\(\int \frac {\sqrt {e x}}{(c+d x) (a x+b x^2)^{3/2}} \, dx\) [190]

Optimal result

Integrand size = 28, antiderivative size = 155 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 b \sqrt {e x}}{a (b c-a d) \sqrt {a x+b x^2}}+\frac {2 d^{3/2} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e} \sqrt {a x+b x^2}}{\sqrt {b c-a d} \sqrt {e x}}\right )}{c (b c-a d)^{3/2}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {e x}}\right )}{a^{3/2} c} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {e x} \left (\sqrt {a} \left (b c \sqrt {b c-a d}+a d^{3/2} \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )-(b c-a d)^{3/2} \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{a^{3/2} c (b c-a d)^{3/2} \sqrt {x (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1261, 96, 25, 174, 73, 218, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 96

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 96

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S 
imp[1/((b*e - a*f)*(d*e - c*f))   Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e 
 + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, 
 x] && LtQ[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 218

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

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 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {2 \sqrt {e x}\, \sqrt {x \left (b x +a \right )}\, \left (d^{2} \operatorname {arctanh}\left (\frac {d \sqrt {\left (b x +a \right ) e}}{\sqrt {e \left (a d -b c \right ) d}}\right ) a \sqrt {a e}\, \sqrt {\left (b x +a \right ) e}-\operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, \sqrt {\left (b x +a \right ) e}\, a d +\operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, \sqrt {\left (b x +a \right ) e}\, b c -b c \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\right )}{x \left (b x +a \right ) a c \sqrt {a e}\, \left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\)	\(211\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 967, normalized size of antiderivative = 6.24 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (123) = 246\).

Time = 0.32 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.41 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=2 \, {\left (\frac {d^{2} \arctan \left (\frac {\sqrt {b e x + a e} d}{\sqrt {b c d e - a d^{2} e}}\right )}{{\left (b c^{2} e^{2} {\left | e \right |} - a c d e^{2} {\left | e \right |}\right )} \sqrt {b c d e - a d^{2} e}} + \frac {b}{{\left (a b c e^{2} {\left | e \right |} - a^{2} d e^{2} {\left | e \right |}\right )} \sqrt {b e x + a e}} + \frac {\arctan \left (\frac {\sqrt {b e x + a e}}{\sqrt {-a e}}\right )}{\sqrt {-a e} a c e^{2} {\left | e \right |}}\right )} e^{4} - \frac {2 \, {\left (\sqrt {a e} \sqrt {-a e} a d^{2} e^{2} \arctan \left (\frac {\sqrt {a e} d}{\sqrt {b c d e - a d^{2} e}}\right ) + \sqrt {b c d e - a d^{2} e} \sqrt {a e} b c e^{2} \arctan \left (\frac {\sqrt {a e}}{\sqrt {-a e}}\right ) - \sqrt {b c d e - a d^{2} e} \sqrt {a e} a d e^{2} \arctan \left (\frac {\sqrt {a e}}{\sqrt {-a e}}\right ) + \sqrt {b c d e - a d^{2} e} \sqrt {-a e} b c e^{2}\right )}}{\sqrt {b c d e - a d^{2} e} \sqrt {a e} \sqrt {-a e} a b c^{2} {\left | e \right |} - \sqrt {b c d e - a d^{2} e} \sqrt {a e} \sqrt {-a e} a^{2} c d {\left | e \right |}} \] Input:

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

Output:

2*(d^2*arctan(sqrt(b*e*x + a*e)*d/sqrt(b*c*d*e - a*d^2*e))/((b*c^2*e^2*abs 
(e) - a*c*d*e^2*abs(e))*sqrt(b*c*d*e - a*d^2*e)) + b/((a*b*c*e^2*abs(e) - 
a^2*d*e^2*abs(e))*sqrt(b*e*x + a*e)) + arctan(sqrt(b*e*x + a*e)/sqrt(-a*e) 
)/(sqrt(-a*e)*a*c*e^2*abs(e)))*e^4 - 2*(sqrt(a*e)*sqrt(-a*e)*a*d^2*e^2*arc 
tan(sqrt(a*e)*d/sqrt(b*c*d*e - a*d^2*e)) + sqrt(b*c*d*e - a*d^2*e)*sqrt(a* 
e)*b*c*e^2*arctan(sqrt(a*e)/sqrt(-a*e)) - sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e 
)*a*d*e^2*arctan(sqrt(a*e)/sqrt(-a*e)) + sqrt(b*c*d*e - a*d^2*e)*sqrt(-a*e 
)*b*c*e^2)/(sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e)*sqrt(-a*e)*a*b*c^2*abs(e) - 
sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e)*sqrt(-a*e)*a^2*c*d*abs(e))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {d}\, \sqrt {b x +a}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a^{2} d +\sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} d^{2}-2 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a b c d +\sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{2} c^{2}-\sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} d^{2}+2 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a b c d -\sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{2} c^{2}-2 a^{2} b c d +2 a \,b^{2} c^{2}\right )}{\sqrt {b x +a}\, a^{2} c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \] Input:

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

Output:

(sqrt(e)*(2*sqrt(d)*sqrt(a + b*x)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d 
)/(sqrt(d)*sqrt( - a*d + b*c)))*a**2*d + sqrt(a)*sqrt(a + b*x)*log(sqrt(a 
+ b*x) - sqrt(a))*a**2*d**2 - 2*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - 
sqrt(a))*a*b*c*d + sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*b**2 
*c**2 - sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**2*d**2 + 2*s 
qrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a*b*c*d - sqrt(a)*sqrt(a 
 + b*x)*log(sqrt(a + b*x) + sqrt(a))*b**2*c**2 - 2*a**2*b*c*d + 2*a*b**2*c 
**2))/(sqrt(a + b*x)*a**2*c*(a**2*d**2 - 2*a*b*c*d + b**2*c**2))