\(\int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx\) [369]

Optimal result

Integrand size = 20, antiderivative size = 143 \[ \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {3 (b c-5 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {2 a (c+d x)^{3/2}}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}+\frac {3 (b c-5 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} \sqrt {d}} \] Output:

3/4*(-5*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3+2*a*(d*x+c)^(3/2)/b^2/(b* 
x+a)^(1/2)+1/2*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b^2+3/4*(-5*a*d+b*c)*(-a*d+b*c) 
*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x} \left (-15 a^2 d+a b (13 c-5 d x)+b^2 x (5 c+2 d x)\right )}{\sqrt {a+b x}}-\frac {6 \left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {d}}}{4 b^{7/2}} \] Input:

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

Output:

((Sqrt[b]*Sqrt[c + d*x]*(-15*a^2*d + a*b*(13*c - 5*d*x) + b^2*x*(5*c + 2*d 
*x)))/Sqrt[a + b*x] - (6*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b 
]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[d])/ 
(4*b^(7/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 60, 60, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(b c-5 a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{b (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(b c-5 a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{b (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(b c-5 a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{b (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(b c-5 a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{b (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(b c-5 a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{b (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)}\)

Input:

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

Output:

(2*a*(c + d*x)^(5/2))/(b*(b*c - a*d)*Sqrt[a + b*x]) + ((b*c - 5*a*d)*((Sqr 
t[a + b*x]*(c + d*x)^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c 
+ d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + 
 d*x])])/(b^(3/2)*Sqrt[d])))/(4*b)))/(b*(b*c - a*d))
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 87

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

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. \(454\) vs. \(2(113)=226\).

Time = 0.24 (sec) , antiderivative size = 455, normalized size of antiderivative = 3.18

method	result	size
default	\(\frac {\sqrt {x d +c}\, \left (15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b \,d^{2} x -18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{3} c^{2} x +4 b^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} d^{2}-18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b c d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c^{2}-10 a b d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+10 b^{2} c x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-30 a^{2} d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+26 a b c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\right )}{8 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {b x +a}\, b^{3}}\)	\(455\)

Input:

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

Output:

1/8*(d*x+c)^(1/2)*(15*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*b*d^2*x-18*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^2*c*d*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^3*c^2*x+4*b^2*d 
*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+15*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^3*d^2-18*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*b*c*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))*a 
*b^2*c^2-10*a*b*d*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+10*b^2*c*x*((b*x+a 
)*(d*x+c))^(1/2)*(d*b)^(1/2)-30*a^2*d*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+ 
26*a*b*c*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(d*b 
)^(1/2)/(b*x+a)^(1/2)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.03 \[ \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\left [\frac {3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{3} d^{2} x^{2} + 13 \, a b^{2} c d - 15 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{5} d x + a b^{4} d\right )}}, -\frac {3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{3} d^{2} x^{2} + 13 \, a b^{2} c d - 15 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{5} d x + a b^{4} d\right )}}\right ] \] Input:

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

Output:

[1/16*(3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d^2 + (b^3*c^2 - 6*a*b^2*c*d + 5 
*a^2*b*d^2)*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*(2*b^3*d^2*x^2 + 13*a*b^2*c*d - 15*a^2*b*d^2 + 5*(b^ 
3*c*d - a*b^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d*x + a*b^4*d), -1 
/8*(3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d^2 + (b^3*c^2 - 6*a*b^2*c*d + 5*a^ 
2*b*d^2)*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* 
(2*b^3*d^2*x^2 + 13*a*b^2*c*d - 15*a^2*b*d^2 + 5*(b^3*c*d - a*b^2*d^2)*x)* 
sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d*x + a*b^4*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(d*x+c)^(3/2)/(b*x+a)^(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. 250 vs. \(2 (113) = 226\).

Time = 0.22 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.75 \[ \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{4} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{5}} + \frac {5 \, b^{10} c d^{2} {\left | b \right |} - 9 \, a b^{9} d^{3} {\left | b \right |}}{b^{14} d^{2}}\right )} - \frac {3 \, {\left (b^{2} c^{2} {\left | b \right |} - 6 \, a b c d {\left | b \right |} + 5 \, a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, \sqrt {b d} b^{4}} + \frac {4 \, {\left (a b^{2} c^{2} d {\left | b \right |} - 2 \, a^{2} b c d^{2} {\left | b \right |} + a^{3} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{3}} \] Input:

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

Output:

1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d*abs(b 
)/b^5 + (5*b^10*c*d^2*abs(b) - 9*a*b^9*d^3*abs(b))/(b^14*d^2)) - 3/8*(b^2* 
c^2*abs(b) - 6*a*b*c*d*abs(b) + 5*a^2*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x 
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^4) + 4*(a*b^2* 
c^2*d*abs(b) - 2*a^2*b*c*d^2*abs(b) + a^3*d^3*abs(b))/((b^2*c - a*b*d - (s 
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*sqrt(b*d) 
*b^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.05 \[ \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {15 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{2}-18 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c d +3 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2}-10 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} d^{2}+11 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b c d -\sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{2}-15 \sqrt {d x +c}\, a^{2} b \,d^{2}+13 \sqrt {d x +c}\, a \,b^{2} c d -5 \sqrt {d x +c}\, a \,b^{2} d^{2} x +5 \sqrt {d x +c}\, b^{3} c d x +2 \sqrt {d x +c}\, b^{3} d^{2} x^{2}}{4 \sqrt {b x +a}\, b^{4} d} \] Input:

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

Output:

(15*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqr 
t(c + d*x))/sqrt(a*d - b*c))*a**2*d**2 - 18*sqrt(d)*sqrt(b)*sqrt(a + b*x)* 
log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c 
*d + 3*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)* 
sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**2 - 10*sqrt(d)*sqrt(b)*sqrt(a + b* 
x)*a**2*d**2 + 11*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b*c*d - sqrt(d)*sqrt(b)* 
sqrt(a + b*x)*b**2*c**2 - 15*sqrt(c + d*x)*a**2*b*d**2 + 13*sqrt(c + d*x)* 
a*b**2*c*d - 5*sqrt(c + d*x)*a*b**2*d**2*x + 5*sqrt(c + d*x)*b**3*c*d*x + 
2*sqrt(c + d*x)*b**3*d**2*x**2)/(4*sqrt(a + b*x)*b**4*d)