\(\int \frac {1}{\sqrt {d+e x} (c d^2-c e^2 x^2)^{5/2}} \, dx\) [202]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {d+7 e x}{24 c d^2 e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}+\frac {35 (d+3 e x)}{192 c^2 d^4 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}-\frac {35 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}}\right )}{64 \sqrt {2} c^{5/2} d^{9/2} e} \] Output:

1/24*(7*e*x+d)/c/d^2/e/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(3/2)+35/192*(3*e* 
x+d)/c^2/d^4/e/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2)-35/128*arctanh(2^(1/ 
2)*c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2))*2^(1/2)/c^(5/2) 
/d^(9/2)/e
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d} \left (43 d^3+161 d^2 e x-35 d e^2 x^2-105 e^3 x^3\right )-105 \sqrt {2} \sqrt {d+e x} \left (d^2-e^2 x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{384 c^2 d^{9/2} e (d-e x) (d+e x)^{3/2} \sqrt {c \left (d^2-e^2 x^2\right )}} \] Input:

Integrate[1/(Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2)),x]
 

Output:

(2*Sqrt[d]*(43*d^3 + 161*d^2*e*x - 35*d*e^2*x^2 - 105*e^3*x^3) - 105*Sqrt[ 
2]*Sqrt[d + e*x]*(d^2 - e^2*x^2)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[d]*Sqrt[d + e 
*x])/Sqrt[d^2 - e^2*x^2]])/(384*c^2*d^(9/2)*e*(d - e*x)*(d + e*x)^(3/2)*Sq 
rt[c*(d^2 - e^2*x^2)])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.57, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {470, 467, 470, 467, 471, 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.

\(\displaystyle \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {7 \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{5/2}}dx}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}dx}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}}dx}{4 d}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}\right )}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}dx}{2 c d}+\frac {\sqrt {d+e x}}{c d e \sqrt {c d^2-c e^2 x^2}}\right )}{4 d}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}\right )}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {e \int \frac {1}{\frac {e^2 \left (c d^2-c e^2 x^2\right )}{d+e x}-2 c d e^2}d\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}}{c d}+\frac {\sqrt {d+e x}}{c d e \sqrt {c d^2-c e^2 x^2}}\right )}{4 d}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}\right )}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {d+e x}}{c d e \sqrt {c d^2-c e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} c^{3/2} d^{3/2} e}\right )}{4 d}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}\right )}{6 c d}+\frac {\sqrt {d+e x}}{3 c d e \left (c d^2-c e^2 x^2\right )^{3/2}}\right )}{8 d}-\frac {1}{4 c d e \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}\)

Input:

Int[1/(Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2)),x]
 

Output:

-1/4*1/(c*d*e*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2)) + (7*(Sqrt[d + e*x] 
/(3*c*d*e*(c*d^2 - c*e^2*x^2)^(3/2)) + (5*(-1/2*1/(c*d*e*Sqrt[d + e*x]*Sqr 
t[c*d^2 - c*e^2*x^2]) + (3*(Sqrt[d + e*x]/(c*d*e*Sqrt[c*d^2 - c*e^2*x^2]) 
- ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])] 
/(Sqrt[2]*c^(3/2)*d^(3/2)*e)))/(4*d)))/(6*c*d)))/(8*d)
 

Defintions of rubi rules used

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 467

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-c)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*a*d*(p + 1))), x] + Simp[c*((n + 2 
*p + 2)/(2*a*(p + 1)))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[p, -1] && LtQ[0, 
n, 1] && IntegerQ[2*p]
 

rule 470

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

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) e^{3} x^{3} \sqrt {c \left (-e x +d \right )}-105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) d \,e^{2} x^{2} \sqrt {c \left (-e x +d \right )}+105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) d^{2} e x \sqrt {c \left (-e x +d \right )}+210 \sqrt {c d}\, e^{3} x^{3}+105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) d^{3} \sqrt {c \left (-e x +d \right )}+70 \sqrt {c d}\, d \,e^{2} x^{2}-322 \sqrt {c d}\, d^{2} e x -86 \sqrt {c d}\, d^{3}\right )}{384 c^{3} \left (e x +d \right )^{\frac {5}{2}} \left (-e x +d \right )^{2} e \,d^{4} \sqrt {c d}}\)	\(263\)

