\(\int \frac {1}{x^2 (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [129]

Optimal result

Integrand size = 40, antiderivative size = 515 \[ \int \frac {1}{x^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {c \left (3 c d^2-a e^2\right )}{a^2 e^2 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {1}{a d e x (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (15 c^2 d^4-10 a c d^2 e^2+7 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 a^2 d^2 e \left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac {\left (45 c^3 d^6-45 a c^2 d^4 e^2+83 a^2 c d^2 e^4-35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 a^2 d^3 e \left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {\left (45 c^4 d^8-60 a c^3 d^6 e^2+346 a^2 c^2 d^4 e^4-340 a^3 c d^2 e^6+105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 a^2 d^4 e \left (c d^2-a e^2\right )^4 (d+e x)}+\frac {\left (3 c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{5/2} d^{9/2} e^{5/2}} \] Output:

-c*(-a*e^2+3*c*d^2)/a^2/e^2/(-a*e^2+c*d^2)/(e*x+d)^2/(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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 
1/2)-1/5*(7*a^2*e^4-10*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^2/d^2/e/(-a*e^2+c*d^2)^2/(e*x+d)^3-1/15*(-35*a^3*e^6+83*a^2 
*c*d^2*e^4-45*a*c^2*d^4*e^2+45*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)^2-1/15*(105*a^4*e^8-340*a^3*c*d^2 
*e^6+346*a^2*c^2*d^4*e^4-60*a*c^3*d^6*e^2+45*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)+(7*a*e^2+3*c*d^2)*a 
rctanh(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^(9/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (45 c^5 d^9 x (d+e x)^3+15 a c^4 d^7 e (d-4 e x) (d+e x)^3+a^5 e^9 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )+2 a^3 c^2 d^3 e^5 \left (45 d^4+255 d^3 e x+145 d^2 e^2 x^2-226 d e^3 x^3-170 e^4 x^4\right )-a^4 c d e^7 \left (60 d^4+515 d^3 e x+637 d^2 e^2 x^2+95 d e^3 x^3-105 e^4 x^4\right )+2 a^2 c^3 d^5 e^3 \left (-30 d^4-45 d^3 e x+195 d^2 e^2 x^2+380 d e^3 x^3+173 e^4 x^4\right )\right )}{\left (c d^2-a e^2\right )^4 x (d+e x)}+15 \left (3 c d^2+7 a e^2\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{15 a^{5/2} d^{9/2} e^{5/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[1/(x^2*(d + e*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 + c*d*x)*(45*c^5*d^9*x*(d + e*x)^3 + 15*a 
*c^4*d^7*e*(d - 4*e*x)*(d + e*x)^3 + a^5*e^9*(15*d^3 + 161*d^2*e*x + 245*d 
*e^2*x^2 + 105*e^3*x^3) + 2*a^3*c^2*d^3*e^5*(45*d^4 + 255*d^3*e*x + 145*d^ 
2*e^2*x^2 - 226*d*e^3*x^3 - 170*e^4*x^4) - a^4*c*d*e^7*(60*d^4 + 515*d^3*e 
*x + 637*d^2*e^2*x^2 + 95*d*e^3*x^3 - 105*e^4*x^4) + 2*a^2*c^3*d^5*e^3*(-3 
0*d^4 - 45*d^3*e*x + 195*d^2*e^2*x^2 + 380*d*e^3*x^3 + 173*e^4*x^4)))/((c* 
d^2 - a*e^2)^4*x*(d + e*x))) + 15*(3*c*d^2 + 7*a*e^2)*(a*e + c*d*x)^(3/2)* 
(d + e*x)^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e 
+ c*d*x])])/(15*a^(5/2)*d^(9/2)*e^(5/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.48, 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.

