\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^3 (d+e x)^2} \, dx\) [26]

Optimal result

Integrand size = 40, antiderivative size = 261 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)^2} \, dx=-\frac {\left (c d^2-15 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a d^3 (d+e x)}-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 d x^2 (d+e x)}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x (d+e x)}+\frac {\left (c^2 d^4+6 a c d^2 e^2-15 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{4 a^{3/2} d^{7/2} e^{3/2}} \] Output:

-1/4*(-15*a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d^3/(e*x+ 
d)-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d/x^2/(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)^(1/2)/x/(e*x+d)+1/4*(-15*a^2*e^4+ 
6*a*c*d^2*e^2+c^2*d^4)*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^(3/2)/d^(7/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-c d^2 x (d+e x)+a e \left (-2 d^2+5 d e x+15 e^2 x^2\right )\right )}{x^2 (d+e x)}+\frac {\left (c^2 d^4+6 a c d^2 e^2-15 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{4 a^{3/2} d^{7/2} e^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {e^2 \left (\frac {\left (\frac {c^2 d^4}{a}-15 a e^4+6 c d^2 e^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}\right )}{4 d^2}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^2}+\frac {2 e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*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*(e^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*x^2) + (e^2*(-((( 
(c*d)/(a*e) - (7*e)/d)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x) + ( 
((c^2*d^4)/a + 6*c*d^2*e^2 - 15*a*e^4)*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)*e^(3/2))))/(4*d^2))/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 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. \(1566\) vs. \(2(233)=466\).

Time = 3.00 (sec) , antiderivative size = 1567, normalized size of antiderivative = 6.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1567\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/x^3/(e*x+d)^2,x,method=_RETURN 
VERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 594, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)^2} \, dx=\left [-\frac {{\left ({\left (c^{2} d^{4} e + 6 \, a c d^{2} e^{3} - 15 \, a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} d^{3} e^{2} + {\left (a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2} + {\left (a c d^{4} e - 5 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (a^{2} d^{4} e^{3} x^{3} + a^{2} d^{5} e^{2} x^{2}\right )}}, -\frac {{\left ({\left (c^{2} d^{4} e + 6 \, a c d^{2} e^{3} - 15 \, a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} d^{3} e^{2} + {\left (a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} x^{2} + {\left (a c d^{4} e - 5 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (a^{2} d^{4} e^{3} x^{3} + a^{2} d^{5} e^{2} x^{2}\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3/(e*x+d)^2,x, algorit 
hm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**3/(e*x+d)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3/(e*x+d)^2,x, algorit 
hm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (233) = 466\).

Time = 0.18 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)^2} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3/(e*x+d)^2,x, algorit 
hm="giac")
 

Output:

1/4*(8*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*e*sgn(1/(e*x + d) 
)*sgn(e)/(d^3*abs(e)) + (c^2*d^4*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + 
6*a*c*d^2*e^2*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) - 15*a^2*e^4*arctan(s 
qrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + sqrt(-a*d*e)*sqrt(c*d*e)*c*d^2 - 15*sqrt( 
-a*d*e)*sqrt(c*d*e)*a*e^2)*sgn(1/(e*x + d))*sgn(e)/(sqrt(-a*d*e)*a*d^3*e*a 
bs(e)) - (c^2*d^4*sgn(1/(e*x + d))*sgn(e) + 6*a*c*d^2*e^2*sgn(1/(e*x + d)) 
*sgn(e) - 15*a^2*e^4*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(c*d*e - c*d^2*e/ 
(e*x + d) + a*e^3/(e*x + d))*d/(sqrt(-a*d*e)*e))/(sqrt(-a*d*e)*a*d^3*e*abs 
(e)) - (sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^2*d^4*e^3*sg 
n(1/(e*x + d))*sgn(e) + 6*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d) 
)*a^2*c*d^2*e^5*sgn(1/(e*x + d))*sgn(e) - 7*sqrt(c*d*e - c*d^2*e/(e*x + d) 
 + a*e^3/(e*x + d))*a^3*e^7*sgn(1/(e*x + d))*sgn(e) + (c*d*e - c*d^2*e/(e* 
x + d) + a*e^3/(e*x + d))^(3/2)*c^2*d^5*sgn(1/(e*x + d))*sgn(e) - 10*(c*d* 
e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c*d^3*e^2*sgn(1/(e*x + d) 
)*sgn(e) + 9*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^2*d*e^4 
*sgn(1/(e*x + d))*sgn(e))/((a*e^3 - (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e* 
x + d))*d)^2*a*d^3*abs(e)))*abs(e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 1897, normalized size of antiderivative = 7.27 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)^2} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3/(e*x+d)^2,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 + 80*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a**2*c*d**3*e**4*x**2 - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
c**2*d**6*e*x - 6*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 + 15*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 + 15*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)*sqr 
t(c)*sqrt(d + e*x))*a**2*c*d**2*e**5*x**3 - 23*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**2*x**2 - 23*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**4*e**3*x**3 - 3* 
sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*...