\(\int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx\) [350]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}} \] Output:

2*c^2*(b*x+a)^(1/2)/d^2/(-a*d+b*c)/(d*x+c)^(1/2)+(b*x+a)^(1/2)*(d*x+c)^(1/ 
2)/b/d^2-(a*d+3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/ 
b^(3/2)/d^(5/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\sqrt {b} \sqrt {d} \sqrt {a+b x} (a d (c+d x)-b c (3 c+d x))+\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2} (-b c+a d) \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*(a*d*(c + d*x) - b*c*(3*c + d*x)) + (3*b^2* 
c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/( 
Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*d^(5/2)*(-(b*c) + a*d)*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 27, 90, 66, 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.

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(92)=184\).

Time = 0.27 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.92

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} d^{3} x +2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b c \,d^{2} x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c^{2} d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} c \,d^{2}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b \,c^{2} d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c^{3}-2 a \,d^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+2 b c d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-2 a c d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+6 b \,c^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\right )}{2 \sqrt {d b}\, b \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, d^{2} \sqrt {x d +c}}\)	\(439\)

Input:

int(x^2/(b*x+a)^(1/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x+a)^(1/2)*(ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+ 
a*d+b*c)/(d*b)^(1/2))*a^2*d^3*x+2*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b*c*d^2*x-3*ln(1/2*(2*b*d*x+2*((b*x+ 
a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^2*c^2*d*x+ln(1/2*(2* 
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*c*d^ 
2+2*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^( 
1/2))*a*b*c^2*d-3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a* 
d+b*c)/(d*b)^(1/2))*b^2*c^3-2*a*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+ 
2*b*c*d*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-2*a*c*d*((b*x+a)*(d*x+c))^(1 
/2)*(d*b)^(1/2)+6*b*c^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2))/(d*b)^(1/2)/b 
/(a*d-b*c)/((b*x+a)*(d*x+c))^(1/2)/d^2/(d*x+c)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (92) = 184\).

Time = 0.14 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.18 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}, \frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}\right ] \] Input:

integrate(x^2/(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*((3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2 - 
a^2*d^3)*x)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 
4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d 
 + a*b*d^2)*x) + 4*(3*b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*sqr 
t(b*x + a)*sqrt(d*x + c))/(b^3*c^2*d^3 - a*b^2*c*d^4 + (b^3*c*d^4 - a*b^2* 
d^5)*x), 1/2*((3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b* 
c*d^2 - a^2*d^3)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d) 
*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)* 
x)) + 2*(3*b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*sqrt(b*x + a)* 
sqrt(d*x + c))/(b^3*c^2*d^3 - a*b^2*c*d^4 + (b^3*c*d^4 - a*b^2*d^5)*x)]
 

Sympy [F]

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

integrate(x**2/(b*x+a)**(1/2)/(d*x+c)**(3/2),x)
 

Output:

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

Maxima [F(-2)]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (92) = 184\).

Time = 0.15 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.80 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {b x + a} {\left (\frac {{\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}} + \frac {3 \, b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b^{3} c + a b^{2} d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2} {\left | b \right |}}}{b^{2}} \] Input:

integrate(x^2/(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

(sqrt(b*x + a)*((b^5*c*d^2 - a*b^4*d^3)*(b*x + a)/(b^3*c*d^3*abs(b) - a*b^ 
2*d^4*abs(b)) + (3*b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3)/(b^3*c*d^3*abs 
(b) - a*b^2*d^4*abs(b)))/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) + (3*b^3*c + 
a*b^2*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a 
*b*d)))/(sqrt(b*d)*d^2*abs(b)))/b^2
 

Mupad [F(-1)]

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

int(x^2/((a + b*x)^(1/2)*(c + d*x)^(3/2)),x)
 

Output:

int(x^2/((a + b*x)^(1/2)*(c + d*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.13 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {4 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c \,d^{2}+4 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{3} x -12 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2} d -4 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x -4 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} c \,d^{2}-4 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{3} x -8 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d -8 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c \,d^{2} x +12 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3}+12 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2} d x -\sqrt {d}\, \sqrt {b}\, a^{2} c \,d^{2}-\sqrt {d}\, \sqrt {b}\, a^{2} d^{3} x +2 \sqrt {d}\, \sqrt {b}\, a b \,c^{2} d +2 \sqrt {d}\, \sqrt {b}\, a b c \,d^{2} x -9 \sqrt {d}\, \sqrt {b}\, b^{2} c^{3}-9 \sqrt {d}\, \sqrt {b}\, b^{2} c^{2} d x}{4 b^{2} d^{3} \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:

int(x^2/(b*x+a)^(1/2)/(d*x+c)^(3/2),x)
 

Output:

(4*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c*d**2 + 4*sqrt(c + d*x)*sqrt(a + b*x)* 
a*b*d**3*x - 12*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c**2*d - 4*sqrt(c + d*x)* 
sqrt(a + b*x)*b**2*c*d**2*x - 4*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) 
 + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*c*d**2 - 4*sqrt(d)*sqrt(b) 
*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2 
*d**3*x - 8*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + 
d*x))/sqrt(a*d - b*c))*a*b*c**2*d - 8*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a 
+ b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c*d**2*x + 12*sqrt(d) 
*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b* 
c))*b**2*c**3 + 12*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sq 
rt(c + d*x))/sqrt(a*d - b*c))*b**2*c**2*d*x - sqrt(d)*sqrt(b)*a**2*c*d**2 
- sqrt(d)*sqrt(b)*a**2*d**3*x + 2*sqrt(d)*sqrt(b)*a*b*c**2*d + 2*sqrt(d)*s 
qrt(b)*a*b*c*d**2*x - 9*sqrt(d)*sqrt(b)*b**2*c**3 - 9*sqrt(d)*sqrt(b)*b**2 
*c**2*d*x)/(4*b**2*d**3*(a*c*d + a*d**2*x - b*c**2 - b*c*d*x))