\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^3 (d+e x)^3} \, dx\) [45]

Optimal result

Integrand size = 40, antiderivative size = 279 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^3 (d+e x)^3} \, dx=\frac {3 \left (c d^2-5 a e^2\right ) \left (c-\frac {a e^2}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a d e (d+e x)}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 x (d+e x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 a d e x^2 (d+e x)^3}-\frac {3 \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{4 \sqrt {a} d^{7/2} \sqrt {e}} \] Output:

3/4*(-5*a*e^2+c*d^2)*(c-a*e^2/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
/a/d/e/(e*x+d)-1/4*(c/a/e-5*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2) 
/x/(e*x+d)^2-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/a/d/e/x^2/(e*x+d) 
^3-3/4*(-5*a*e^2+c*d^2)*(-a*e^2+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))/a^(1/2)/d^(7/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 10.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^3 (d+e x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {d} \left (c d^2 x (5 d+13 e x)+a e \left (2 d^2-5 d e x-15 e^2 x^2\right )\right )}{x^2 (d+e x)}-\frac {3 \left (c^2 d^4-6 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{4 d^{7/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[d]*(c*d^2*x*(5*d + 13*e*x) + a*e*( 
2*d^2 - 5*d*e*x - 15*e^2*x^2)))/(x^2*(d + e*x))) - (3*(c^2*d^4 - 6*a*c*d^2 
*e^2 + 5*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqr 
t[d + e*x])])/(Sqrt[a]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(4*d^(7/ 
2))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1214, 25, 2181, 27, 1228, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {\int -\frac {a e^5 \left (a e \left (5 c d^2-7 a e^2\right )+2 d \left (\frac {2 a^2 e^4}{d^2}-5 a c e^2+2 c^2 d^2\right ) x\right )}{2 d x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^4}-\frac {2 e \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \int \frac {a e \left (5 c d^2-7 a e^2\right )+2 d \left (\frac {2 a^2 e^4}{d^2}-5 a c e^2+2 c^2 d^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^2}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^4}-\frac {2 e \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}-\frac {\left (5 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^4}-\frac {2 e \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {e^4 \left (-\frac {3 \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d}-\frac {\left (5 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^4}-\frac {2 e \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {e^4 \left (-\frac {3 \left (c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}}-\frac {\left (5 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^4}-\frac {2 e \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

Input:

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

Output:

(-2*e*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^3*(d 
 + e*x)) + (-1/2*(a*e^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2* 
x^2) + (e^4*(-(((5*c*d^2 - 7*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e 
*x^2])/(d*x)) - (3*(c*d^2 - 5*a*e^2)*(c*d^2 - 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])])/(2*Sqrt[a]*d^(3/2)*Sqrt[e])))/(4*d^2))/e^4
 

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 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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 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. \(3142\) vs. \(2(251)=502\).

Time = 3.94 (sec) , antiderivative size = 3143, normalized size of antiderivative = 11.27

method	result	size
default	\(\text {Expression too large to display}\)	\(3143\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^3 (d+e x)^3} \, dx=\left [\frac {3 \, {\left ({\left (c^{2} d^{4} e - 6 \, a c d^{2} e^{3} + 5 \, a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} - 6 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} d^{3} e^{2} + {\left (13 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2} + 5 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (a d^{4} e^{2} x^{3} + a d^{5} e x^{2}\right )}}, \frac {3 \, {\left ({\left (c^{2} d^{4} e - 6 \, a c d^{2} e^{3} + 5 \, a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} - 6 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} d^{3} e^{2} + {\left (13 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2} + 5 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (a d^{4} e^{2} x^{3} + a d^{5} e x^{2}\right )}}\right ] \] Input:

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

Output:

[1/16*(3*((c^2*d^4*e - 6*a*c*d^2*e^3 + 5*a^2*e^5)*x^3 + (c^2*d^5 - 6*a*c*d 
^3*e^2 + 5*a^2*d*e^4)*x^2)*sqrt(a*d*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) - 4*(2*a^2*d^3*e^2 + (13*a*c*d^3*e^2 - 15*a^2*d*e^4)*x^2 + 5*(a*c*d^4* 
e - a^2*d^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^4*e^ 
2*x^3 + a*d^5*e*x^2), 1/8*(3*((c^2*d^4*e - 6*a*c*d^2*e^3 + 5*a^2*e^5)*x^3 
+ (c^2*d^5 - 6*a*c*d^3*e^2 + 5*a^2*d*e^4)*x^2)*sqrt(-a*d*e)*arctan(1/2*sqr 
t(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)*sqr 
t(-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)) - 2 
*(2*a^2*d^3*e^2 + (13*a*c*d^3*e^2 - 15*a^2*d*e^4)*x^2 + 5*(a*c*d^4*e - a^2 
*d^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^4*e^2*x^3 + 
 a*d^5*e*x^2)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^3/(e*x+d)^3,x, algorit 
hm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^3/(e*x+d)^3,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,[0,3,9]%%%},[2,4]%%%}+%%%{%%%{-4,[1,5,7]%%%},[2,3]%% 
%}+%%%{%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 1897, normalized size of antiderivative = 6.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^3 (d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*d**3*e**4 + 50*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**3*d**2*e**5*x + 150*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
*3*d*e**6*x**2 - 12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**5*e**2 - 20* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**4*e**3*x - 40*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a**2*c*d**3*e**4*x**2 - 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
*c**2*d**6*e*x - 78*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**2*x**2 
+ 75*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 
**6*x**2 + 75*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**7*x**3 - 45*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**4*x**2 - 45*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqr 
t(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*s 
qrt(c)*sqrt(d + e*x))*a**2*c*d**2*e**5*x**3 - 39*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*c**2*d**5*e**2*x**2 - 39*sqrt(e)*sqr 
t(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**3*x**3 + 
9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c...