Input:

int(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384*(c*(-e^2*x^2+d^2))^(1/2)/c^3*(-105*2^(1/2)*arctanh(1/2*(c*(-e*x+d)) 
^(1/2)*2^(1/2)/(c*d)^(1/2))*e^3*x^3*(c*(-e*x+d))^(1/2)-105*2^(1/2)*arctanh 
(1/2*(c*(-e*x+d))^(1/2)*2^(1/2)/(c*d)^(1/2))*d*e^2*x^2*(c*(-e*x+d))^(1/2)+ 
105*2^(1/2)*arctanh(1/2*(c*(-e*x+d))^(1/2)*2^(1/2)/(c*d)^(1/2))*d^2*e*x*(c 
*(-e*x+d))^(1/2)+210*(c*d)^(1/2)*e^3*x^3+105*2^(1/2)*arctanh(1/2*(c*(-e*x+ 
d))^(1/2)*2^(1/2)/(c*d)^(1/2))*d^3*(c*(-e*x+d))^(1/2)+70*(c*d)^(1/2)*d*e^2 
*x^2-322*(c*d)^(1/2)*d^2*e*x-86*(c*d)^(1/2)*d^3)/(e*x+d)^(5/2)/(-e*x+d)^2/ 
e/d^4/(c*d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, \sqrt {2} {\left (e^{5} x^{5} + d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 2 \, d^{3} e^{2} x^{2} + d^{4} e x + d^{5}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, {\left (105 \, d e^{3} x^{3} + 35 \, d^{2} e^{2} x^{2} - 161 \, d^{3} e x - 43 \, d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{768 \, {\left (c^{3} d^{5} e^{6} x^{5} + c^{3} d^{6} e^{5} x^{4} - 2 \, c^{3} d^{7} e^{4} x^{3} - 2 \, c^{3} d^{8} e^{3} x^{2} + c^{3} d^{9} e^{2} x + c^{3} d^{10} e\right )}}, \frac {105 \, \sqrt {2} {\left (e^{5} x^{5} + d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 2 \, d^{3} e^{2} x^{2} + d^{4} e x + d^{5}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{2 \, {\left (c d e x + c d^{2}\right )}}\right ) - 2 \, {\left (105 \, d e^{3} x^{3} + 35 \, d^{2} e^{2} x^{2} - 161 \, d^{3} e x - 43 \, d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{384 \, {\left (c^{3} d^{5} e^{6} x^{5} + c^{3} d^{6} e^{5} x^{4} - 2 \, c^{3} d^{7} e^{4} x^{3} - 2 \, c^{3} d^{8} e^{3} x^{2} + c^{3} d^{9} e^{2} x + c^{3} d^{10} e\right )}}\right ] \] Input:

integrate(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/768*(105*sqrt(2)*(e^5*x^5 + d*e^4*x^4 - 2*d^2*e^3*x^3 - 2*d^3*e^2*x^2 + 
 d^4*e*x + d^5)*sqrt(c*d)*log(-(c*e^2*x^2 - 2*c*d*e*x - 3*c*d^2 + 2*sqrt(2 
)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d 
^2)) - 4*(105*d*e^3*x^3 + 35*d^2*e^2*x^2 - 161*d^3*e*x - 43*d^4)*sqrt(-c*e 
^2*x^2 + c*d^2)*sqrt(e*x + d))/(c^3*d^5*e^6*x^5 + c^3*d^6*e^5*x^4 - 2*c^3* 
d^7*e^4*x^3 - 2*c^3*d^8*e^3*x^2 + c^3*d^9*e^2*x + c^3*d^10*e), 1/384*(105* 
sqrt(2)*(e^5*x^5 + d*e^4*x^4 - 2*d^2*e^3*x^3 - 2*d^3*e^2*x^2 + d^4*e*x + d 
^5)*sqrt(-c*d)*arctan(1/2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*sqrt 
(e*x + d)/(c*d*e*x + c*d^2)) - 2*(105*d*e^3*x^3 + 35*d^2*e^2*x^2 - 161*d^3 
*e*x - 43*d^4)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d))/(c^3*d^5*e^6*x^5 + 
c^3*d^6*e^5*x^4 - 2*c^3*d^7*e^4*x^3 - 2*c^3*d^8*e^3*x^2 + c^3*d^9*e^2*x + 
c^3*d^10*e)]
 

