\(\int \frac {1}{x (d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [84]

Optimal result

Integrand size = 40, antiderivative size = 125 \[ \int \frac {1}{x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d \left (c d^2-a e^2\right ) (d+e x)}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {a} d^{3/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (-\frac {\sqrt {d} e^{3/2} (a e+c d x)}{c d^2-a e^2}-\frac {\sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a}}\right )}{d^{3/2} \sqrt {e} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1214, 25, 27, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.09

method	result	size
default	\(-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{d \sqrt {a d e}}+\frac {2 \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{d \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\)	\(136\)

Input:

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

Output:

-1/d/(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/d/(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 = 0.30 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.63 \[ \int \frac {1}{x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e^{2} - {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right )}{2 \, {\left (a c d^{5} e - a^{2} d^{3} e^{3} + {\left (a c d^{4} e^{2} - a^{2} d^{2} e^{4}\right )} x\right )}}, -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e^{2} - {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right )}{a c d^{5} e - a^{2} d^{3} e^{3} + {\left (a c d^{4} e^{2} - a^{2} d^{2} e^{4}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

int(1/(x*(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*(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 = 0.31 (sec) , antiderivative size = 763, normalized size of antiderivative = 6.10 \[ \int \frac {1}{x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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