\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^2 (d+e x)} \, dx\) [53]

Optimal result

Integrand size = 40, antiderivative size = 318 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx=\frac {\left (c^2 d^4+28 a c d^2 e^2+19 a^2 e^4+2 c d e \left (c d^2+7 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e}-\frac {(3 a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 x}-\frac {\left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {c} \sqrt {d} e^{3/2}}-a^{3/2} \sqrt {d} e^{3/2} \left (5 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right ) \] Output:

1/8*(c^2*d^4+28*a*c*d^2*e^2+19*a^2*e^4+2*c*d*e*(7*a*e^2+c*d^2)*x)*(a*d*e+( 
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e-1/3*(-c*d*x+3*a*e)*(a*d*e+(a*e^2+c*d^2)* 
x+c*d*e*x^2)^(3/2)/x-1/8*(-5*a^3*e^6-45*a^2*c*d^2*e^4-15*a*c^2*d^4*e^2+c^3 
*d^6)*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^(1/2)/d^(1/2)/e^(3/2)-a^(3/2)*d^(1/2)*e^(3/2)*(3*a*e^2+5*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))
 

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (3 a^2 e^3 (-8 d+11 e x)+2 a c d e^2 x (34 d+13 e x)+c^2 d^2 x \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )-24 a^{3/2} \sqrt {c} d e^3 \left (5 c d^2+3 a e^2\right ) x \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )-3 \left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 a^3 e^6\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{24 \sqrt {c} \sqrt {d} e^{3/2} x \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^2*(d + e*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]*(3*a^2*e^3*(-8*d + 11*e*x) + 2*a*c*d*e^2*x*(34*d + 13*e*x) 
 + c^2*d^2*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2)) - 24*a^(3/2)*Sqrt[c]*d*e^3*(5 
*c*d^2 + 3*a*e^2)*x*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*S 
qrt[d + e*x])] - 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3* 
e^6)*x*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]) 
]))/(24*Sqrt[c]*Sqrt[d]*e^(3/2)*x*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1215, 1230, 25, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {1}{2} \int -\frac {\left (a e \left (5 c d^2+3 a e^2\right )+c d \left (c d^2+7 a e^2\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x}dx-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \int \frac {\left (a e \left (5 c d^2+3 a e^2\right )+c d \left (c d^2+7 a e^2\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x}dx-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{2} \left (\frac {\left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\int -\frac {c d \left (8 a^2 d e^3 \left (5 c d^2+3 a e^2\right )-\left (c^3 d^6-15 a c^2 e^2 d^4-45 a^2 c e^4 d^2-5 a^3 e^6\right ) x\right )}{2 x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {8 a^2 d e^3 \left (5 c d^2+3 a e^2\right )-\left (c^3 d^6-15 a c^2 e^2 d^4-45 a^2 c e^4 d^2-5 a^3 e^6\right ) x}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 e}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{2} \left (\frac {8 a^2 d e^3 \left (3 a e^2+5 c d^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 e}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {8 a^2 d e^3 \left (3 a e^2+5 c d^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-2 \left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{8 e}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {8 a^2 d e^3 \left (3 a e^2+5 c d^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {e}}}{8 e}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {-16 a^2 d e^3 \left (3 a e^2+5 c d^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\frac {\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {e}}}{8 e}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right )}{4 e}+\frac {-8 a^{3/2} \sqrt {d} e^{5/2} \left (3 a e^2+5 c d^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac {\left (-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {c} \sqrt {d} \sqrt {e}}}{8 e}\right )-\frac {(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x}\)

Input:

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

Output:

-1/3*((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/x + ( 
((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) + (-(((c^3*d^6 - 15*a*c^2* 
d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*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*a^(3/2)*Sqrt[d]*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])])/(8*e))/2
 

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 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2075\) vs. \(2(280)=560\).

Time = 2.83 (sec) , antiderivative size = 2076, normalized size of antiderivative = 6.53

method	result	size
default	\(\text {Expression too large to display}\)	\(2076\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^2/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

1/d*(-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 = 8.35 (sec) , antiderivative size = 1717, normalized size of antiderivative = 5.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/96*(3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt 
(c*d*e)*x*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*sq 
rt(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) - 24*(5*a*c^2*d^3*e^3 + 3*a^2*c*d*e 
^5)*sqrt(a*d*e)*x*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*(8*c^3*d^3*e^ 
3*x^3 - 24*a^2*c*d^2*e^4 + 2*(7*c^3*d^4*e^2 + 13*a*c^2*d^2*e^4)*x^2 + (3*c 
^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + 
(c*d^2 + a*e^2)*x))/(c*d*e^2*x), 1/48*(3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45* 
a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(-c*d*e)*x*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)) + 12*(5*a*c^2*d^3*e^3 + 
 3*a^2*c*d*e^5)*sqrt(a*d*e)*x*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) + 2*( 
8*c^3*d^3*e^3*x^3 - 24*a^2*c*d^2*e^4 + 2*(7*c^3*d^4*e^2 + 13*a*c^2*d^2*e^4 
)*x^2 + (3*c^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*x), 1/96*(48*(5*a*c^2*d^3*e^3 + 3 
*a^2*c*d*e^5)*sqrt(-a*d*e)*x*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, c^{2} d^{2} e x + \frac {7 \, c^{4} d^{5} e^{2} + 13 \, a c^{3} d^{3} e^{4}}{c^{2} d^{2} e^{2}}\right )} x + \frac {3 \, c^{4} d^{6} e + 68 \, a c^{3} d^{4} e^{3} + 33 \, a^{2} c^{2} d^{2} e^{5}}{c^{2} d^{2} e^{2}}\right )} + \frac {{\left (5 \, a^{2} c d^{3} e^{2} + 3 \, a^{3} d e^{4}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e}} + \frac {{\left (c^{3} d^{6} - 15 \, a c^{2} d^{4} e^{2} - 45 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} e} - \frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{2} c d^{3} e^{2} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{3} d e^{4} + 2 \, \sqrt {c d e} a^{3} d^{2} e^{3}}{a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d),x, algorithm 
="giac")
 

Output:

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*c^2*d^2*e*x + (7*c^ 
4*d^5*e^2 + 13*a*c^3*d^3*e^4)/(c^2*d^2*e^2))*x + (3*c^4*d^6*e + 68*a*c^3*d 
^4*e^3 + 33*a^2*c^2*d^2*e^5)/(c^2*d^2*e^2)) + (5*a^2*c*d^3*e^2 + 3*a^3*d*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) + 1/16*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d 
^2*e^4 - 5*a^3*e^6)*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)*e) - ((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^2 + (s 
qrt(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^4 + 2* 
sqrt(c*d*e)*a^3*d^2*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)
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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