\(\int \frac {(d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^3} \, dx\) [7]

Optimal result

Integrand size = 38, antiderivative size = 227 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx=-\frac {\left (2 a d e+\left (c d^2+5 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a e x^2}+2 \sqrt {c} \sqrt {d} e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac {\left (c^2 d^4-6 a c d^2 e^2-3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{3/2} \sqrt {d} e^{3/2}} \] Output:

-1/4*(2*a*d*e+(5*a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a 
/e/x^2+2*c^(1/2)*d^(1/2)*e^(3/2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/( 
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/4*(-3*a^2*e^4-6*a*c*d^2*e^2+c^2* 
d^4)*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 = 0.90 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (-\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (c d^2 x+a e (2 d+5 e x)\right )+\left (c^2 d^4-6 a c d^2 e^2-3 a^2 e^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+8 a^{3/2} \sqrt {c} d e^3 x^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{4 a^{3/2} \sqrt {d} e^{3/2} x^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d 
*x]*Sqrt[d + e*x]*(c*d^2*x + a*e*(2*d + 5*e*x))) + (c^2*d^4 - 6*a*c*d^2*e^ 
2 - 3*a^2*e^4)*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sq 
rt[d + e*x])] + 8*a^(3/2)*Sqrt[c]*d*e^3*x^2*ArcTanh[(Sqrt[e]*Sqrt[a*e + c* 
d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])]))/(4*a^(3/2)*Sqrt[d]*e^(3/2)*x^2*Sq 
rt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1229, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int \frac {d \left (c^2 d^4-6 a c e^2 d^2-8 a c e^3 x d-3 a^2 e^4\right )}{2 x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {c^2 d^4-6 a c e^2 d^2-8 a c e^3 x d-3 a^2 e^4}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\left (-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-8 a c d e^3 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\left (-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-16 a c d e^3 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\left (-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-8 a \sqrt {c} \sqrt {d} e^{5/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-2 \left (-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-8 a \sqrt {c} \sqrt {d} e^{5/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\left (-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {a} \sqrt {d} \sqrt {e}}-8 a \sqrt {c} \sqrt {d} e^{5/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (x \left (5 a e^2+c d^2\right )+2 a d e\right )}{4 a e x^2}\)

Input:

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

Output:

-1/4*((2*a*d*e + (c*d^2 + 5*a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d 
*e*x^2])/(a*e*x^2) - (-8*a*Sqrt[c]*Sqrt[d]*e^(5/2)*ArcTanh[(c*d^2 + a*e^2 
+ 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2])] - ((c^2*d^4 - 6*a*c*d^2*e^2 - 3*a^2*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])])/(Sqrt[a]*Sqrt[d]*Sqrt[e]))/(8*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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1034\) vs. \(2(193)=386\).

Time = 2.25 (sec) , antiderivative size = 1035, normalized size of antiderivative = 4.56

method	result	size
default	\(\text {Expression too large to display}\)	\(1035\)

Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/x^3,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)^(3/2)-1/4*(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)^(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)))+1/2*c/a*((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)))+e*(-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*l 
n((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*...
 

Fricas [A] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 1375, normalized size of antiderivative = 6.06 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(8*sqrt(c*d*e)*a^2*d*e^3*x^2*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c 
*d^2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d* 
e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - (c^2*d^4 
 - 6*a*c*d^2*e^2 - 3*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 + (a*c*d^3*e + 5*a^2*d*e^3)*x)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d*e^2*x^2), -1/16*(16*sqrt(-c*d*e)*a^ 
2*d*e^3*x^2*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d* 
e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^ 
3*e + a*c*d*e^3)*x)) + (c^2*d^4 - 6*a*c*d^2*e^2 - 3*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 + (a*c*d^3*e 
+ 5*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d*e^2* 
x^2), 1/8*(4*sqrt(c*d*e)*a^2*d*e^3*x^2*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6 
*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2* 
c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - (c^2 
*d^4 - 6*a*c*d^2*e^2 - 3*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...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3,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. 626 vs. \(2 (193) = 386\).

Time = 0.20 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx=-\frac {c d e^{2} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{\sqrt {c d e}} - \frac {{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{4 \, \sqrt {-a d e} a e} + \frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a c^{2} d^{5} e + 2 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{2} c d^{3} e^{3} - 3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{3} d e^{5} - 8 \, \sqrt {c d e} a^{3} d^{2} e^{4} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} c^{2} d^{4} + 10 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a c d^{2} e^{2} + 5 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{2} e^{4} + 8 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a c d^{3} e + 16 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{2} d e^{3}}{4 \, {\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )}^{2} a e} \] Input:

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

Output:

-c*d*e^2*log(abs(-c*d^2 - a*e^2 - 2*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))))/sqrt(c*d*e) - 1/4*(c^2*d^4 - 6*a*c*d 
^2*e^2 - 3*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*e) + 1/4*((sqrt(c*d*e)*x - s 
qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a*c^2*d^5*e + 2*(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 - 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^3*d^2*e^4 + (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 + 10*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x 
 + a*d*e))^3*a*c*d^2*e^2 + 5*(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*c*d^3*e + 16*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 - (sq 
rt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^2*a*e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1089, normalized size of antiderivative = 4.80 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*d**2*e**4 - 10*sqrt(d + e*x)*sq 
rt(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 - 12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**3*e**3*x - 2*sqrt( 
d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**5*e*x + 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))*a**3*e**6*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*c*d**2*e**4*x**2 + 5*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 
 - 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**3*d**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**3 
*e**6*x**2 + 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqr 
t(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 + 5*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 - 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...