\(\int \frac {x \sqrt {a x+b x^2}}{(c+d x)^2} \, dx\) [39]

Optimal result

Integrand size = 22, antiderivative size = 148 \[ \int \frac {x \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\frac {2 \sqrt {a x+b x^2}}{d^2}-\frac {x \sqrt {a x+b x^2}}{d (c+d x)}-\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^3 \sqrt {b c-a d}} \] Output:

2*(b*x^2+a*x)^(1/2)/d^2-x*(b*x^2+a*x)^(1/2)/d/(d*x+c)-(-a*d+4*b*c)*arctanh 
(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(1/2)/d^3+c^(1/2)*(-3*a*d+4*b*c)*arctanh(( 
-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/d^3/(-a*d+b*c)^(1/2)
 

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.30 \[ \int \frac {x \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {d (-b c+a d) \sqrt {x} (2 c+d x)}{c+d x}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {1+\frac {b x}{a}}}-\frac {\sqrt {c} (4 b c-3 a d) \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a+b x}}\right )}{d^3 (-b c+a d) \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(a + b*x)]*((d*(-(b*c) + a*d)*Sqrt[x]*(2*c + d*x))/(c + d*x) + ((4 
*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[ 
a]*Sqrt[b]*Sqrt[1 + (b*x)/a]) - (Sqrt[c]*(4*b*c - 3*a*d)*Sqrt[b*c - a*d]*A 
rcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/Sqrt[a + b*x])) 
/(d^3*(-(b*c) + a*d)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1230, 1269, 1091, 219, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {\sqrt {a x+b x^2} (2 c+d x)}{d^2 (c+d x)}-\frac {\int \frac {2 a c+(4 b c-a d) x}{(c+d x) \sqrt {b x^2+a x}}dx}{2 d^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\sqrt {a x+b x^2} (2 c+d x)}{d^2 (c+d x)}-\frac {\frac {(4 b c-a d) \int \frac {1}{\sqrt {b x^2+a x}}dx}{d}-\frac {c (4 b c-3 a d) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{2 d^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\sqrt {a x+b x^2} (2 c+d x)}{d^2 (c+d x)}-\frac {\frac {2 (4 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{d}-\frac {c (4 b c-3 a d) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{2 d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a x+b x^2} (2 c+d x)}{d^2 (c+d x)}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (4 b c-a d)}{\sqrt {b} d}-\frac {c (4 b c-3 a d) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{2 d^2}\)

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*c + d*x)*Sqrt[a*x + b*x^2])/(d^2*(c + d*x)) - ((2*(4*b*c - a*d)*ArcTan 
h[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*d) - (Sqrt[c]*(4*b*c - 3*a*d)*A 
rcTanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2 
])])/(d*Sqrt[b*c - a*d]))/(2*d^2)
 

Defintions of rubi rules used

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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {4 \left (\left (d x +c \right ) \left (b c -\frac {3 a d}{4}\right ) \sqrt {b}\, c \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )-\frac {\left (\frac {\left (d x +c \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{2}+d \left (\frac {d x}{2}+c \right ) \sqrt {x \left (b x +a \right )}\, \sqrt {b}\right ) \sqrt {c \left (a d -b c \right )}}{2}\right )}{\sqrt {b}\, \sqrt {c \left (a d -b c \right )}\, d^{3} \left (d x +c \right )}\)	\(141\)
risch	\(\frac {x \left (b x +a \right )}{d^{2} \sqrt {x \left (b x +a \right )}}+\frac {\frac {\left (a d -4 b c \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {2 c \left (2 a d -3 b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {2 c^{2} \left (a d -b c \right ) \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}}{2 d^{2}}\)	\(447\)
default	\(\frac {\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{d^{2}}-\frac {c \left (\frac {d^{2} \left (b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}\right )^{\frac {3}{2}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{2 c \left (a d -b c \right )}-\frac {2 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )+\frac {a d -2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{4 b}+\frac {\left (-\frac {4 b c \left (a d -b c \right )}{d^{2}}-\frac {\left (a d -2 b c \right )^{2}}{d^{2}}\right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{c \left (a d -b c \right )}\right )}{d^{3}}\)	\(875\)

Input:

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

Output:

