\(\int \frac {(d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x} \, dx\) [5]

Optimal result

Integrand size = 38, antiderivative size = 223 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx=\frac {\left (5 c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d}+\frac {\left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{3/2} d^{3/2} \sqrt {e}}-2 \sqrt {a} d^{3/2} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right ) \] Output:

1/4*(2*c*d*e*x+a*e^2+5*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d+ 
1/4*(-a^2*e^4+6*a*c*d^2*e^2+3*c^2*d^4)*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))/c^(3/2)/d^(3/2)/e^(1/2)-2*a^ 
(1/2)*d^(3/2)*e^(1/2)*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))
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (a e^2+c d (5 d+2 e x)\right )-8 \sqrt {a} c^{3/2} d^3 e \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+\left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{4 c^{3/2} d^{3/2} \sqrt {e} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1231, 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.

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1231

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

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 [A] (verified)

Time = 2.09 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.48

method	result	size
default	\(e \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )+d \left (\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d e \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{\sqrt {a d e}}\right )\)	\(330\)

Input:

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

Output:

e*(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))+ 
d*((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))
 

Fricas [A] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 1337, normalized size of antiderivative = 6.00 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x,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 [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 6.58 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx=d\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-d\,\ln \left (\frac {a\,e^2}{2}+\frac {c\,d^2}{2}+\frac {\sqrt {a\,d\,e}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x}+\frac {a\,d\,e}{x}\right )\,\sqrt {a\,d\,e}+e\,\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+\frac {d\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (c\,d^2+a\,e^2\right )}{2\,\sqrt {c\,d\,e}}-\frac {e\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.92 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx=\frac {\sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{3}+5 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} x +4 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{3} e +4 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{3} e -4 \sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c^{2} d^{3} e -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}+6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4}}{4 c^{2} d^{2} e} \] Input:

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

Output:

(sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d*e**3 + 5*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*c**2*d**3*e + 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**2*e**2*x + 4 
*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sq 
rt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**2*d**3*e 
+ 4*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**3 
*e - 4*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)*c**2*d**3*e - sqrt 
(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt( 
d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*e**4 + 6*sqrt(e)*sqrt(d)*sqrt(c)*log 
((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - 
 c*d**2))*a*c*d**2*e**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e 
+ c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**4 
)/(4*c**2*d**2*e)