\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^6 (d+e x)} \, dx\) [57]

Optimal result

Integrand size = 40, antiderivative size = 272 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}} \] Output:

3/128*(-a*e^2+c*d^2)^3*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(1/2)/a^2/d^3/e^2/x^2-1/16*(c/a/e-e/d^2)*(2*a*d*e+(a*e^2+c*d^2)*x 
)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^4-1/5*(a*d*e+(a*e^2+c*d^2)*x+c 
*d*e*x^2)^(5/2)/d/x^5-3/128*(-a*e^2+c*d^2)^5*arctanh(a^(1/2)*e^(1/2)*(e*x+ 
d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(5/2)/d^(7/2)/e^(5/2 
)
 

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (d+e x)^3 \left (15 a^4 e^4-\frac {15 d^4 (a e+c d x)^4}{(d+e x)^4}+\frac {70 a d^3 e (a e+c d x)^3}{(d+e x)^3}+\frac {128 a^2 d^2 e^2 (a e+c d x)^2}{(d+e x)^2}-\frac {70 a^3 d e^3 (a e+c d x)}{d+e x}\right )}{\left (-c d^2+a e^2\right )^5 x^5 (a e+c d x)}+\frac {15 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 a^{5/2} d^{7/2} e^{5/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x]
 

Output:

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[a]*Sqrt[d]* 
Sqrt[e]*(d + e*x)^3*(15*a^4*e^4 - (15*d^4*(a*e + c*d*x)^4)/(d + e*x)^4 + ( 
70*a*d^3*e*(a*e + c*d*x)^3)/(d + e*x)^3 + (128*a^2*d^2*e^2*(a*e + c*d*x)^2 
)/(d + e*x)^2 - (70*a^3*d*e^3*(a*e + c*d*x))/(d + e*x)))/((-(c*d^2) + a*e^ 
2)^5*x^5*(a*e + c*d*x))) + (15*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a 
]*Sqrt[e]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(640*a^ 
(5/2)*d^(7/2)*e^(5/2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1215, 1228, 1152, 1152, 1154, 219}

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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx\)

\(\Big \downarrow \) 1215

\(\displaystyle \int \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{x^6}dx\)

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^3}dx}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x]
 

Output:

-1/5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d*x^5) + ((c*d^2 - a*e 
^2)*(-1/8*((2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2)^(3/2))/(a*d*e*x^4) - (3*(c*d^2 - a*e^2)^2*(-1/4*((2*a*d*e + (c*d^2 
+ a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x^2) + ((c 
*d^2 - a*e^2)^2*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*S 
qrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*a^(3/2)*d^(3/2)*e 
^(3/2))))/(16*a*d*e)))/(2*d)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1215

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

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(19538\) vs. \(2(244)=488\).

Time = 4.34 (sec) , antiderivative size = 19539, normalized size of antiderivative = 71.83

method	result	size
default	\(\text {Expression too large to display}\)	\(19539\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^6/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 10.52 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d),x, algorithm 
="fricas")
 

Output:

