\(\int \frac {d+e x}{x^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [78]

Optimal result

Integrand size = 38, antiderivative size = 195 \[ \int \frac {d+e x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a e x^2}+\frac {\left (3 c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 d e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{5/2} d^{3/2} e^{5/2}} \] Output:

-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/e/x^2+1/4*(-a*e^2+3*c*d^2)* 
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d/e^2/x-1/4*(-a*e^2+c*d^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 = 10.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (-3 c^2 d^3 x^2+a c d e x (-d+e x)+a^2 e^2 (2 d+e x)\right )-\left (3 c^2 d^4-2 a c d^2 e^2-a^2 e^4\right ) x^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{4 a^{5/2} d^{3/2} e^{5/2} x^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1237, 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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Defintions of rubi rules used

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 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 1237

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)/((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*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(398\) vs. \(2(171)=342\).

Time = 2.33 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.05

method	result	size
default	\(d \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 a d e \,x^{2}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{4 a d e}+\frac {c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a \sqrt {a d e}}\right )+e \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )\)	\(399\)

Input:

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

Output:

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*(-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))
 

Fricas [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.30 \[ \int \frac {d+e x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {{\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {a d e} x^{2} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} d^{2} e^{2} - {\left (3 \, a c d^{3} e - a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, a^{3} d^{2} e^{3} x^{2}}, \frac {{\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-a d e} x^{2} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} e^{2} - {\left (3 \, a c d^{3} e - a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, a^{3} d^{2} e^{3} x^{2}}\right ] \] Input:

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

Output:

[-1/16*((3*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*sqrt(a*d*e)*x^2*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*d^2*e^2 - (3*a*c*d^3*e - a^2*d*e^3)* 
x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^2*e^3*x^2), 1/8*((3 
*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*sqrt(-a*d*e)*x^2*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 
*d^2*e^2 - (3*a*c*d^3*e - a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
a*e^2)*x))/(a^3*d^2*e^3*x^2)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (171) = 342\).

Time = 0.14 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.61 \[ \int \frac {d+e x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {{\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{4 \, \sqrt {-a d e} a^{2} d e^{2}} + \frac {5 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a c^{2} d^{5} e + 10 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{2} c d^{3} e^{3} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{3} d e^{5} + 8 \, \sqrt {c d e} a^{2} c d^{4} e^{2} - 3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} c^{2} d^{4} + 2 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a c d^{2} e^{2} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{2} e^{4} + 8 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{2} d e^{3}}{4 \, {\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )}^{2} a^{2} d e^{2}} \] Input:

integrate((e*x+d)/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm 
="giac")
 

Output:

1/4*(3*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*arctan(-(sqrt(c*d*e)*x - sqrt(c* 
d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a^2*d*e^ 
2) + 1/4*(5*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))* 
a*c^2*d^5*e + 10*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d 
*e))*a^2*c*d^3*e^3 + (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + 
 a*d*e))*a^3*d*e^5 + 8*sqrt(c*d*e)*a^2*c*d^4*e^2 - 3*(sqrt(c*d*e)*x - sqrt 
(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*c^2*d^4 + 2*(sqrt(c*d*e)*x - sq 
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a*c*d^2*e^2 + (sqrt(c*d*e)*x 
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^2*e^4 + 8*sqrt(c*d*e)*( 
sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^2*d*e^3)/ 
((a*d*e - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2) 
^2*a^2*d*e^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 966, normalized size of antiderivative = 4.95 \[ \int \frac {d+e x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*d**2*e**4 - 2*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**3*d*e**5*x - 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d* 
*4*e**2 + 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**3*e**3*x + 6*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*a*c**2*d**5*e*x - 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**3*e**6*x**2 - 3*sqrt(e)*sqrt(d)*sqrt(a)*l 
og(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*d**2*e**4*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))*a*c**2*d**4*e**2*x**2 + 3*s 
qrt(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**3*d**6*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))*a**3*e**6* 
x**2 - 3*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sq 
rt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2 
*c*d**2*e**4*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))*a*c**2*d**4*e**2*x**2 + 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)*...