\(\int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} (a+b x^2)^2} \, dx\) [931]

Optimal result

Integrand size = 26, antiderivative size = 487 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 b c^2+4 a d^2}{2 a^2 c \left (b c^2+a d^2\right ) e \sqrt {e x} \sqrt {c+d x}}-\frac {d \left (5 b^2 c^4+7 a b c^2 d^2+8 a^2 d^4\right ) \sqrt {e x}}{2 a^2 c^2 \left (b c^2+a d^2\right )^2 e^2 \sqrt {c+d x}}+\frac {b (c-d x)}{2 a \left (b c^2+a d^2\right ) e \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )}-\frac {b \left (5 b^{3/2} c^3+2 \sqrt {-a} b c^2 d+11 a \sqrt {b} c d^2+8 \sqrt {-a} a d^3\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {b \left (5 b^{3/2} c^3-2 \sqrt {-a} b c^2 d+11 a \sqrt {b} c d^2-8 \sqrt {-a} a d^3\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )^2 e^{3/2}} \] Output:

-1/2*(4*a*d^2+5*b*c^2)/a^2/c/(a*d^2+b*c^2)/e/(e*x)^(1/2)/(d*x+c)^(1/2)-1/2 
*d*(8*a^2*d^4+7*a*b*c^2*d^2+5*b^2*c^4)*(e*x)^(1/2)/a^2/c^2/(a*d^2+b*c^2)^2 
/e^2/(d*x+c)^(1/2)+1/2*b*(-d*x+c)/a/(a*d^2+b*c^2)/e/(e*x)^(1/2)/(d*x+c)^(1 
/2)/(b*x^2+a)-1/4*b*(5*b^(3/2)*c^3+2*(-a)^(1/2)*b*c^2*d+11*a*b^(1/2)*c*d^2 
+8*(-a)^(1/2)*a*d^3)*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a 
)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(9/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/( 
a*d^2+b*c^2)^2/e^(3/2)+1/4*b*(5*b^(3/2)*c^3-2*(-a)^(1/2)*b*c^2*d+11*a*b^(1 
/2)*c*d^2-8*(-a)^(1/2)*a*d^3)*arctanh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x) 
^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(9/4)/(b^(1/2)*c+(-a)^(1/2)* 
d)^(1/2)/(a*d^2+b*c^2)^2/e^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.39 (sec) , antiderivative size = 1081, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(-2*(5*b^3*c^4*x^2*(c + d*x) + 4*a^3*d^4*(c + 2*d*x) + 4*a^2*b*d^2*(2*c 
^3 + 2*c^2*d*x + c*d^2*x^2 + 2*d^3*x^3) + a*b^2*c^2*(4*c^3 + 6*c^2*d*x + 9 
*c*d^2*x^2 + 7*d^3*x^3)) - 4*b*c^2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(a + b*x^ 
2)*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 
+ b*#1^4 & , (b^2*c^5*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1 
] + 18*a*b*c^3*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 
 16*a^2*c*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b^ 
2*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 4*a*b*c^2 
*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 4*a^2*d^4* 
Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b^2*c^3*Log[c + 
 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 + 2*a*b*c*d^2*Log[c + 
2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^3 - 3*b*c^2*#1 - 
8*a*d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ] - b*c^2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x 
]*(a + b*x^2)*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 
4*b*c*#1^3 + b*#1^4 & , (b^2*c^5*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c 
+ d*x] - #1] - 61*a*b*c^3*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d 
*x] - #1] - 64*a^2*c*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - 
 #1] - 2*b^2*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 
+ 2*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 
 16*a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + ...
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1014\) vs. \(2(487)=974\).

