\(\int \frac {(a+\frac {b}{c+d x})^{3/2}}{x} \, dx\) [24]

Optimal result

Integrand size = 19, antiderivative size = 98 \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{x} \, dx=\frac {2 b \sqrt {a+\frac {b}{c+d x}}}{c}+2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x}}}{\sqrt {a}}\right )-\frac {2 (b+a c)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x}}}{\sqrt {b+a c}}\right )}{c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{x} \, dx=\frac {2 b \sqrt {\frac {b+a c+a d x}{c+d x}}}{c}-\frac {2 (-b-a c)^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {-b-a c} \sqrt {\frac {b+a c+a d x}{c+d x}}}{b+a c}\right )}{c^{3/2}}+2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x}{c+d x}}}{\sqrt {a}}\right ) \] Input:

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

Output:

(2*b*Sqrt[(b + a*c + a*d*x)/(c + d*x)])/c - (2*(-b - a*c)^(3/2)*ArcTan[(Sq 
rt[c]*Sqrt[-b - a*c]*Sqrt[(b + a*c + a*d*x)/(c + d*x)])/(b + a*c)])/c^(3/2 
) + 2*a^(3/2)*ArcTanh[Sqrt[(b + a*c + a*d*x)/(c + d*x)]/Sqrt[a]]
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {896, 25, 941, 948, 25, 95, 25, 174, 73, 221}

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.

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

\(\Big \downarrow \) 896

\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{d x}d(c+d x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int -\frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{d x}d(c+d x)\)

\(\Big \downarrow \) 941

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 95

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*b*Sqrt[a + b/(c + d*x)])/c - (-2*a^(3/2)*c*ArcTanh[Sqrt[a + b/(c + d*x) 
]/Sqrt[a]] + (2*(b + a*c)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b/(c + d*x)])/Sq 
rt[b + a*c]])/Sqrt[c])/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 95

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p - 1)/(b*d*(p - 1))), x] + Simp[1/(b*d)   Int[(b 
*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*((e + f*x)^(p - 2)/((a + 
b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

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

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 941

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym 
bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, 
 n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !IntegerQ[p])
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2583\) vs. \(2(80)=160\).

Time = 0.14 (sec) , antiderivative size = 2584, normalized size of antiderivative = 26.37

method	result	size
default	\(\text {Expression too large to display}\)	\(2584\)

Input:

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

Output:

