\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^2 (d+e x)^4} \, dx\) [66]

Optimal result

Integrand size = 40, antiderivative size = 269 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=-\frac {\left (2 c d^2-3 a e^2\right ) \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d^2 e (d+e x)}-\frac {a e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d x (d+e x)^2}-\frac {a^{3/2} e^{3/2} \left (5 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{d^{5/2}}+\frac {2 c^{5/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (2 c^2 d^4 x-4 a c d^2 e^2 x+a^2 e^3 (d+3 e x)\right )+a^{3/2} e^3 \left (-5 c d^2+3 a e^2\right ) x (d+e x) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+2 c^{5/2} d^5 x (d+e x) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{d^{5/2} e^{3/2} x \sqrt {a e+c d x} (d+e x)^{3/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)^4), 
x]
 

Output:

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

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1214, 25, 2181, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx\)

\(\Big \downarrow \) 1214

\(\displaystyle -\frac {\int -\frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}+c^3 d^3 x^2 e^5}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 e (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}+c^3 d^3 x^2 e^5}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 e (d+e x)}\)

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {e^5 \left (a^2 e^3 \left (5 c d^2-3 a e^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 c^{5/2} d^{9/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {e}}\right )}{2 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 e (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {e^5 \left (\frac {2 c^{5/2} d^{9/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {e}}-2 a^2 e^3 \left (5 c d^2-3 a e^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{2 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 e (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {e^5 \left (\frac {2 c^{5/2} d^{9/2} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {e}}-\frac {a^{3/2} e^{5/2} \left (5 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {d}}\right )}{2 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 e (d+e x)}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)^4),x]
 

Output:

(-2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*e* 
(d + e*x)) + (-((a^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2 
*x)) + (e^5*((2*c^(5/2)*d^(9/2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr 
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/Sqrt[e 
] - (a^(3/2)*e^(5/2)*(5*c*d^2 - 3*a*e^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[d]))/(2*d^2))/e^6
 

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 1214

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] 
/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[ExpandToS 
um[(((-d)^n*(-2*c*d + b*e)^(-m - 1))/(e^n*x^n) - ((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + 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] && ILtQ[m, 0] && ILtQ[n, 0] && 
EqQ[m + p, -3/2]
 

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]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3548\) vs. \(2(237)=474\).

Time = 4.39 (sec) , antiderivative size = 3549, normalized size of antiderivative = 13.19

method	result	size
default	\(\text {Expression too large to display}\)	\(3549\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^2/(e*x+d)^4,x,method=_RETURN 
VERBOSE)
 

Output:

1/d^4*(-1/a/d/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)+5/2*(a*e^2+c*d^2 
)/a/d/e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)+1/2*(a*e^2+c*d^2)*(1/ 
8*(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)^(3/2)/c/d/e+3/ 
16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(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)))+a*d*e*(1/3*(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)*(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))+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))))+6*c/a*(1/12*(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)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/8*(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)^(3/2)/c/d/e+3/16*(4 
*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(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...
 

Fricas [A] (verification not implemented)

Time = 2.41 (sec) , antiderivative size = 1562, normalized size of antiderivative = 5.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d)^4,x, algorit 
hm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**2/(e*x+d)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{2}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d)^4,x, algorit 
hm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^4*x^2), 
 x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d)^4,x, algorit 
hm="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,[0,0,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[6,6 
]%%%}+%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^2\,{\left (d+e\,x\right )}^4} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)^4),x)
                                                                                    
                                                                                    
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 1711, normalized size of antiderivative = 6.36 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d)^4,x)
 

Output:

( - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*d**2*e**6 - 18*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**3*d*e**7*x - 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d 
**4*e**4 + 18*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**3*e**5*x - 4*sqrt( 
d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**3*x - 4*sqrt(d + e*x)*sqrt(a*e + 
 c*d*x)*c**3*d**7*e*x - 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)*sq 
rt(d + e*x))*a**3*d*e**7*x - 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**3*e**8*x**2 + 12*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**3*e**5*x + 12*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**6*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**5*e**3*x + 5*sq 
rt(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** 
4*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* 
*3*d*e**7*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) +...