\(\int \frac {1}{x (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [128]

Optimal result

Integrand size = 40, antiderivative size = 389 \[ \int \frac {1}{x (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d}{a e \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (5 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 a d \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {2 \left (15 c^2 d^4+14 a c d^2 e^2-5 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 a d^2 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {2 \left (15 c^3 d^6+73 a c^2 d^4 e^2-55 a^2 c d^2 e^4+15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 a d^3 \left (c d^2-a e^2\right )^4 (d+e x)}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} d^{7/2} e^{3/2}} \] Output:

2*c*d/a/e/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
+2/5*(a*e^2+5*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d/(-a*e^2+c 
*d^2)^2/(e*x+d)^3+2/15*(-5*a^2*e^4+14*a*c*d^2*e^2+15*c^2*d^4)*(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d^2/(-a*e^2+c*d^2)^3/(e*x+d)^2+2/15*(15*a^3* 
e^6-55*a^2*c*d^2*e^4+73*a*c^2*d^4*e^2+15*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c 
*d*e*x^2)^(1/2)/a/d^3/(-a*e^2+c*d^2)^4/(e*x+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^(3/2)/d^(7/2)/e^ 
(3/2)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (15 c^4 d^7 (d+e x)^3+a^4 e^8 \left (23 d^2+35 d e x+15 e^2 x^2\right )+a c^3 d^5 e^3 x \left (90 d^2+160 d e x+73 e^2 x^2\right )+a^2 c^2 d^3 e^4 \left (90 d^3+80 d^2 e x-56 d e^2 x^2-55 e^3 x^3\right )-a^3 c d e^6 \left (80 d^3+106 d^2 e x+20 d e^2 x^2-15 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4 (d+e x)}-15 (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{15 a^{3/2} d^{7/2} e^{3/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

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

Output:

(2*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(15*c^4*d^7*(d + e*x)^3 + a^4*e 
^8*(23*d^2 + 35*d*e*x + 15*e^2*x^2) + a*c^3*d^5*e^3*x*(90*d^2 + 160*d*e*x 
+ 73*e^2*x^2) + a^2*c^2*d^3*e^4*(90*d^3 + 80*d^2*e*x - 56*d*e^2*x^2 - 55*e 
^3*x^3) - a^3*c*d*e^6*(80*d^3 + 106*d^2*e*x + 20*d*e^2*x^2 - 15*e^3*x^3))) 
/((c*d^2 - a*e^2)^4*(d + e*x)) - 15*(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])]))/(15*a 
^(3/2)*d^(7/2)*e^(3/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.24, 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.

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

\(\Big \downarrow \) 1259

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*e)/(5*d*(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]) - (4*c*e)/(5*(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)/(3*d^2*(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]) + (16*c^2*d*e*(c*d^2 + a*e^2 + 2*c*d*e* 
x))/(5*(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*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*d*(c*d^2 - a*e^2)^3*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^3*e*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2]) - ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])]/(a^(3/2)*d^(7/2)*e^(3/2))
 

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 [A] (verified)

Time = 2.70 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}}{d^{2}}-\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}}{d^{2}}-\frac {-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e d}\)	\(590\)

Input:

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

Output:

1/d^2*(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))-1/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))-1/e/d*(-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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1107 vs. \(2 (361) = 722\).