Sympy [F]

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

integrate(1/(e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(5/2),x)
 

Output:

Integral(1/((-c*(-d + e*x)*(d + e*x))**(5/2)*sqrt(d + e*x)), x)
 

Maxima [F]

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

integrate(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((-c*e^2*x^2 + c*d^2)^(5/2)*sqrt(e*x + d)), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {105 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} c^{2} d^{4}} + \frac {16 \, {\left (9 \, {\left (e x + d\right )} c - 20 \, c d\right )}}{{\left ({\left (e x + d\right )} c - 2 \, c d\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{4}} - \frac {6 \, {\left (26 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 11 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}\right )}}{{\left (e x + d\right )}^{2} c^{4} d^{4}}}{384 \, e} \] Input:

integrate(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(5/2),x, algorithm="giac")
 

Output:

1/384*(105*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-(e*x + d)*c + 2*c*d)/sqrt(-c*d 
))/(sqrt(-c*d)*c^2*d^4) + 16*(9*(e*x + d)*c - 20*c*d)/(((e*x + d)*c - 2*c* 
d)*sqrt(-(e*x + d)*c + 2*c*d)*c^2*d^4) - 6*(26*sqrt(-(e*x + d)*c + 2*c*d)* 
c*d - 11*(-(e*x + d)*c + 2*c*d)^(3/2))/((e*x + d)^2*c^4*d^4))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,d^2-c\,e^2\,x^2\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \] Input:

int(1/((c*d^2 - c*e^2*x^2)^(5/2)*(d + e*x)^(1/2)),x)
 

Output:

int(1/((c*d^2 - c*e^2*x^2)^(5/2)*(d + e*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d^{3}+105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d^{2} e x -105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) d \,e^{2} x^{2}-105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}-\sqrt {d}\, \sqrt {2}\right ) e^{3} x^{3}-105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d^{3}-105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d^{2} e x +105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) d \,e^{2} x^{2}+105 \sqrt {d}\, \sqrt {-e x +d}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-e x +d}+\sqrt {d}\, \sqrt {2}\right ) e^{3} x^{3}+172 d^{4}+644 d^{3} e x -140 d^{2} e^{2} x^{2}-420 d \,e^{3} x^{3}\right )}{768 \sqrt {-e x +d}\, c^{3} d^{5} e \left (-e^{3} x^{3}-d \,e^{2} x^{2}+d^{2} e x +d^{3}\right )} \] Input:

int(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(5/2),x)
 

Output:

(sqrt(c)*(105*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) - sqrt(d)*sq 
rt(2))*d**3 + 105*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) - sqrt(d 
)*sqrt(2))*d**2*e*x - 105*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) 
- sqrt(d)*sqrt(2))*d*e**2*x**2 - 105*sqrt(d)*sqrt(d - e*x)*sqrt(2)*log(sqr 
t(d - e*x) - sqrt(d)*sqrt(2))*e**3*x**3 - 105*sqrt(d)*sqrt(d - e*x)*sqrt(2 
)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d**3 - 105*sqrt(d)*sqrt(d - e*x)*sq 
rt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d**2*e*x + 105*sqrt(d)*sqrt(d - 
 e*x)*sqrt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*d*e**2*x**2 + 105*sqrt( 
d)*sqrt(d - e*x)*sqrt(2)*log(sqrt(d - e*x) + sqrt(d)*sqrt(2))*e**3*x**3 + 
172*d**4 + 644*d**3*e*x - 140*d**2*e**2*x**2 - 420*d*e**3*x**3))/(768*sqrt 
(d - e*x)*c**3*d**5*e*(d**3 + d**2*e*x - d*e**2*x**2 - e**3*x**3))