Time = 3.81 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {b} c-2 \sqrt {-a} d}{4 a^2 c \left (\sqrt {b} c-\sqrt {-a} d\right ) e \sqrt {e x} \sqrt {c+d x}}-\frac {b \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{9/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} e^{3/2}}-\frac {3 b \left (\sqrt {b} c-2 \sqrt {-a} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{5/2} e^{3/2}}+\frac {3 b \left (\sqrt {b} c+2 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{5/2} e^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{9/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} e^{3/2}}-\frac {d \left (\sqrt {-a} \sqrt {b} c+2 a d\right ) \sqrt {e x}}{2 a^2 c^2 \left (\sqrt {-a} \sqrt {b} c+a d\right ) e^2 \sqrt {c+d x}}-\frac {d \left (\sqrt {-a} \sqrt {b} c-2 a d\right ) \sqrt {e x}}{2 a^2 c^2 \left (\sqrt {-a} \sqrt {b} c-a d\right ) e^2 \sqrt {c+d x}}-\frac {d \left (3 b c^2-4 \sqrt {-a} \sqrt {b} d c-4 a d^2\right ) \sqrt {e x}}{4 a^2 c^2 \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 \sqrt {c+d x}}-\frac {d \left (3 b c^2+4 \sqrt {-a} \sqrt {b} d c-4 a d^2\right ) \sqrt {e x}}{4 a^2 c^2 \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 \sqrt {c+d x}}-\frac {3 \sqrt {b} c+2 \sqrt {-a} d}{4 a^2 c \left (\sqrt {b} c+\sqrt {-a} d\right ) e \sqrt {e x} \sqrt {c+d x}}-\frac {1}{a^2 c e \sqrt {e x} \sqrt {c+d x}}-\frac {\sqrt {b}}{4 a \left (\sqrt {-a} \sqrt {b} c-a d\right ) e \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}-\frac {\sqrt {b}}{4 a \left (\sqrt {-a} \sqrt {b} c+a d\right ) e \sqrt {e x} \left (\sqrt {b} x+\sqrt {-a}\right ) \sqrt {c+d x}}\)

Input:

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

Output:

-(1/(a^2*c*e*Sqrt[e*x]*Sqrt[c + d*x])) - (3*Sqrt[b]*c - 2*Sqrt[-a]*d)/(4*a 
^2*c*(Sqrt[b]*c - Sqrt[-a]*d)*e*Sqrt[e*x]*Sqrt[c + d*x]) - (3*Sqrt[b]*c + 
2*Sqrt[-a]*d)/(4*a^2*c*(Sqrt[b]*c + Sqrt[-a]*d)*e*Sqrt[e*x]*Sqrt[c + d*x]) 
 - (d*(Sqrt[-a]*Sqrt[b]*c - 2*a*d)*Sqrt[e*x])/(2*a^2*c^2*(Sqrt[-a]*Sqrt[b] 
*c - a*d)*e^2*Sqrt[c + d*x]) - (d*(Sqrt[-a]*Sqrt[b]*c + 2*a*d)*Sqrt[e*x])/ 
(2*a^2*c^2*(Sqrt[-a]*Sqrt[b]*c + a*d)*e^2*Sqrt[c + d*x]) - (d*(3*b*c^2 - 4 
*Sqrt[-a]*Sqrt[b]*c*d - 4*a*d^2)*Sqrt[e*x])/(4*a^2*c^2*(b*c^2 - 2*Sqrt[-a] 
*Sqrt[b]*c*d - a*d^2)*e^2*Sqrt[c + d*x]) - (d*(3*b*c^2 + 4*Sqrt[-a]*Sqrt[b 
]*c*d - 4*a*d^2)*Sqrt[e*x])/(4*a^2*c^2*(b*c^2 + 2*Sqrt[-a]*Sqrt[b]*c*d - a 
*d^2)*e^2*Sqrt[c + d*x]) - Sqrt[b]/(4*a*(Sqrt[-a]*Sqrt[b]*c - a*d)*e*Sqrt[ 
e*x]*(Sqrt[-a] - Sqrt[b]*x)*Sqrt[c + d*x]) - Sqrt[b]/(4*a*(Sqrt[-a]*Sqrt[b 
]*c + a*d)*e*Sqrt[e*x]*(Sqrt[-a] + Sqrt[b]*x)*Sqrt[c + d*x]) - (3*b*(Sqrt[ 
b]*c - 2*Sqrt[-a]*d)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a) 
^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4*(-a)^(9/4)*(Sqrt[b]*c - Sqrt[-a]*d)^(5/ 
2)*e^(3/2)) - (b*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/ 
4)*Sqrt[e]*Sqrt[c + d*x])])/(2*(-a)^(9/4)*(Sqrt[b]*c - Sqrt[-a]*d)^(3/2)*e 
^(3/2)) + (b*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)* 
Sqrt[e]*Sqrt[c + d*x])])/(2*(-a)^(9/4)*(Sqrt[b]*c + Sqrt[-a]*d)^(3/2)*e^(3 
/2)) + (3*b*(Sqrt[b]*c + 2*Sqrt[-a]*d)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]* 
d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4*(-a)^(9/4)*(Sqrt[...
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2075\) vs. \(2(401)=802\).