1/2*(-3*((a*c+b)*c)^(1/2)*ln(1/2*(2*a*d^2*x+2*a*c*d+2*((a*d*x+a*c+b)*(d*x+ 
c))^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a*b*c^3*d+3*ln(1/2*(2*a*d^2*x+ 
2*a*c*d+2*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/( 
a*d^2)^(1/2))*((a*c+b)*c)^(1/2)*a*b*c*d^3*x^2-4*(a*d^2)^(1/2)*ln((2*a*d*x* 
c+2*a*c^2+b*d*x+2*((a*c+b)*c)^(1/2)*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^ 
(1/2)+2*b*c)/x)*a*b*c^2*d^2*x^2-3*((a*c+b)*c)^(1/2)*ln(1/2*(2*a*d^2*x+2*a* 
c*d+2*((a*d*x+a*c+b)*(d*x+c))^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a*b* 
c*d^3*x^2+2*ln(1/2*(2*a*d^2*x+2*a*c*d+2*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b 
*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*((a*c+b)*c)^(1/2)*b^2*c*d^2*x+ 
2*ln(1/2*(2*a*d^2*x+2*a*c*d+2*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)* 
(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*((a*c+b)*c)^(1/2)*a^2*c^2*d^3*x^2+4*(a*d 
^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)*(a*d^2)^(1/2)*((a*c+b)*c)^(1/2)*a* 
c^2*d*x-12*(a*d^2)^(1/2)*((a*c+b)*c)^(1/2)*((a*d*x+a*c+b)*(d*x+c))^(1/2)*a 
*c^2*d*x+4*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)*(a*d^2)^(1/2)*((a*c 
+b)*c)^(1/2)*b*c*d*x+ln(1/2*(2*a*d^2*x+2*a*c*d+2*(a*d^2*x^2+2*a*c*d*x+a*c^ 
2+b*d*x+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*((a*c+b)*c)^(1/2)*b^2 
*c^2*d-2*(a*d^2)^(1/2)*ln((2*a*d*x*c+2*a*c^2+b*d*x+2*((a*c+b)*c)^(1/2)*(a* 
d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)+2*b*c)/x)*b^2*c*d^2*x^2+2*(a*d^2* 
x^2+2*a*c*d*x+a*c^2+b*d*x+b*c)^(1/2)*(a*d^2)^(1/2)*((a*c+b)*c)^(1/2)*b*d^2 
*x^2+3*ln(1/2*(2*a*d^2*x+2*a*c*d+2*(a*d^2*x^2+2*a*c*d*x+a*c^2+b*d*x+b*c...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 591, normalized size of antiderivative = 6.03 \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{x} \, dx=\left [\frac {a^{\frac {3}{2}} c \log \left (2 \, a d x + 2 \, a c + 2 \, {\left (d x + c\right )} \sqrt {a} \sqrt {\frac {a d x + a c + b}{d x + c}} + b\right ) + {\left (a c + b\right )} \sqrt {\frac {a c + b}{c}} \log \left (-\frac {2 \, a c^{2} + {\left (2 \, a c + b\right )} d x + 2 \, b c - 2 \, {\left (c d x + c^{2}\right )} \sqrt {\frac {a d x + a c + b}{d x + c}} \sqrt {\frac {a c + b}{c}}}{x}\right ) + 2 \, b \sqrt {\frac {a d x + a c + b}{d x + c}}}{c}, -\frac {2 \, \sqrt {-a} a c \arctan \left (\frac {{\left (d x + c\right )} \sqrt {-a} \sqrt {\frac {a d x + a c + b}{d x + c}}}{a d x + a c + b}\right ) - {\left (a c + b\right )} \sqrt {\frac {a c + b}{c}} \log \left (-\frac {2 \, a c^{2} + {\left (2 \, a c + b\right )} d x + 2 \, b c - 2 \, {\left (c d x + c^{2}\right )} \sqrt {\frac {a d x + a c + b}{d x + c}} \sqrt {\frac {a c + b}{c}}}{x}\right ) - 2 \, b \sqrt {\frac {a d x + a c + b}{d x + c}}}{c}, \frac {a^{\frac {3}{2}} c \log \left (2 \, a d x + 2 \, a c + 2 \, {\left (d x + c\right )} \sqrt {a} \sqrt {\frac {a d x + a c + b}{d x + c}} + b\right ) + 2 \, {\left (a c + b\right )} \sqrt {-\frac {a c + b}{c}} \arctan \left (\frac {c \sqrt {\frac {a d x + a c + b}{d x + c}} \sqrt {-\frac {a c + b}{c}}}{a c + b}\right ) + 2 \, b \sqrt {\frac {a d x + a c + b}{d x + c}}}{c}, -\frac {2 \, {\left (\sqrt {-a} a c \arctan \left (\frac {{\left (d x + c\right )} \sqrt {-a} \sqrt {\frac {a d x + a c + b}{d x + c}}}{a d x + a c + b}\right ) - {\left (a c + b\right )} \sqrt {-\frac {a c + b}{c}} \arctan \left (\frac {c \sqrt {\frac {a d x + a c + b}{d x + c}} \sqrt {-\frac {a c + b}{c}}}{a c + b}\right ) - b \sqrt {\frac {a d x + a c + b}{d x + c}}\right )}}{c}\right ] \] Input:

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

Output:

[(a^(3/2)*c*log(2*a*d*x + 2*a*c + 2*(d*x + c)*sqrt(a)*sqrt((a*d*x + a*c + 
b)/(d*x + c)) + b) + (a*c + b)*sqrt((a*c + b)/c)*log(-(2*a*c^2 + (2*a*c + 
b)*d*x + 2*b*c - 2*(c*d*x + c^2)*sqrt((a*d*x + a*c + b)/(d*x + c))*sqrt((a 
*c + b)/c))/x) + 2*b*sqrt((a*d*x + a*c + b)/(d*x + c)))/c, -(2*sqrt(-a)*a* 
c*arctan((d*x + c)*sqrt(-a)*sqrt((a*d*x + a*c + b)/(d*x + c))/(a*d*x + a*c 
 + b)) - (a*c + b)*sqrt((a*c + b)/c)*log(-(2*a*c^2 + (2*a*c + b)*d*x + 2*b 
*c - 2*(c*d*x + c^2)*sqrt((a*d*x + a*c + b)/(d*x + c))*sqrt((a*c + b)/c))/ 
x) - 2*b*sqrt((a*d*x + a*c + b)/(d*x + c)))/c, (a^(3/2)*c*log(2*a*d*x + 2* 
a*c + 2*(d*x + c)*sqrt(a)*sqrt((a*d*x + a*c + b)/(d*x + c)) + b) + 2*(a*c 
+ b)*sqrt(-(a*c + b)/c)*arctan(c*sqrt((a*d*x + a*c + b)/(d*x + c))*sqrt(-( 
a*c + b)/c)/(a*c + b)) + 2*b*sqrt((a*d*x + a*c + b)/(d*x + c)))/c, -2*(sqr 
t(-a)*a*c*arctan((d*x + c)*sqrt(-a)*sqrt((a*d*x + a*c + b)/(d*x + c))/(a*d 
*x + a*c + b)) - (a*c + b)*sqrt(-(a*c + b)/c)*arctan(c*sqrt((a*d*x + a*c + 
 b)/(d*x + c))*sqrt(-(a*c + b)/c)/(a*c + b)) - b*sqrt((a*d*x + a*c + b)/(d 
*x + c)))/c]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate((a + b/(d*x + c))^(3/2)/x, x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 753, normalized size of antiderivative = 7.68 \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^{3/2}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

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