Time = 9.49 (sec) , antiderivative size = 2234, normalized size of antiderivative = 5.74 \[ \int \frac {1}{x (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30*(15*(a*c^4*d^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c* 
d^5*e^7 + a^5*d^3*e^9 + (c^5*d^9*e^3 - 4*a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 
 - 4*a^3*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8* 
e^4 + 14*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)* 
x^3 + 3*(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5* 
e^7 - 3*a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a 
^2*c^3*d^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x 
)*sqrt(a*d*e)*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*(15*a*c^4*d^11*e 
+ 90*a^3*c^2*d^7*e^5 - 80*a^4*c*d^5*e^7 + 23*a^5*d^3*e^9 + (15*a*c^4*d^8*e 
^4 + 73*a^2*c^3*d^6*e^6 - 55*a^3*c^2*d^4*e^8 + 15*a^4*c*d^2*e^10)*x^3 + (4 
5*a*c^4*d^9*e^3 + 160*a^2*c^3*d^7*e^5 - 56*a^3*c^2*d^5*e^7 - 20*a^4*c*d^3* 
e^9 + 15*a^5*d*e^11)*x^2 + (45*a*c^4*d^10*e^2 + 90*a^2*c^3*d^8*e^4 + 80*a^ 
3*c^2*d^6*e^6 - 106*a^4*c*d^4*e^8 + 35*a^5*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a 
*d*e + (c*d^2 + a*e^2)*x))/(a^3*c^4*d^15*e^3 - 4*a^4*c^3*d^13*e^5 + 6*a^5* 
c^2*d^11*e^7 - 4*a^6*c*d^9*e^9 + a^7*d^7*e^11 + (a^2*c^5*d^13*e^5 - 4*a^3* 
c^4*d^11*e^7 + 6*a^4*c^3*d^9*e^9 - 4*a^5*c^2*d^7*e^11 + a^6*c*d^5*e^13)*x^ 
4 + (3*a^2*c^5*d^14*e^4 - 11*a^3*c^4*d^12*e^6 + 14*a^4*c^3*d^10*e^8 - 6*a^ 
5*c^2*d^8*e^10 - a^6*c*d^6*e^12 + a^7*d^4*e^14)*x^3 + 3*(a^2*c^5*d^15*e...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (361) = 722\).

Time = 0.19 (sec) , antiderivative size = 998, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/15*((15*c^4*d^4*e^3/(sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a 
*sgn(1/(e*x + d))*sgn(e)) + 15*(c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a^2*c^2* 
d^4*e^7 - 4*a^3*c*d^2*e^9 + a^4*e^11)*arctan(sqrt(c*d*e - c*d^2*e/(e*x + d 
) + a*e^3/(e*x + d))*d/(sqrt(-a*d*e)*e))/(sqrt(-a*d*e)*a*d^3*e*sgn(1/(e*x 
+ d))*sgn(e)) + (90*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^2* 
d^6*e^4*sgn(1/(e*x + d))^4*sgn(e)^4 - 60*sqrt(c*d*e - c*d^2*e/(e*x + d) + 
a*e^3/(e*x + d))*a*c*d^4*e^6*sgn(1/(e*x + d))^4*sgn(e)^4 + 15*sqrt(c*d*e - 
 c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*d^2*e^8*sgn(1/(e*x + d))^4*sgn(e 
)^4 - 20*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c*d^5*e^3*sgn 
(1/(e*x + d))^4*sgn(e)^4 + 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d)) 
^(3/2)*a*d^3*e^5*sgn(1/(e*x + d))^4*sgn(e)^4 + 3*(c*d*e - c*d^2*e/(e*x + d 
) + a*e^3/(e*x + d))^(5/2)*d^4*e^2*sgn(1/(e*x + d))^4*sgn(e)^4)/(d^5*sgn(1 
/(e*x + d))^5*sgn(e)^5))*e^4*abs(e)/(c*d^2*e^2 - a*e^4)^4 - (15*sqrt(c*d*e 
)*c^4*d^8*abs(e)*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) - 60*sqrt(c*d*e)*a 
*c^3*d^6*e^2*abs(e)*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + 90*sqrt(c*d*e 
)*a^2*c^2*d^4*e^4*abs(e)*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) - 60*sqrt( 
c*d*e)*a^3*c*d^2*e^6*abs(e)*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + 15*sq 
rt(c*d*e)*a^4*e^8*abs(e)*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + 15*sqrt( 
-a*d*e)*c^4*d^7*e*abs(e) + 73*sqrt(-a*d*e)*a*c^3*d^5*e^3*abs(e) - 55*sqrt( 
-a*d*e)*a^2*c^2*d^3*e^5*abs(e) + 15*sqrt(-a*d*e)*a^3*c*d*e^7*abs(e))*sg...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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