\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^3} \, dx\) [215]

Optimal result

Integrand size = 37, antiderivative size = 160 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^2}-\frac {3 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {e} \left (-2 a e^2+c d (3 d+e x)\right )}{(a e+c d x) (d+e x)^2}-\frac {3 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{e^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1125, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e* 
x) - (c*d*(-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] + (3*(c*d^2 - a*e^ 
2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d 
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[c]*Sqrt[d]*Sqrt[e])))/e^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1125

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 
2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[(1/Sqrt[a + b*x + c*x^2])*Expan 
dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x 
), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && ILtQ[m, 0] && EqQ[m + p, -3/2]
 

rule 1160

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs. \(2(142)=284\).

Time = 2.10 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.45

method	result	size
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}}{e^{3}}\)	\(392\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^3,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^3*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^ 
(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a 
*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c*d^2)*(1/3*(d*e*c*(x+d/e)^2+(a* 
e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c* 
d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2) 
^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/ 
e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\left [-\frac {3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {\frac {c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, {\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {c d}{e}} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{4 \, {\left (e^{3} x + d e^{2}\right )}}, \frac {3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {-\frac {c d}{e}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {c d}{e}}}{2 \, {\left (c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{2 \, {\left (e^{3} x + d e^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(3*(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(c*d/e)*log(8*c^2*d^2 
*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + 
a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^ 
3*e + a*c*d*e^3)*x) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*e 
*x + 3*c*d^2 - 2*a*e^2))/(e^3*x + d*e^2), 1/2*(3*(c*d^3 - a*d*e^2 + (c*d^2 
*e - a*e^3)*x)*sqrt(-c*d/e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a 
*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + a*c*d^2 
*e + (c^2*d^3 + a*c*d*e^2)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2 
)*x)*(c*d*e*x + 3*c*d^2 - 2*a*e^2))/(e^3*x + d*e^2)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^3,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(e*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} c d}{e^{2}} + \frac {3 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2 \, \sqrt {c d e} e^{2}} + \frac {2 \, {\left (\sqrt {c d e} c^{2} d^{4} - 2 \, \sqrt {c d e} a c d^{2} e^{2} + \sqrt {c d e} a^{2} e^{4}\right )}}{\sqrt {c d e} {\left ({\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} e + \sqrt {c d e} d\right )} e^{2}} \] Input:

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

Output:

sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*c*d/e^2 + 3/2*(c^2*d^3 - a*c*d 
*e^2)*log(abs(-c*d^2 - a*e^2 - 2*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))))/(sqrt(c*d*e)*e^2) + 2*(sqrt(c*d*e)*c^2* 
d^4 - 2*sqrt(c*d*e)*a*c*d^2*e^2 + sqrt(c*d*e)*a^2*e^4)/(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))*e + sqrt(c*d*e)*d 
)*e^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {-8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,e^{3}+12 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c \,d^{2} e +4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c d \,e^{2} x +12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a d \,e^{2}+12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{3} x -12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{3}-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2} e x -9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a d \,e^{2}-9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a \,e^{3} x +9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{3}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{2} e x}{4 e^{3} \left (e x +d \right )} \] Input:

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

Output:

( - 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*e**3 + 12*sqrt(d + e*x)*sqrt(a*e + 
 c*d*x)*c*d**2*e + 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c*d*e**2*x + 12*sqrt( 
e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d 
 + e*x))/sqrt(a*e**2 - c*d**2))*a*d*e**2 + 12*sqrt(e)*sqrt(d)*sqrt(c)*log( 
(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - 
c*d**2))*a*e**3*x - 12*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d 
*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c*d**3 - 12*sq 
rt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqr 
t(d + e*x))/sqrt(a*e**2 - c*d**2))*c*d**2*e*x - 9*sqrt(e)*sqrt(d)*sqrt(c)* 
a*d*e**2 - 9*sqrt(e)*sqrt(d)*sqrt(c)*a*e**3*x + 9*sqrt(e)*sqrt(d)*sqrt(c)* 
c*d**3 + 9*sqrt(e)*sqrt(d)*sqrt(c)*c*d**2*e*x)/(4*e**3*(d + e*x))