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

\(\Big \downarrow \) 1259

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{a^{3/2} d^{9/2} \sqrt {e}}+\frac {3 \left (a e^2+c d^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} d^{9/2} e^{5/2}}-\frac {4 \left (a^2 e^4+c d e x \left (a e^2+c d^2\right )+c^2 d^4\right )}{a d^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (3 a^2 e^4-2 a c d^2 e^2+3 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 d^4 e^2 x \left (c d^2-a e^2\right )^2}+\frac {2 \left (a^2 e^4+c d e x \left (a e^2+c d^2\right )+c^2 d^4\right )}{a d^3 e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c e^2 \left (a e^2+c d^2+2 c d e x\right )}{3 d^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {4 c e^2}{5 d (d+e x) \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2 e^2}{5 d^2 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {4 e^2}{3 d^3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[1/(x^2*(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^2)/(5*d^2*(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^2)/(5*d*(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]) + (4*e^2)/(3*d^3*(c*d^2 - a*e^2)*(d + e*x)*S 
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (16*c^2*e^2*(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]) - (16*c*e^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*d^2*(c*d^2 - a*e^2)^3*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*(c^2*d^4 + a^2*e^4 + c*d*e* 
(c*d^2 + a*e^2)*x))/(a*d^4*(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^3 
*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^4*e^2*(c*d^2 - a*e^2)^2*x) + (2*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^(9/2)*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]*Sqrt[a*d*e + (c*d^2 + a*e 
^2)*x + c*d*e*x^2])])/(2*a^(5/2)*d^(9/2)*e^(5/2))
 

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 = 3.02 (sec) , antiderivative size = 943, normalized size of antiderivative = 1.83

method	result	size
default	\(\frac {-\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}}}{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 )}}{d^{2}}-\frac {2 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 )}{d^{3}}+\frac {2 e \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 )}{d^{3}}\)	\(943\)

Input:

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

Output:

1/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)*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/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)))-2/d^3*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))+2*e/d^3 
*(-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. 1321 vs. \(2 (485) = 970\).

Time = 23.56 (sec) , antiderivative size = 2662, normalized size of antiderivative = 5.17 \[ \int \frac {1}{x^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/60*(15*((3*c^6*d^11*e^3 - 5*a*c^5*d^9*e^5 - 10*a^2*c^4*d^7*e^7 + 30*a^3 
*c^3*d^5*e^9 - 25*a^4*c^2*d^3*e^11 + 7*a^5*c*d*e^13)*x^5 + (9*c^6*d^12*e^2 
 - 12*a*c^5*d^10*e^4 - 35*a^2*c^4*d^8*e^6 + 80*a^3*c^3*d^6*e^8 - 45*a^4*c^ 
2*d^4*e^10 - 4*a^5*c*d^2*e^12 + 7*a^6*e^14)*x^4 + 3*(3*c^6*d^13*e - 2*a*c^ 
5*d^11*e^3 - 15*a^2*c^4*d^9*e^5 + 20*a^3*c^3*d^7*e^7 + 5*a^4*c^2*d^5*e^9 - 
 18*a^5*c*d^3*e^11 + 7*a^6*d*e^13)*x^3 + (3*c^6*d^14 + 4*a*c^5*d^12*e^2 - 
25*a^2*c^4*d^10*e^4 + 65*a^4*c^2*d^6*e^8 - 68*a^5*c*d^4*e^10 + 21*a^6*d^2* 
e^12)*x^2 + (3*a*c^5*d^13*e - 5*a^2*c^4*d^11*e^3 - 10*a^3*c^3*d^9*e^5 + 30 
*a^4*c^2*d^7*e^7 - 25*a^5*c*d^5*e^9 + 7*a^6*d^3*e^11)*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^2*c^4*d^12*e^2 - 60*a^3*c^3*d^ 
10*e^4 + 90*a^4*c^2*d^8*e^6 - 60*a^5*c*d^6*e^8 + 15*a^6*d^4*e^10 + (45*a*c 
^5*d^10*e^4 - 60*a^2*c^4*d^8*e^6 + 346*a^3*c^3*d^6*e^8 - 340*a^4*c^2*d^4*e 
^10 + 105*a^5*c*d^2*e^12)*x^4 + (135*a*c^5*d^11*e^3 - 165*a^2*c^4*d^9*e^5 
+ 760*a^3*c^3*d^7*e^7 - 452*a^4*c^2*d^5*e^9 - 95*a^5*c*d^3*e^11 + 105*a^6* 
d*e^13)*x^3 + (135*a*c^5*d^12*e^2 - 135*a^2*c^4*d^10*e^4 + 390*a^3*c^3*d^8 
*e^6 + 290*a^4*c^2*d^6*e^8 - 637*a^5*c*d^4*e^10 + 245*a^6*d^2*e^12)*x^2 + 
(45*a*c^5*d^13*e - 15*a^2*c^4*d^11*e^3 - 90*a^3*c^3*d^9*e^5 + 510*a^4*c^2* 
d^7*e^7 - 515*a^5*c*d^5*e^9 + 161*a^6*d^3*e^11)*x)*sqrt(c*d*e*x^2 + a*d...
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4890 vs. \(2 (485) = 970\).

