) + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b^2 - 33*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))*B*a^4*c - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b*c + 48*B*a^4* 
b*sqrt(c) - 48*A*a^3*b^2*sqrt(c) + 32*A*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^2 - a)^3*a^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{x^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx=\frac {-16 \sqrt {c \,x^{2}+b x +a}\, a^{3}-4 \sqrt {c \,x^{2}+b x +a}\, a^{2} b x +32 \sqrt {c \,x^{2}+b x +a}\, a^{2} c \,x^{2}+6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x^{2}+12 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b c \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{3} x^{3}-12 \sqrt {a}\, \mathrm {log}\left (x \right ) a b c \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{3} x^{3}}{48 a^{3} x^{3}} \] Input:

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

Output:

( - 16*sqrt(a + b*x + c*x**2)*a**3 - 4*sqrt(a + b*x + c*x**2)*a**2*b*x + 3 
2*sqrt(a + b*x + c*x**2)*a**2*c*x**2 + 6*sqrt(a + b*x + c*x**2)*a*b**2*x** 
2 + 12*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b*c*x** 
3 + 3*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**3*x**3 
- 12*sqrt(a)*log(x)*a*b*c*x**3 - 3*sqrt(a)*log(x)*b**3*x**3)/(48*a**3*x**3 
)