\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^2 (d+e x)^2} \, dx\) [25]

Optimal result

Integrand size = 40, antiderivative size = 185 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)^2} \, dx=\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{a d e x (d+e x)^2}-\frac {\left (c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{\sqrt {a} d^{5/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {d} (d+3 e x)}{x (d+e x)}+\frac {\left (-c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{d^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1214, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*e*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*(d + e*x)) + (-(Sq 
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d*x)) - ((c*d - (3*a*e^2)/d)*Ar 
cTanh[(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]*Sqrt[e]))/d
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(919\) vs. \(2(167)=334\).

Time = 2.76 (sec) , antiderivative size = 920, normalized size of antiderivative = 4.97

method	result	size
default	\(\frac {-\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\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 ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d e \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 )}{\sqrt {a d e}}\right )}{2 a d e}+\frac {2 c \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{a}}{d^{2}}+\frac {-\frac {2 \left (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 )^{\frac {3}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d e c \left (\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 {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\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 )}{2 \sqrt {d e c}}\right )}{a \,e^{2}-c \,d^{2}}}{d^{2}}-\frac {2 e \left (\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 ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d e \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 )}{\sqrt {a d e}}\right )}{d^{3}}+\frac {2 e \left (\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 {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\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 )}{2 \sqrt {d e c}}\right )}{d^{3}}\)	\(920\)

Input:

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

Output:

1/d^2*(-1/a/d/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+1/2*(a*e^2+c*d^2 
)/a/d/e*((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)*ln((1/2 
*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^ 
(1/2))/(d*e*c)^(1/2)-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*c/a*(1/4*(2*c*d* 
e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c* 
d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1 
/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))+1/d^2*(-2/(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))^(3/2)+2*d*e*c 
/(a*e^2-c*d^2)*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/2*(a*e^2-c 
*d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^ 
2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))-2/d^3*e*((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)*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/ 
(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)-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/d^3*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d 
/e))^(1/2)+1/2*(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c 
)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.36 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)^2} \, dx=\left [-\frac {\sqrt {a d e} {\left ({\left (c d^{2} e - 3 \, a e^{3}\right )} x^{2} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x\right )} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a d e^{2} x + a d^{2} e\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left (a d^{3} e^{2} x^{2} + a d^{4} e x\right )}}, \frac {\sqrt {-a d e} {\left ({\left (c d^{2} e - 3 \, a e^{3}\right )} x^{2} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (3 \, a d e^{2} x + a d^{2} e\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (a d^{3} e^{2} x^{2} + a d^{4} e x\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((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. 395 vs. \(2 (167) = 334\).

Time = 0.16 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)^2} \, dx=-{\left (\frac {{\left (c d^{2} \arctan \left (\frac {\sqrt {c d e} d}{\sqrt {-a d e} e}\right ) - 3 \, a e^{2} \arctan \left (\frac {\sqrt {c d e} d}{\sqrt {-a d e} e}\right ) - 3 \, \sqrt {-a d e} \sqrt {c d e}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {-a d e} d^{2} {\left | e \right |}} + \frac {2 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2} {\left | e \right |}} - \frac {{\left (c d^{2} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 3 \, a e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} d}{\sqrt {-a d e} e}\right )}{\sqrt {-a d e} d^{2} e {\left | e \right |}} - \frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c d^{2} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{{\left (a e^{3} - {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )} d\right )} d^{2} {\left | e \right |}}\right )} {\left | e \right |} \] Input:

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

Output:

-((c*d^2*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) - 3*a*e^2*arctan(sqrt(c*d* 
e)*d/(sqrt(-a*d*e)*e)) - 3*sqrt(-a*d*e)*sqrt(c*d*e))*sgn(1/(e*x + d))*sgn( 
e)/(sqrt(-a*d*e)*d^2*abs(e)) + 2*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e 
*x + d))*sgn(1/(e*x + d))*sgn(e)/(d^2*abs(e)) - (c*d^2*e*sgn(1/(e*x + d))* 
sgn(e) - 3*a*e^3*sgn(1/(e*x + d))*sgn(e))*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)*d^2*e*abs(e)) - 
 (sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c*d^2*e*sgn(1/(e*x + d 
))*sgn(e) - sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*e^3*sgn(1/ 
(e*x + d))*sgn(e))/((a*e^3 - (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d)) 
*d)*d^2*abs(e)))*abs(e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 932, normalized size of antiderivative = 5.04 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*d**2*e**3 - 18*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*d*e**4*x - 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**4 
*e - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**3*e**2*x - 9*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*d*e**4*x - 9*sqrt(e)*sq 
rt(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*e**5*x**2 + 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))*c**2*d**5*x + 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))*c**2*d**4*e*x**2 - 
 9*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*d*e** 
4*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqr 
t(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2* 
e**5*x**2 + 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))*c 
**2*d**5*x + 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))* 
c**2*d**4*e*x**2 + 9*sqrt(e)*sqrt(d)*sqrt(a)*log(2*sqrt(e)*sqrt(d)*sqrt...