Time = 0.29 (sec) , antiderivative size = 4890, normalized size of antiderivative = 9.50 \[ \int \frac {1}{x^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/15*(e^7*(2*(150*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^18* 
d^42*e^30*sgn(1/(e*x + d))^4*sgn(e)^4 - 2550*sqrt(c*d*e - c*d^2*e/(e*x + d 
) + a*e^3/(e*x + d))*a*c^17*d^40*e^32*sgn(1/(e*x + d))^4*sgn(e)^4 + 20445* 
sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c^16*d^38*e^34*sgn(1 
/(e*x + d))^4*sgn(e)^4 - 102720*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e* 
x + d))*a^3*c^15*d^36*e^36*sgn(1/(e*x + d))^4*sgn(e)^4 + 362400*sqrt(c*d*e 
 - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c^14*d^34*e^38*sgn(1/(e*x + d) 
)^4*sgn(e)^4 - 953400*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^ 
5*c^13*d^32*e^40*sgn(1/(e*x + d))^4*sgn(e)^4 + 1938300*sqrt(c*d*e - c*d^2* 
e/(e*x + d) + a*e^3/(e*x + d))*a^6*c^12*d^30*e^42*sgn(1/(e*x + d))^4*sgn(e 
)^4 - 3113760*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^7*c^11*d 
^28*e^44*sgn(1/(e*x + d))^4*sgn(e)^4 + 4006860*sqrt(c*d*e - c*d^2*e/(e*x + 
 d) + a*e^3/(e*x + d))*a^8*c^10*d^26*e^46*sgn(1/(e*x + d))^4*sgn(e)^4 - 41 
61300*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^9*c^9*d^24*e^48* 
sgn(1/(e*x + d))^4*sgn(e)^4 + 3496350*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e 
^3/(e*x + d))*a^10*c^8*d^22*e^50*sgn(1/(e*x + d))^4*sgn(e)^4 - 2371200*sqr 
t(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^11*c^7*d^20*e^52*sgn(1/(e 
*x + d))^4*sgn(e)^4 + 1288560*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x 
+ d))*a^12*c^6*d^18*e^54*sgn(1/(e*x + d))^4*sgn(e)^4 - 553560*sqrt(c*d*e - 
 c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^13*c^5*d^16*e^56*sgn(1/(e*x + d...
 

Mupad [F(-1)]

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

int(1/(x^2*(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^2*(d + e*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 = 20.66 (sec) , antiderivative size = 7938, normalized size of antiderivative = 15.41 \[ \int \frac {1}{x^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 315*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)*sq 
rt(d + e*x))*a**6*d**3*e**12*x - 945*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**6*d**2*e**13*x**2 - 945*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**6*d*e**14*x**3 - 315*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) + sq 
rt(d)*sqrt(c)*sqrt(d + e*x))*a**6*e**15*x**4 + 1230*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**5*c*d**5*e**10 
*x + 3690*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**5*c*d**4*e**11*x**2 + 3690*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**5*c*d**3*e**12*x**3 + 
 1230*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**5*c*d**2*e**13*x**4 - 1725*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a...