\(\int \frac {(d+e x)^{7/2}}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [174]

Optimal result

Integrand size = 37, antiderivative size = 147 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2}+\frac {2 (d+e x)^{5/2}}{5 c d}-\frac {2 \left (c d^2-a e^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x} \left (15 a^2 e^4-5 a c d e^2 (7 d+e x)+c^2 d^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )}{15 c^3 d^3}-\frac {2 \left (-c d^2+a e^2\right )^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{7/2} d^{7/2}} \] Input:

Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
 

Output:

(2*Sqrt[d + e*x]*(15*a^2*e^4 - 5*a*c*d*e^2*(7*d + e*x) + c^2*d^2*(23*d^2 + 
 11*d*e*x + 3*e^2*x^2)))/(15*c^3*d^3) - (2*(-(c*d^2) + a*e^2)^(5/2)*ArcTan 
[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(7/2)*d^(7/2) 
)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1121, 60, 60, 60, 73, 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.

\(\displaystyle \int \frac {(d+e x)^{7/2}}{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\)

\(\Big \downarrow \) 1121

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
 

Output:

(2*(d + e*x)^(5/2))/(5*c*d) + ((d^2 - (a*e^2)/c)*((2*(d + e*x)^(3/2))/(3*c 
*d) + ((d^2 - (a*e^2)/c)*((2*Sqrt[d + e*x])/(c*d) - (2*(d^2 - (a*e^2)/c)*A 
rcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c]*d^(3 
/2)*Sqrt[c*d^2 - a*e^2])))/d))/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 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 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 1121

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] 
/; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int 
egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
 

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (a \,e^{2}-c \,d^{2}\right )^{3} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )-\sqrt {e x +d}\, \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, \left (\frac {\left (d^{2} e^{2} x^{2}+\frac {11}{3} d^{3} e x +\frac {23}{3} d^{4}\right ) c^{2}}{5}-\frac {7 e^{2} \left (d +\frac {e x}{7}\right ) a d c}{3}+a^{2} e^{4}\right )\right )}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, d^{3} c^{3}}\)	\(146\)
risch	\(\frac {2 \left (3 x^{2} c^{2} d^{2} e^{2}-5 x a c d \,e^{3}+11 x \,c^{2} d^{3} e +15 a^{2} e^{4}-35 a c \,d^{2} e^{2}+23 c^{2} d^{4}\right ) \sqrt {e x +d}}{15 d^{3} c^{3}}-\frac {2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{d^{3} c^{3} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\)	\(170\)
derivativedivides	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a^{2} e^{4} \sqrt {e x +d}-4 a c \,d^{2} e^{2} \sqrt {e x +d}+2 c^{2} d^{4} \sqrt {e x +d}}{d^{3} c^{3}}+\frac {2 \left (-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{3} d^{3} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\)	\(194\)
default	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a^{2} e^{4} \sqrt {e x +d}-4 a c \,d^{2} e^{2} \sqrt {e x +d}+2 c^{2} d^{4} \sqrt {e x +d}}{d^{3} c^{3}}+\frac {2 \left (-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{3} d^{3} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\)	\(194\)

Input:

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x,method=_RETURNVERBOS 
E)
 

Output:

-2*((a*e^2-c*d^2)^3*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^2))^(1/2))-(e 
*x+d)^(1/2)*(c*d*(a*e^2-c*d^2))^(1/2)*(1/5*(d^2*e^2*x^2+11/3*d^3*e*x+23/3* 
d^4)*c^2-7/3*e^2*(d+1/7*e*x)*a*d*c+a^2*e^4))/(c*d*(a*e^2-c*d^2))^(1/2)/d^3 
/c^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 23 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + {\left (11 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, c^{3} d^{3}}, -\frac {2 \, {\left (15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 23 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + {\left (11 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{15 \, c^{3} d^{3}}\right ] \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fr 
icas")
 

Output:

