\(\int \frac {d+e x}{x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [93]

Optimal result

Integrand size = 38, antiderivative size = 209 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {1}{a e x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c^2 d^4-a^2 e^4+c d e \left (3 c d^2-a e^2\right ) x}{a^2 d e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3 c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{5/2} d^{3/2} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.91 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (a^2 e^3-3 c^2 d^3 x+a c d e (-d+e x)\right )+\left (3 c^2 d^4-2 a c d^2 e^2-a^2 e^4\right ) x \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{a^{5/2} d^{3/2} e^{5/2} \left (c d^2-a e^2\right ) x \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)/(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]*(d + e*x)*(a^2*e^3 - 3*c^2*d^3*x + a*c*d*e*(-d + 
e*x)) + (3*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*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)*d^(3/2)*e^(5/2)*(c*d^2 - a*e^2)*x*Sqrt[(a*e + c*d*x)*(d + e*x)] 
)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.

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

\(\Big \downarrow \) 1212

\(\displaystyle -c^2 d^2 e^4 \int -\frac {a e-c d x}{a^2 c^2 d^2 e^6 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {2 c^2 d^2 (d+e x)}{a^2 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle c^2 d^2 e^4 \int \frac {a e-c d x}{a^2 c^2 d^2 e^6 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {2 c^2 d^2 (d+e x)}{a^2 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*c^2*d^2*(d + e*x))/(a^2*e^2*(c*d^2 - a*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]/(d*x)) 
 + ((3*c*d^2 + a*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqr 
t[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[a]*d^( 
3/2)*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. \(563\) vs. \(2(191)=382\).

Time = 2.07 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.70

method	result	size
default	\(d \left (-\frac {1}{a d e x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+e \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )\)	\(564\)

Input:

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

Output:

d*(-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))+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 = 1.04 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.08 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [\frac {\sqrt {a d e} {\left ({\left (3 \, c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x^{2} + {\left (3 \, a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x\right )} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (a^{2} c d^{3} e^{2} - a^{3} d e^{4} + {\left (3 \, a c^{2} d^{4} e - a^{2} c d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left ({\left (a^{3} c^{2} d^{5} e^{3} - a^{4} c d^{3} e^{5}\right )} x^{2} + {\left (a^{4} c d^{4} e^{4} - a^{5} d^{2} e^{6}\right )} x\right )}}, -\frac {\sqrt {-a d e} {\left ({\left (3 \, c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x^{2} + {\left (3 \, a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (a^{2} c d^{3} e^{2} - a^{3} d e^{4} + {\left (3 \, a c^{2} d^{4} e - a^{2} c d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left ({\left (a^{3} c^{2} d^{5} e^{3} - a^{4} c d^{3} e^{5}\right )} x^{2} + {\left (a^{4} c d^{4} e^{4} - a^{5} d^{2} e^{6}\right )} x\right )}}\right ] \] Input:

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

Output:

[1/4*(sqrt(a*d*e)*((3*c^3*d^5 - 2*a*c^2*d^3*e^2 - a^2*c*d*e^4)*x^2 + (3*a* 
c^2*d^4*e - 2*a^2*c*d^2*e^3 - a^3*e^5)*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*(a^2*c*d^3*e^2 - a^3*d*e^4 + (3*a*c^2*d^4*e - a^2*c*d^2*e^3)*x) 
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^3*c^2*d^5*e^3 - a^4*c*d^ 
3*e^5)*x^2 + (a^4*c*d^4*e^4 - a^5*d^2*e^6)*x), -1/2*(sqrt(-a*d*e)*((3*c^3* 
d^5 - 2*a*c^2*d^3*e^2 - a^2*c*d*e^4)*x^2 + (3*a*c^2*d^4*e - 2*a^2*c*d^2*e^ 
3 - a^3*e^5)*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*(a^2*c*d^3*e^2 - a^3*d*e^4 + (3*a*c^2*d^4*e 
 - a^2*c*d^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^3*c^ 
2*d^5*e^3 - a^4*c*d^3*e^5)*x^2 + (a^4*c*d^4*e^4 - a^5*d^2*e^6)*x)]
 

Sympy [F]

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

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

Maxima [F(-2)]

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

integrate((e*x+d)/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm 
="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

int((d + e*x)/(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)/(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 = 0.39 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.69 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)/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 - 2*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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*c*d**2*e**2*x + 3*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 - 2*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)*sq 
rt(c)*sqrt(d + e*x))*a*c*d**2*e**2*x + 3*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)*sqrt(c)*sqrt(d + e*x)*sq 
rt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)*a**2*e**4*x + 2*sqrt( 
e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sqrt(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)*a*c*d**2*e** 
2*x - 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sqrt(d)*sq 
rt(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e...