\(\int \frac {1}{x^2 (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [137]

Optimal result

Integrand size = 40, antiderivative size = 647 \[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {c \left (3 c d^2-a e^2\right )}{a^2 e^2 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {1}{a d e x (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (21 c^2 d^4-14 a c d^2 e^2+9 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 a^2 d^2 e \left (c d^2-a e^2\right )^2 (d+e x)^4}-\frac {3 \left (35 c^3 d^6-35 a c^2 d^4 e^2+53 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 a^2 d^3 e \left (c d^2-a e^2\right )^3 (d+e x)^3}-\frac {\left (105 c^4 d^8-140 a c^3 d^6 e^2+422 a^2 c^2 d^4 e^4-364 a^3 c d^2 e^6+105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 a^2 d^4 e \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {\left (105 c^5 d^{10}-175 a c^4 d^8 e^2+1474 a^2 c^3 d^6 e^4-2198 a^3 c^2 d^4 e^6+1365 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 a^2 d^5 e \left (c d^2-a e^2\right )^5 (d+e x)}+\frac {3 \left (c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{5/2} d^{11/2} e^{5/2}} \] Output:

-c*(-a*e^2+3*c*d^2)/a^2/e^2/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*e*x^2)^(1/2)-1/a/d/e/x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 
1/2)-1/7*(9*a^2*e^4-14*a*c*d^2*e^2+21*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^(1/2)/a^2/d^2/e/(-a*e^2+c*d^2)^2/(e*x+d)^4-3/35*(-21*a^3*e^6+53*a^2 
*c*d^2*e^4-35*a*c^2*d^4*e^2+35*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ 
(1/2)/a^2/d^3/e/(-a*e^2+c*d^2)^3/(e*x+d)^3-1/35*(105*a^4*e^8-364*a^3*c*d^2 
*e^6+422*a^2*c^2*d^4*e^4-140*a*c^3*d^6*e^2+105*c^4*d^8)*(a*d*e+(a*e^2+c*d^ 
2)*x+c*d*e*x^2)^(1/2)/a^2/d^4/e/(-a*e^2+c*d^2)^4/(e*x+d)^2-1/35*(-315*a^5* 
e^10+1365*a^4*c*d^2*e^8-2198*a^3*c^2*d^4*e^6+1474*a^2*c^3*d^6*e^4-175*a*c^ 
4*d^8*e^2+105*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^5/e/ 
(-a*e^2+c*d^2)^5/(e*x+d)+3*(3*a*e^2+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^(11/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-105 c^6 d^{11} x (d+e x)^4-35 a c^5 d^9 e (d-5 e x) (d+e x)^4+a^6 e^{11} \left (35 d^4+528 d^3 e x+1218 d^2 e^2 x^2+1050 d e^3 x^3+315 e^4 x^4\right )+a^2 c^4 d^7 e^3 \left (175 d^5+350 d^4 e x-1750 d^3 e^2 x^2-5250 d^2 e^3 x^3-4809 d e^4 x^4-1474 e^5 x^5\right )+a^4 c^2 d^3 e^7 \left (350 d^5+3675 d^4 e x+6328 d^3 e^2 x^2+2062 d^2 e^3 x^3-2366 d e^4 x^4-1365 e^5 x^5\right )-a^5 c d e^9 \left (175 d^5+2289 d^4 e x+4790 d^3 e^2 x^2+3346 d^2 e^3 x^3+315 d e^4 x^4-315 e^5 x^5\right )+2 a^3 c^3 d^5 e^5 \left (-175 d^5-1225 d^4 e x-1050 d^3 e^2 x^2+1834 d^2 e^3 x^3+2953 d e^4 x^4+1099 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 x (d+e x)^2}+105 \left (c d^2+3 a e^2\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{35 a^{5/2} d^{11/2} e^{5/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[1/(x^2*(d + e*x)^3*(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 + c*d*x)*(-105*c^6*d^11*x*(d + e*x)^4 - 35* 
a*c^5*d^9*e*(d - 5*e*x)*(d + e*x)^4 + a^6*e^11*(35*d^4 + 528*d^3*e*x + 121 
8*d^2*e^2*x^2 + 1050*d*e^3*x^3 + 315*e^4*x^4) + a^2*c^4*d^7*e^3*(175*d^5 + 
 350*d^4*e*x - 1750*d^3*e^2*x^2 - 5250*d^2*e^3*x^3 - 4809*d*e^4*x^4 - 1474 
*e^5*x^5) + a^4*c^2*d^3*e^7*(350*d^5 + 3675*d^4*e*x + 6328*d^3*e^2*x^2 + 2 
062*d^2*e^3*x^3 - 2366*d*e^4*x^4 - 1365*e^5*x^5) - a^5*c*d*e^9*(175*d^5 + 
2289*d^4*e*x + 4790*d^3*e^2*x^2 + 3346*d^2*e^3*x^3 + 315*d*e^4*x^4 - 315*e 
^5*x^5) + 2*a^3*c^3*d^5*e^5*(-175*d^5 - 1225*d^4*e*x - 1050*d^3*e^2*x^2 + 
1834*d^2*e^3*x^3 + 2953*d*e^4*x^4 + 1099*e^5*x^5)))/((c*d^2 - a*e^2)^5*x*( 
d + e*x)^2) + 105*(c*d^2 + 3*a*e^2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*Ar 
cTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(35*a^ 
(5/2)*d^(11/2)*e^(5/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 1011, normalized size of antiderivative = 1.56, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1259, 2009}

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.