Time = 0.64 (sec) , antiderivative size = 2076, normalized size of antiderivative = 4.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(2076\)
default	\(\text {Expression too large to display}\)	\(8835\)

Input:

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

Output:

-2/c^2/a^2*(d*x+c)^(1/2)/e/(e*x)^(1/2)+(-2/c^2*b^2*d^4/((-a*b)^(1/2)*d+b*c 
)^2/((-a*b)^(1/2)*d-b*c)^2/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+1/4 
/a^2*(-a*b)^(1/2)/((-a*b)^(1/2)*d+b*c)/e/(a*d-c*(-a*b)^(1/2))*b/(x-(-a*b)^ 
(1/2)/b)*(d*e*(x-(-a*b)^(1/2)/b)^2+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1 
/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)+1/4/a*b/((-a*b)^(1/2)*d+b*c)/(a*d-c 
*(-a*b)^(1/2))/(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d-c*(-a*b)^(1 
/2))/b+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)+2*(-e*(a*d-c*(-a*b)^( 
1/2))/b)^(1/2)*(d*e*(x-(-a*b)^(1/2)/b)^2+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a 
*b)^(1/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2))/(x-(-a*b)^(1/2)/b))*d-1/8/a^ 
2*c*(-a*b)^(1/2)/((-a*b)^(1/2)*d+b*c)/(a*d-c*(-a*b)^(1/2))/(-e*(a*d-c*(-a* 
b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/2)*d+b*c 
)/b*(x-(-a*b)^(1/2)/b)+2*(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x-(-a*b)^ 
(1/2)/b)^2+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^( 
1/2))/b)^(1/2))/(x-(-a*b)^(1/2)/b))*b+1/4/a^2*(-a*b)^(1/2)/((-a*b)^(1/2)*d 
-b*c)/e/(a*d+c*(-a*b)^(1/2))*b/(x+(-a*b)^(1/2)/b)*(d*e*(x+(-a*b)^(1/2)/b)^ 
2+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1/2)/b)-e*(a*d+c*(-a*b)^(1/2))/b) 
^(1/2)-1/4/a*b/((-a*b)^(1/2)*d-b*c)/(a*d+c*(-a*b)^(1/2))/(-e*(a*d+c*(-a*b) 
^(1/2))/b)^(1/2)*ln((-2*e*(a*d+c*(-a*b)^(1/2))/b+e*(-2*(-a*b)^(1/2)*d+b*c) 
/b*(x+(-a*b)^(1/2)/b)+2*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x+(-a*b)^( 
1/2)/b)^2+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1/2)/b)-e*(a*d+c*(-a*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6579 vs. \(2 (402) = 804\).

Time = 6.69 (sec) , antiderivative size = 6579, normalized size of antiderivative = 13.51 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\int \frac {1}{\sqrt {x}\, \sqrt {d x +c}\, a^{2} c x +\sqrt {x}\, \sqrt {d x +c}\, a^{2} d \,x^{2}+2 \sqrt {x}\, \sqrt {d x +c}\, a b c \,x^{3}+2 \sqrt {x}\, \sqrt {d x +c}\, a b d \,x^{4}+\sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,x^{5}+\sqrt {x}\, \sqrt {d x +c}\, b^{2} d \,x^{6}}d x}{\sqrt {e}\, e} \] Input:

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

Output:

int(1/(sqrt(x)*sqrt(c + d*x)*a**2*c*x + sqrt(x)*sqrt(c + d*x)*a**2*d*x**2 
+ 2*sqrt(x)*sqrt(c + d*x)*a*b*c*x**3 + 2*sqrt(x)*sqrt(c + d*x)*a*b*d*x**4 
+ sqrt(x)*sqrt(c + d*x)*b**2*c*x**5 + sqrt(x)*sqrt(c + d*x)*b**2*d*x**6),x 
)/(sqrt(e)*e)