\(\int \frac {1}{x^3 (d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [86]

Optimal result

Integrand size = 40, antiderivative size = 296 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 d^3 e \left (c d^2-a e^2\right ) (d+e x)}-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d e x^2 (d+e x)}+\frac {\left (\frac {5 a}{d^2}+\frac {3 c}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 x (d+e x)}-\frac {3 \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{5/2} d^{7/2} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (3 c^3 d^5 x^2 (d+e x)+a^3 e^4 \left (2 d^2-5 d e x-15 e^2 x^2\right )+a c^2 d^3 e x \left (d^2+5 d e x+4 e^2 x^2\right )-a^2 c d e^2 \left (2 d^3-4 d^2 e x+d e^2 x^2+15 e^3 x^3\right )\right )-3 \left (c^3 d^6+a c^2 d^4 e^2+3 a^2 c d^2 e^4-5 a^3 e^6\right ) x^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{4 a^{5/2} d^{7/2} e^{5/2} \left (c d^2-a e^2\right ) x^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^3*d^5*x^2*(d + e*x) + a^3*e^4*(2*d^2 - 5*d*e 
*x - 15*e^2*x^2) + a*c^2*d^3*e*x*(d^2 + 5*d*e*x + 4*e^2*x^2) - a^2*c*d*e^2 
*(2*d^3 - 4*d^2*e*x + d*e^2*x^2 + 15*e^3*x^3)) - 3*(c^3*d^6 + a*c^2*d^4*e^ 
2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTa 
nh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(4*a^(5/2 
)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)*x^2*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1214, 25, 2181, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {\int \frac {\frac {7 a e^2}{d}+2 \left (c-\frac {2 a e^2}{d^2}\right ) x e+3 c d}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\frac {7 a e^2}{d}+2 \left (c-\frac {2 a e^2}{d^2}\right ) x e+3 c d}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {3 \left (\frac {c^2 d^2}{a}+\frac {5 a e^4}{d^2}+2 c e^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}-\frac {\left (\frac {3 c}{a e}+\frac {7 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{4 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {3 \left (\frac {c^2 d^2}{a}+\frac {5 a e^4}{d^2}+2 c e^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}-\frac {\left (\frac {3 c}{a e}+\frac {7 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{4 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(a*d^2*e*x^2) - (2*e^3*Sq 
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^3*(c*d^2 - a*e^2)*(d + e*x)) 
 - (-((((3*c)/(a*e) + (7*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2])/x) + (3*((c^2*d^2)/a + 2*c*e^2 + (5*a*e^4)/d^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*Sqrt[a]*Sqrt[d]*e^(3/2)))/(4*a*d*e)
 

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 1214

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 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[ExpandToS 
um[(((-d)^n*(-2*c*d + b*e)^(-m - 1))/(e^n*x^n) - ((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + 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] && ILtQ[m, 0] && ILtQ[n, 0] && 
EqQ[m + p, -3/2]
 

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]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(268)=536\).

Time = 2.91 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.84

method	result	size
default	\(\frac {-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 a d e \,x^{2}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{4 a d e}+\frac {c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a \sqrt {a d e}}}{d}-\frac {e^{2} \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{d^{3} \sqrt {a d e}}-\frac {e \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{d^{2}}+\frac {2 e^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{d^{3} \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\)	\(545\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.94 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.68 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*(3*((c^3*d^6*e + a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 5*a^3*e^7)*x^3 + 
(c^3*d^7 + a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 - 5*a^3*d*e^6)*x^2)*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*(2*a^2*c*d^5*e^2 - 2*a^3*d^3*e 
^4 - (3*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - 15*a^3*d*e^6)*x^2 - (3*a*c^2*d^6 
*e + 2*a^2*c*d^4*e^3 - 5*a^3*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
 a*e^2)*x))/((a^3*c*d^6*e^4 - a^4*d^4*e^6)*x^3 + (a^3*c*d^7*e^3 - a^4*d^5* 
e^5)*x^2), 1/8*(3*((c^3*d^6*e + a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 5*a^3*e^ 
7)*x^3 + (c^3*d^7 + a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 - 5*a^3*d*e^6)*x^2)*sq 
rt(-a*d*e)*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*(2*a^2*c*d^5*e^2 - 2*a^3*d^3*e^4 - (3*a*c^2*d^5* 
e^2 + 4*a^2*c*d^3*e^4 - 15*a^3*d*e^6)*x^2 - (3*a*c^2*d^6*e + 2*a^2*c*d^4*e 
^3 - 5*a^3*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^3* 
c*d^6*e^4 - a^4*d^4*e^6)*x^3 + (a^3*c*d^7*e^3 - a^4*d^5*e^5)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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,[0,3,9]%%%},[2,4]%%%}+%%%{%%%{-4,[1,5,7]%%%},[2,3]%% 
%}+%%%{%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 2075, normalized size of antiderivative = 7.01 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d**3*e**6 + 50*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**4*d**2*e**7*x + 150*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
*4*d*e**8*x**2 + 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**5*e**4 + 10*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**4*e**5*x + 50*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a**3*c*d**3*e**6*x**2 + 12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
*2*c**2*d**7*e**2 - 42*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**6*e**3 
*x - 54*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e**4*x**2 - 18*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**8*e*x - 18*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*a*c**3*d**7*e**2*x**2 + 75*sqrt(e)*sqrt(d)*sqrt(a)*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)*sq 
rt(c)*sqrt(d + e*x))*a**4*d*e**8*x**2 + 75*sqrt(e)*sqrt(d)*sqrt(a)*log(sqr 
t(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*e**9*x**3 - 42*sqrt(e)*sqrt(d)*sqrt(a)* 
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*c**2*d**5*e**4*x**2 - 42*sqrt(e) 
*sqrt(d)*sqrt(a)*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*c**2*d**4*e**5* 
x**3 - 24*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*s 
qrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c 
**3*d**7*e**2*x**2 - 24*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + ...