\(\int \frac {(d+e x)^3}{x (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [107]

Optimal result

Integrand size = 40, antiderivative size = 197 \[ \int \frac {(d+e x)^3}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (c d^2-a e^2\right ) (d+e x)}{a c d e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{3/2} d^{3/2}}-\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} e^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.35 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.84 \[ \int \frac {(d+e x)^3}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \left (\sqrt {c} \sqrt {d} \left (\sqrt {a} \sqrt {e} \left (-c d^2+a e^2\right ) (d+e x)+\sqrt {d} \left (\sqrt {a} e-i \sqrt {c d^2-a e^2}\right ) \sqrt {c d^2-2 a e^2-2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d^2-2 a e^2-2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x}}{\sqrt {d} \sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )+\sqrt {d} \left (\sqrt {a} e+i \sqrt {c d^2-a e^2}\right ) \sqrt {c d^2-2 a e^2+2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d^2-2 a e^2+2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x}}{\sqrt {d} \sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )\right )+2 a^{3/2} e^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )\right )}{a^{3/2} c^{3/2} d^{3/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1212, 25, 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.

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

\(\Big \downarrow \) 1212

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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. \(699\) vs. \(2(167)=334\).

Time = 2.26 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.55

method	result	size
default	\(d^{3} \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}\right )+e^{3} \left (-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{d e c \sqrt {d e c}}\right )+\frac {6 d^{2} e \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+3 d \,e^{2} \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )\)	\(700\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 1360, normalized size of antiderivative = 6.90 \[ \int \frac {(d+e x)^3}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

integrate((e*x+d)^3/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:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

int((d + e*x)^3/(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 = 0.30 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^3}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{3}+\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c^{2} d^{3}-\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \sqrt {c d x +a e}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c^{2} d^{3}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{3}-2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} e^{3}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{2} e -2 \sqrt {e x +d}\, a^{2} c d \,e^{3}+2 \sqrt {e x +d}\, a \,c^{2} d^{3} e}{\sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2}} \] Input:

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