\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^4 (d+e x)^2} \, dx\) [27]

Optimal result

Integrand size = 40, antiderivative size = 370 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)^2} \, dx=\frac {\left (3 c^2 d^4+10 a c d^2 e^2-105 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^4 e (d+e x)}-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3 (d+e x)}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2 (d+e x)}+\frac {\left (3 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x (d+e x)}-\frac {\left (c^3 d^6+3 a c^2 d^4 e^2+15 a^2 c d^2 e^4-35 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{8 a^{5/2} d^{9/2} e^{5/2}} \] Output:

1/24*(-105*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(1/2)/a^2/d^4/e/(e*x+d)-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d 
/x^3/(e*x+d)-1/12*(c/a/e-7*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
x^2/(e*x+d)+1/24*(-7*a*e^2+3*c*d^2)*(5*a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x 
+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/x/(e*x+d)-1/8*(-35*a^3*e^6+15*a^2*c*d^2*e^4+ 
3*a*c^2*d^4*e^2+c^3*d^6)*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^(5/2)/d^(9/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-3 c^2 d^4 x^2 (d+e x)+2 a c d^2 e x \left (d^2-4 d e x-5 e^2 x^2\right )+a^2 e^2 \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )\right )}{x^3 (d+e x)}-\frac {3 \left (c^3 d^6+3 a c^2 d^4 e^2+15 a^2 c d^2 e^4-35 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 a^{5/2} d^{9/2} e^{5/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-3*c^2*d^4*x^2 
*(d + e*x) + 2*a*c*d^2*e*x*(d^2 - 4*d*e*x - 5*e^2*x^2) + a^2*e^2*(8*d^3 - 
14*d^2*e*x + 35*d*e^2*x^2 + 105*e^3*x^3)))/(x^3*(d + e*x))) - (3*(c^3*d^6 
+ 3*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 35*a^3*e^6)*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])))/(24*a^(5/2)*d^(9/2)*e^(5/2))
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.98, 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, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {\int -\frac {\frac {6 a \left (c d^2-a e^2\right ) x^2 e^5}{d^3}-2 a \left (5 c-\frac {3 a e^2}{d^2}\right ) x e^4+\frac {a \left (c d^2-11 a e^2\right ) e^3}{d}}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\frac {6 a \left (c d^2-a e^2\right ) x^2 e^5}{d^3}-2 a \left (5 c-\frac {3 a e^2}{d^2}\right ) x e^4+a \left (c d-\frac {11 a e^2}{d}\right ) e^3}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {-\frac {\int \frac {a e^3 \left (\left (c d^2-3 a e^2\right ) \left (3 c d^2+19 a e^2\right )+2 d e \left (\frac {12 a^2 e^4}{d^2}-23 a c e^2+c^2 d^2\right ) x\right )}{2 d x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {e^2 \left (c-\frac {11 a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {e^2 \int \frac {\left (c d^2-3 a e^2\right ) \left (3 c d^2+19 a e^2\right )+2 d e \left (\frac {12 a^2 e^4}{d^2}-23 a c e^2+c^2 d^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^2}-\frac {e^2 \left (c-\frac {11 a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {-\frac {e^2 \left (-\frac {3 \left (-35 a^3 e^6+15 a^2 c d^2 e^4+3 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (c d^2-3 a e^2\right ) \left (19 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {e^2 \left (c-\frac {11 a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {-\frac {e^2 \left (\frac {3 \left (-35 a^3 e^6+15 a^2 c d^2 e^4+3 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (c d^2-3 a e^2\right ) \left (19 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {e^2 \left (c-\frac {11 a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {-\frac {e^2 \left (\frac {3 \left (-35 a^3 e^6+15 a^2 c d^2 e^4+3 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (c d^2-3 a e^2\right ) \left (19 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {e^2 \left (c-\frac {11 a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a d e}-\frac {e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^2}-\frac {2 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

Input:

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

Output:

(-2*e^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^4*(d + e*x)) + (-1 
/3*(e^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*x^3) + (-1/2*(e^ 
2*(c - (11*a*e^2)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x^2 - 
(e^2*(-(((c*d^2 - 3*a*e^2)*(3*c*d^2 + 19*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^ 
2)*x + c*d*e*x^2])/(a*d*e*x)) + (3*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 15*a^2*c*d 
^2*e^4 - 35*a^3*e^6)*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*a^(3/2)*d^(3 
/2)*e^(3/2))))/(4*d^2))/(6*a*d*e))/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. \(2274\) vs. \(2(338)=676\).