[1/15*(15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt((c*d^2 - a*e^2)/(c*d))* 
log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/ 
(c*d)))/(c*d*x + a*e)) + 2*(3*c^2*d^2*e^2*x^2 + 23*c^2*d^4 - 35*a*c*d^2*e^ 
2 + 15*a^2*e^4 + (11*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^3*d^3), 
 -2/15*(15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-(c*d^2 - a*e^2)/(c*d) 
)*arctan(-sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) 
- (3*c^2*d^2*e^2*x^2 + 23*c^2*d^4 - 35*a*c*d^2*e^2 + 15*a^2*e^4 + (11*c^2* 
d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^3*d^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="ma 
xima")
 

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*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{3} d^{3}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{4} d^{4} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{5} + 15 \, \sqrt {e x + d} c^{4} d^{6} - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{2} - 30 \, \sqrt {e x + d} a c^{3} d^{4} e^{2} + 15 \, \sqrt {e x + d} a^{2} c^{2} d^{2} e^{4}\right )}}{15 \, c^{5} d^{5}} \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="gi 
ac")
 

Output:

2*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*arctan(sqrt(e*x 
+ d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/(sqrt(-c^2*d^3 + a*c*d*e^2)*c^3*d^3) 
+ 2/15*(3*(e*x + d)^(5/2)*c^4*d^4 + 5*(e*x + d)^(3/2)*c^4*d^5 + 15*sqrt(e* 
x + d)*c^4*d^6 - 5*(e*x + d)^(3/2)*a*c^3*d^3*e^2 - 30*sqrt(e*x + d)*a*c^3* 
d^4*e^2 + 15*sqrt(e*x + d)*a^2*c^2*d^2*e^4)/(c^5*d^5)
 

Mupad [B] (verification not implemented)

Time = 5.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,{\left (d+e\,x\right )}^{5/2}}{5\,c\,d}+\frac {2\,{\left (a\,e^2-c\,d^2\right )}^2\,\sqrt {d+e\,x}}{c^3\,d^3}-\frac {2\,\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,c^2\,d^2}-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,{\left (a\,e^2-c\,d^2\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}\right )\,{\left (a\,e^2-c\,d^2\right )}^{5/2}}{c^{7/2}\,d^{7/2}} \] Input:

int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)
 

Output:

(2*(d + e*x)^(5/2))/(5*c*d) + (2*(a*e^2 - c*d^2)^2*(d + e*x)^(1/2))/(c^3*d 
^3) - (2*(a*e^2 - c*d^2)*(d + e*x)^(3/2))/(3*c^2*d^2) - (2*atan((c^(1/2)*d 
^(1/2)*(a*e^2 - c*d^2)^(5/2)*(d + e*x)^(1/2))/(a^3*e^6 - c^3*d^6 + 3*a*c^2 
*d^4*e^2 - 3*a^2*c*d^2*e^4))*(a*e^2 - c*d^2)^(5/2))/(c^(7/2)*d^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {-2 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}+4 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2}-2 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4}+2 \sqrt {e x +d}\, a^{2} c d \,e^{4}-\frac {14 \sqrt {e x +d}\, a \,c^{2} d^{3} e^{2}}{3}-\frac {2 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{3} x}{3}+\frac {46 \sqrt {e x +d}\, c^{3} d^{5}}{15}+\frac {22 \sqrt {e x +d}\, c^{3} d^{4} e x}{15}+\frac {2 \sqrt {e x +d}\, c^{3} d^{3} e^{2} x^{2}}{5}}{c^{4} d^{4}} \] Input:

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
 

Output:

(2*( - 15*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/( 
sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**2*e**4 + 30*sqrt(d)*sqrt(c)*sqr 
t(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - 
 c*d**2)))*a*c*d**2*e**2 - 15*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan(( 
sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*c**2*d**4 + 15 
*sqrt(d + e*x)*a**2*c*d*e**4 - 35*sqrt(d + e*x)*a*c**2*d**3*e**2 - 5*sqrt( 
d + e*x)*a*c**2*d**2*e**3*x + 23*sqrt(d + e*x)*c**3*d**5 + 11*sqrt(d + e*x 
)*c**3*d**4*e*x + 3*sqrt(d + e*x)*c**3*d**3*e**2*x**2))/(15*c**4*d**4)