-4/b^(1/2)/(c*(a*d-b*c))^(1/2)*((d*x+c)*(b*c-3/4*a*d)*b^(1/2)*c*arctan((x* 
(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))-1/2*(1/2*(d*x+c)*(a*d-4*b*c)*arcta 
nh((x*(b*x+a))^(1/2)/x/b^(1/2))+d*(1/2*d*x+c)*(x*(b*x+a))^(1/2)*b^(1/2))*( 
c*(a*d-b*c))^(1/2))/d^3/(d*x+c)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 797, normalized size of antiderivative = 5.39 \[ \int \frac {x \sqrt {a x+b x^2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*((4*b*c^2 - a*c*d + (4*b*c*d - a*d^2)*x)*sqrt(b)*log(2*b*x + a + 2*s 
qrt(b*x^2 + a*x)*sqrt(b)) + (4*b^2*c^2 - 3*a*b*c*d + (4*b^2*c*d - 3*a*b*d^ 
2)*x)*sqrt(c/(b*c - a*d))*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x) 
*(b*c - a*d)*sqrt(c/(b*c - a*d)))/(d*x + c)) - 2*(b*d^2*x + 2*b*c*d)*sqrt( 
b*x^2 + a*x))/(b*d^4*x + b*c*d^3), 1/2*(2*(4*b^2*c^2 - 3*a*b*c*d + (4*b^2* 
c*d - 3*a*b*d^2)*x)*sqrt(-c/(b*c - a*d))*arctan(-sqrt(b*x^2 + a*x)*(b*c - 
a*d)*sqrt(-c/(b*c - a*d))/(b*c*x + a*c)) - (4*b*c^2 - a*c*d + (4*b*c*d - a 
*d^2)*x)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(b*d^2*x 
 + 2*b*c*d)*sqrt(b*x^2 + a*x))/(b*d^4*x + b*c*d^3), 1/2*(2*(4*b*c^2 - a*c* 
d + (4*b*c*d - a*d^2)*x)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + 
 a)) - (4*b^2*c^2 - 3*a*b*c*d + (4*b^2*c*d - 3*a*b*d^2)*x)*sqrt(c/(b*c - a 
*d))*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(c/( 
b*c - a*d)))/(d*x + c)) + 2*(b*d^2*x + 2*b*c*d)*sqrt(b*x^2 + a*x))/(b*d^4* 
x + b*c*d^3), ((4*b^2*c^2 - 3*a*b*c*d + (4*b^2*c*d - 3*a*b*d^2)*x)*sqrt(-c 
/(b*c - a*d))*arctan(-sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(-c/(b*c - a*d))/( 
b*c*x + a*c)) + (4*b*c^2 - a*c*d + (4*b*c*d - a*d^2)*x)*sqrt(-b)*arctan(sq 
rt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (b*d^2*x + 2*b*c*d)*sqrt(b*x^2 + a*x 
))/(b*d^4*x + b*c*d^3)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

Integral(x*sqrt(x*(a + b*x))/(c + d*x)**2, x)
 

Maxima [F(-2)]

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

integrate(x*(b*x^2+a*x)^(1/2)/(d*x+c)^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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 714, normalized size of antiderivative = 4.82 \[ \int \frac {x \sqrt {a x+b x^2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - 
 sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b*c*d + 3*sqrt(c)*sqrt(a*d 
- b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqr 
t(b))/(sqrt(c)*sqrt(b)))*a*b*d**2*x - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt 
(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sq 
rt(b)))*b**2*c**2 - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt 
(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**2*c*d*x 
 + 3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) 
 + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b*c*d + 3*sqrt(c)*sqrt(a* 
d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*s 
qrt(b))/(sqrt(c)*sqrt(b)))*a*b*d**2*x - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sq 
rt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)* 
sqrt(b)))*b**2*c**2 - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sq 
rt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**2*c*d 
*x + 2*sqrt(x)*sqrt(a + b*x)*a*b*c*d**2 + sqrt(x)*sqrt(a + b*x)*a*b*d**3*x 
 - 2*sqrt(x)*sqrt(a + b*x)*b**2*c**2*d - sqrt(x)*sqrt(a + b*x)*b**2*c*d**2 
*x + sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*c*d**2 + 
sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*d**3*x - 5*sqr 
t(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b*c**2*d - 5*sqrt(b) 
*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b*c*d**2*x + 4*sqrt(b...