\(\int \frac {(d+e x)^2}{x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [100]

Optimal result

Integrand size = 40, antiderivative size = 165 \[ \int \frac {(d+e x)^2}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 c d (d+e x)}{a^2 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a^2 e^2 x}+\frac {\left (3 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{5/2} \sqrt {d} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^2}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (a e+3 c d x) (d+e x)+\left (3 c d^2-a e^2\right ) x \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{a^{5/2} \sqrt {d} e^{5/2} x \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1212, 25, 27, 1228, 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.

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

\(\Big \downarrow \) 1212

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1212

Int[((x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.))/((a_) + (b_.)*(x_) + (c_.)*(x_) 
^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*d - b*e)^n*((d + e 
*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c*x^2])), x] - Simp[e^2/c^(m + 
n - 1)   Int[ExpandToSum[(c^(m + n - 1)*(d + e*x)^(m - 1) - ((c*d - b*e)^n* 
(2*c*d - b*e)^(m - 1))/(e^n*x^n))/(c*d - b*e - c*e*x), x]/(Sqrt[a + b*x + c 
*x^2]/x^n), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^ 
2, 0] && IGtQ[m, 0] && ILtQ[n, 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. \(644\) vs. \(2(147)=294\).

Time = 1.53 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.91

method	result	size
default	\(\frac {2 e^{2} \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+d^{2} \left (-\frac {1}{a d e x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+2 d e \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )\)	\(645\)

Input:

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

Output:

2*e^2*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(1/2)+d^2*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 
*e)^(1/2)-3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2 
+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/a/d/e/(a*d*e)^(1/2)*l 
n((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2))/x))-4*c/a*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2 
)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+2*d*e*(1/a/d/e/(a*d*e+(a*e^2+c* 
d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c 
*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/a/d/e/ 
(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c* 
d^2)*x+c*d*x^2*e)^(1/2))/x))
 

Fricas [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.84 \[ \int \frac {(d+e x)^2}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {\sqrt {a d e} {\left ({\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x^{2} + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} x\right )} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a c d^{2} e x + a^{2} d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left (a^{3} c d^{2} e^{3} x^{2} + a^{4} d e^{4} x\right )}}, -\frac {\sqrt {-a d e} {\left ({\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x^{2} + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} x\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (3 \, a c d^{2} e x + a^{2} d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (a^{3} c d^{2} e^{3} x^{2} + a^{4} d e^{4} x\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^2/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[1,1,0]%%%},[6,0]%%%}+%%%{%%{[%%%{-2,[0,0,1]%%%},0]: 
[1,0,%%%{
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.41 \[ \int \frac {(d+e x)^2}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a^{2} e^{4} x -9 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{4} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a^{2} e^{4} x -9 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{4} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a^{2} e^{4} x +9 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c^{2} d^{4} x +10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} d \,e^{3} x +6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{3} e x -2 \sqrt {e x +d}\, a^{3} d \,e^{4}-6 \sqrt {e x +d}\, a^{2} c \,d^{3} e^{2}-6 \sqrt {e x +d}\, a^{2} c \,d^{2} e^{3} x -18 \sqrt {e x +d}\, a \,c^{2} d^{4} e x}{2 \sqrt {c d x +a e}\, a^{3} d \,e^{3} x \left (a \,e^{2}+3 c \,d^{2}\right )} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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)*sqrt(d + 
e*x))*a**2*e**4*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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)*sqrt(d + e*x))*c**2*d**4*x + sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e 
+ c*d*x)*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)*sqrt(d + e*x))*a**2*e**4*x - 9*sqrt(e)*sqrt 
(d)*sqrt(a)*sqrt(a*e + c*d*x)*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)*sqrt(d + e*x))*c**2*d* 
*4*x - sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sqrt(d)*sqr 
t(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)* 
a**2*e**4*x + 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sq 
rt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2* 
c*d*e*x)*c**2*d**4*x + 10*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*d 
*e**3*x + 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**3*e*x - 2*sqr 
t(d + e*x)*a**3*d*e**4 - 6*sqrt(d + e*x)*a**2*c*d**3*e**2 - 6*sqrt(d + e*x 
)*a**2*c*d**2*e**3*x - 18*sqrt(d + e*x)*a*c**2*d**4*e*x)/(2*sqrt(a*e + c*d 
*x)*a**3*d*e**3*x*(a*e**2 + 3*c*d**2))