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

\(\Big \downarrow \) 1259

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {128 d e^2 \left (c d^2+2 c e x d+a e^2\right ) c^3}{35 \left (c d^2-a e^2\right )^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\frac {32 e^2 \left (c d^2+2 c e x d+a e^2\right ) c^2}{5 d \left (c d^2-a e^2\right )^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {32 e^2 c^2}{35 \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\frac {8 e^2 \left (c d^2+2 c e x d+a e^2\right ) c}{d^3 \left (c d^2-a e^2\right )^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {8 e^2 c}{5 d^2 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {16 e^2 c}{35 d \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {3 \left (c d^2+a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{2 a^{5/2} d^{11/2} e^{5/2}}+\frac {3 \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{a^{3/2} d^{11/2} \sqrt {e}}-\frac {\left (3 c^2 d^4-2 a c e^2 d^2+3 a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{a^2 d^5 e^2 \left (c d^2-a e^2\right )^2 x}+\frac {2 \left (c^2 d^4+c e \left (c d^2+a e^2\right ) x d+a^2 e^4\right )}{a d^4 e \left (c d^2-a e^2\right )^2 x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\frac {6 \left (c^2 d^4+c e \left (c d^2+a e^2\right ) x d+a^2 e^4\right )}{a d^5 \left (c d^2-a e^2\right )^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {2 e^2}{d^4 \left (c d^2-a e^2\right ) (d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {4 e^2}{5 d^3 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {2 e^2}{7 d^2 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\)

Input:

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

Output:

(2*e^2)/(7*d^2*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
+ c*d*e*x^2]) + (16*c*e^2)/(35*d*(c*d^2 - a*e^2)^2*(d + e*x)^2*Sqrt[a*d*e 
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (4*e^2)/(5*d^3*(c*d^2 - a*e^2)*(d + e* 
x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*e^2)/(35*(c*d^ 
2 - a*e^2)^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (8*c 
*e^2)/(5*d^2*(c*d^2 - a*e^2)^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2]) + (2*e^2)/(d^4*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + 
 a*e^2)*x + c*d*e*x^2]) - (128*c^3*d*e^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(35* 
(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (32*c^2*e 
^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(5*d*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 
 + a*e^2)*x + c*d*e*x^2]) - (8*c*e^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(d^3*(c* 
d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (6*(c^2*d^4 
+ a^2*e^4 + c*d*e*(c*d^2 + a*e^2)*x))/(a*d^5*(c*d^2 - a*e^2)^2*Sqrt[a*d*e 
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*(c^2*d^4 + a^2*e^4 + c*d*e*(c*d^2 + 
 a*e^2)*x))/(a*d^4*e*(c*d^2 - a*e^2)^2*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2]) - ((3*c^2*d^4 - 2*a*c*d^2*e^2 + 3*a^2*e^4)*Sqrt[a*d*e + (c*d^2 
 + a*e^2)*x + c*d*e*x^2])/(a^2*d^5*e^2*(c*d^2 - a*e^2)^2*x) + (3*ArcTanh[( 
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])])/(a^(3/2)*d^(11/2)*Sqrt[e]) + (3*(c*d^2 + a*e^ 
2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqr...
 

Defintions of rubi rules used

rule 1259

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1253\) vs. \(2(613)=1226\).

Time = 3.72 (sec) , antiderivative size = 1254, normalized size of antiderivative = 1.94

method	result	size
default	\(\text {Expression too large to display}\)	\(1254\)

Input:

int(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETU 
RNVERBOSE)
 

Output:

1/d^3*(-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)*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))-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))+1/e/d^2*(-2/7/(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))^(1/2)-8/7*d*e*c/(a*e^2-c*d^2)*(-2/5/(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))^(1/2)-6/5*d*e*c/(a*e^2-c*d^ 
2)*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/ 
2)+8/3*d*e*c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^ 
2+(a*e^2-c*d^2)*(x+d/e))^(1/2))))-3/d^4*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))+3*e/d^4*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2 
+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a 
*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))+2/d^3*(-2/5/(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))^(1/2)-6/5*d*e 
*c/(a*e^2-c*d^2)*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1837 vs. \(2 (613) = 1226\).

Time = 52.43 (sec) , antiderivative size = 3694, normalized size of antiderivative = 5.71 \[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:

integrate(1/x**2/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(x**2*((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} x^{2}} \,d x } \] Input:

integrate(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="maxima")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3*x^2 
), x)
 

Giac [F]

\[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} x^{2}} \,d x } \] Input:

integrate(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="giac")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3*x^2 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (d+e\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int(1/(x^2*(d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
 

Output:

int(1/(x^2*(d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (e x +d \right )^{3} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}d x \] Input:

int(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

int(1/x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)