[-1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2 
*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(a*d*e)*x^5*log((8*a^2*d^2*e^2 
+ (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c* 
d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + 
 a^2*d*e^3)*x)/x^2) + 4*(128*a^5*d^5*e^5 - (15*a*c^4*d^9*e - 70*a^2*c^3*d^ 
7*e^3 - 128*a^3*c^2*d^5*e^5 + 70*a^4*c*d^3*e^7 - 15*a^5*d*e^9)*x^4 + 2*(5* 
a^2*c^3*d^8*e^2 + 233*a^3*c^2*d^6*e^4 + 23*a^4*c*d^4*e^6 - 5*a^5*d^2*e^8)* 
x^3 + 8*(31*a^3*c^2*d^7*e^3 + 64*a^4*c*d^5*e^5 + a^5*d^3*e^7)*x^2 + 16*(21 
*a^4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^ 
2)*x))/(a^3*d^4*e^3*x^5), 1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2* 
c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(-a*d*e 
)*x^5*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c 
*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e 
+ a^2*d*e^3)*x)) - 2*(128*a^5*d^5*e^5 - (15*a*c^4*d^9*e - 70*a^2*c^3*d^7*e 
^3 - 128*a^3*c^2*d^5*e^5 + 70*a^4*c*d^3*e^7 - 15*a^5*d*e^9)*x^4 + 2*(5*a^2 
*c^3*d^8*e^2 + 233*a^3*c^2*d^6*e^4 + 23*a^4*c*d^4*e^6 - 5*a^5*d^2*e^8)*x^3 
 + 8*(31*a^3*c^2*d^7*e^3 + 64*a^4*c*d^5*e^5 + a^5*d^3*e^7)*x^2 + 16*(21*a^ 
4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)* 
x))/(a^3*d^4*e^3*x^5)]
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**6/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d),x, algorithm 
="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^6), x 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2449 vs. \(2 (244) = 488\).

Time = 0.20 (sec) , antiderivative size = 2449, normalized size of antiderivative = 9.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d),x, algorithm 
="giac")
 

Output:

3/128*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^ 
6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + 
c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a^2*d^3*e^2) - 1/6 
40*(15*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c 
^5*d^14*e^4 - 75*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d 
*e))*a^5*c^4*d^12*e^6 + 150*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a* 
e^2*x + a*d*e))*a^6*c^3*d^10*e^8 + 1130*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + 
c*d^2*x + a*e^2*x + a*d*e))*a^7*c^2*d^8*e^10 + 75*(sqrt(c*d*e)*x - sqrt(c* 
d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c*d^6*e^12 - 15*(sqrt(c*d*e)*x - 
 sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^9*d^4*e^14 + 256*sqrt(c*d* 
e)*a^7*c^2*d^9*e^9 - 70*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2* 
x + a*d*e))^3*a^3*c^5*d^13*e^3 + 350*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d 
^2*x + a*e^2*x + a*d*e))^3*a^4*c^4*d^11*e^5 + 5700*(sqrt(c*d*e)*x - sqrt(c 
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^5*c^3*d^9*e^7 + 7100*(sqrt(c*d* 
e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^6*c^2*d^7*e^9 + 22 
10*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c*d 
^5*e^11 + 70*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)) 
^3*a^8*d^3*e^13 + 2560*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))^2*a^6*c^2*d^8*e^8 + 2560*sqrt(c*d*e)*(sqrt(c*d*e)*x 
 - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^7*c*d^6*e^10 + 128*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^6\,\left (d+e\,x\right )} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)), x)
 

Reduce [B] (verification not implemented)

Time = 16.31 (sec) , antiderivative size = 1904, normalized size of antiderivative = 7.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d),x)
 

Output:

( - 256*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**5*e**7 - 352*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**6*d**4*e**8*x - 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**6*d**3*e**9*x**2 + 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**2*e**10*x* 
*3 - 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d*e**11*x**4 - 256*sqrt(d + e 
*x)*sqrt(a*e + c*d*x)*a**5*c*d**7*e**5 - 1024*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*a**5*c*d**6*e**6*x - 1040*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**5* 
e**7*x**2 - 72*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**4*e**8*x**3 + 110 
*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**3*e**9*x**4 - 672*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**4*c**2*d**8*e**4*x - 1520*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*a**4*c**2*d**7*e**5*x**2 - 1024*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4 
*c**2*d**6*e**6*x**3 - 116*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**5* 
e**7*x**4 - 496*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**9*e**3*x**2 - 
 952*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**8*e**4*x**3 - 396*sqrt(d 
 + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**7*e**5*x**4 - 20*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a**2*c**4*d**10*e**2*x**3 - 110*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*a**2*c**4*d**9*e**3*x**4 + 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d 
**11*e*x**4 - 15*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - s 
qrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e* 
x))*a**6*e**12*x**5 + 60*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c* 
d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*...