\(\int \frac {d+e x}{x (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [92]

Optimal result

Integrand size = 38, antiderivative size = 127 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d (d+e x)}{a e \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} \sqrt {d} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {a} c d^{3/2} \sqrt {e} (d+e x)}{c d^2-a e^2}-\sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{a^{3/2} \sqrt {d} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1212, 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.

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

\(\Big \downarrow \) 1212

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(111)=222\).

Time = 1.95 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.26

method	result	size
default	\(\frac {2 e \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+d \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )\)	\(287\)

Input:

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

Output:

2*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)+d*(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (111) = 222\).

Time = 0.38 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.77 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a c d^{2} e + {\left (a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right )}{2 \, {\left (a^{3} c d^{3} e^{3} - a^{4} d e^{5} + {\left (a^{2} c^{2} d^{4} e^{2} - a^{3} c d^{2} e^{4}\right )} x\right )}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a c d^{2} e + {\left (a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right )}{a^{3} c d^{3} e^{3} - a^{4} d e^{5} + {\left (a^{2} c^{2} d^{4} e^{2} - a^{3} c d^{2} e^{4}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)/x/(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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 455, normalized size of antiderivative = 3.58 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2}+\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a \,e^{2}+\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c \,d^{2}-2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a d e -2 \sqrt {e x +d}\, a c \,d^{2} e}{\sqrt {c d x +a e}\, a^{2} d \,e^{2} \left (a \,e^{2}-c \,d^{2}\right )} \] Input:

int((e*x+d)/x/(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*e**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)*sqr 
t(c)*sqrt(d + e*x))*c*d**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)*sqrt(c)*sqrt(d + e*x))*a*e**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)*sqrt(c)*sqrt(d + e*x))*c*d**2 - sqrt(e)*sqrt(d)*s 
qrt(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*e**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)*sqr 
t(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)*c*d**2 - 2*sqrt(e)*sqr 
t(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*d*e - 2*sqrt(d + e*x)*a*c*d**2*e)/(sqrt(a 
*e + c*d*x)*a**2*d*e**2*(a*e**2 - c*d**2))