Time = 3.92 (sec) , antiderivative size = 2275, normalized size of antiderivative = 6.15

method	result	size
default	\(\text {Expression too large to display}\)	\(2275\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/x^4/(e*x+d)^2,x,method=_RETURN 
VERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 4.05 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)^2} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d)^2,x, algorit 
hm="fricas")
 

Output:

[-1/96*(3*((c^3*d^6*e + 3*a*c^2*d^4*e^3 + 15*a^2*c*d^2*e^5 - 35*a^3*e^7)*x 
^4 + (c^3*d^7 + 3*a*c^2*d^5*e^2 + 15*a^2*c*d^3*e^4 - 35*a^3*d*e^6)*x^3)*sq 
rt(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*(8*a^3*d^4*e^3 - (3*a 
*c^2*d^5*e^2 + 10*a^2*c*d^3*e^4 - 105*a^3*d*e^6)*x^3 - (3*a*c^2*d^6*e + 8* 
a^2*c*d^4*e^3 - 35*a^3*d^2*e^5)*x^2 + 2*(a^2*c*d^5*e^2 - 7*a^3*d^3*e^4)*x) 
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^5*e^4*x^4 + a^3*d^6*e 
^3*x^3), 1/48*(3*((c^3*d^6*e + 3*a*c^2*d^4*e^3 + 15*a^2*c*d^2*e^5 - 35*a^3 
*e^7)*x^4 + (c^3*d^7 + 3*a*c^2*d^5*e^2 + 15*a^2*c*d^3*e^4 - 35*a^3*d*e^6)* 
x^3)*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*(8*a^3*d^4*e^3 - (3*a*c^2*d^5*e^2 + 10*a^ 
2*c*d^3*e^4 - 105*a^3*d*e^6)*x^3 - (3*a*c^2*d^6*e + 8*a^2*c*d^4*e^3 - 35*a 
^3*d^2*e^5)*x^2 + 2*(a^2*c*d^5*e^2 - 7*a^3*d^3*e^4)*x)*sqrt(c*d*e*x^2 + a* 
d*e + (c*d^2 + a*e^2)*x))/(a^3*d^5*e^4*x^4 + a^3*d^6*e^3*x^3)]
 

Sympy [F(-1)]

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

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**4/(e*x+d)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(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^4), x 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (338) = 676\).

Time = 0.21 (sec) , antiderivative size = 1133, normalized size of antiderivative = 3.06 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d)^2,x, algorit 
hm="giac")
 

Output:

-1/24*(48*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*e^2*sgn(1/(e*x 
 + d))*sgn(e)/(d^4*abs(e)) + (3*c^3*d^6*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e) 
*e)) + 9*a*c^2*d^4*e^2*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) + 45*a^2*c*d 
^2*e^4*arctan(sqrt(c*d*e)*d/(sqrt(-a*d*e)*e)) - 105*a^3*e^6*arctan(sqrt(c* 
d*e)*d/(sqrt(-a*d*e)*e)) + 3*sqrt(-a*d*e)*sqrt(c*d*e)*c^2*d^4 + 10*sqrt(-a 
*d*e)*sqrt(c*d*e)*a*c*d^2*e^2 - 105*sqrt(-a*d*e)*sqrt(c*d*e)*a^2*e^4)*sgn( 
1/(e*x + d))*sgn(e)/(sqrt(-a*d*e)*a^2*d^4*e^2*abs(e)) - 3*(c^3*d^6*sgn(1/( 
e*x + d))*sgn(e) + 3*a*c^2*d^4*e^2*sgn(1/(e*x + d))*sgn(e) + 15*a^2*c*d^2* 
e^4*sgn(1/(e*x + d))*sgn(e) - 35*a^3*e^6*sgn(1/(e*x + d))*sgn(e))*arctan(s 
qrt(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^2*d^4*e^2*abs(e)) - (3*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/( 
e*x + d))*a^2*c^3*d^6*e^6*sgn(1/(e*x + d))*sgn(e) + 9*sqrt(c*d*e - c*d^2*e 
/(e*x + d) + a*e^3/(e*x + d))*a^3*c^2*d^4*e^8*sgn(1/(e*x + d))*sgn(e) + 45 
*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c*d^2*e^10*sgn(1/(e 
*x + d))*sgn(e) - 57*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^5 
*e^12*sgn(1/(e*x + d))*sgn(e) + 8*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x 
+ d))^(3/2)*a*c^3*d^7*e^3*sgn(1/(e*x + d))*sgn(e) - 24*(c*d*e - c*d^2*e/(e 
*x + d) + a*e^3/(e*x + d))^(3/2)*a^2*c^2*d^5*e^5*sgn(1/(e*x + d))*sgn(e) - 
 120*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^3*c*d^3*e^7*sgn 
(1/(e*x + d))*sgn(e) + 136*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d)...
 

Mupad [F(-1)]

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

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^4*(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^4*(d + e*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 2498, normalized size of antiderivative = 6.75 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)^2} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d)^2,x)
 

Output:

( - 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d**4*e**5 + 196*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**4*d**3*e**6*x - 490*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**4*d**2*e**7*x**2 - 1470*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d*e**8*x** 
3 - 80*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**6*e**3 + 112*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**3*c*d**5*e**4*x - 238*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*a**3*c*d**4*e**5*x**2 - 910*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**3 
*e**6*x**3 - 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**7*e**2*x + 12 
2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**6*e**3*x**2 + 142*sqrt(d + 
e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e**4*x**3 + 30*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*a*c**3*d**8*e*x**2 + 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3* 
d**7*e**2*x**3 - 735*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**4*d*e**8*x**3 - 735*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**4*e**9*x**4 - 210*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*c*d**3*e**6*x**3 - 210*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*c*d**2*e**7*x**4 + 288*sqrt( 
e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(...