\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^5 (d+e x)} \, dx\) [19]

Optimal result

Integrand size = 40, antiderivative size = 372 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a^{7/2} d^{9/2} e^{7/2}} \] Output:

-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d/x^4-1/24*(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^3+1/96*(-35*a^2*e^4+6*a*c*d^2*e^2+ 
5*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/x^2-1/192*( 
-105*a^3*e^6+25*a^2*c*d^2*e^4+17*a*c^2*d^4*e^2+15*c^3*d^6)*(a*d*e+(a*e^2+c 
*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^4/e^3/x+1/64*(-a*e^2+c*d^2)*(35*a^3*e^6+15* 
a^2*c*d^2*e^4+9*a*c^2*d^4*e^2+5*c^3*d^6)*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))/a^(7/2)/d^(9/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 10.39 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-15 c^3 d^6 x^3+a c^2 d^4 e x^2 (10 d-17 e x)+a^2 c d^2 e^2 x \left (-8 d^2+12 d e x-25 e^2 x^2\right )+a^3 e^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{x^4}+\frac {3 \left (5 c^4 d^8+4 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-35 a^4 e^8\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 a^{7/2} d^{9/2} e^{7/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-15*c^3*d^6*x^3 
+ a*c^2*d^4*e*x^2*(10*d - 17*e*x) + a^2*c*d^2*e^2*x*(-8*d^2 + 12*d*e*x - 2 
5*e^2*x^2) + a^3*e^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3))) 
/x^4 + (3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2* 
e^6 - 35*a^4*e^8)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqr 
t[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*a^(7/2)*d^(9/2)*e^( 
7/2))
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1215, 1237, 27, 1237, 27, 1237, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {c d^2-6 c e x d-7 a e^2}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {\int \frac {5 c^2 d^4+6 a c e^2 d^2+4 c e \left (c d^2-7 a e^2\right ) x d-35 a^2 e^4}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {5 c^2 d^4+6 a c e^2 d^2+4 c e \left (c d^2-7 a e^2\right ) x d-35 a^2 e^4}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {-\frac {\int \frac {15 c^3 d^6+17 a c^2 e^2 d^4+25 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4+6 a c e^2 d^2-35 a^2 e^4\right ) x d-105 a^3 e^6}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (\frac {5 c^2 d^4}{a}-35 a e^4+6 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d e x^2}}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {15 c^3 d^6+17 a c^2 e^2 d^4+25 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4+6 a c e^2 d^2-35 a^2 e^4\right ) x d-105 a^3 e^6}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\left (\frac {5 c^2 d^4}{a}-35 a e^4+6 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d e x^2}}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c^2 d^4}{a}-35 a e^4+6 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d e x^2}}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c^2 d^4}{a}-35 a e^4+6 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d e x^2}}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c^2 d^4}{a}-35 a e^4+6 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d e x^2}}{6 a d e}-\frac {\left (\frac {c d}{a e}-\frac {7 e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 x^3}}{8 d}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}\)

Input:

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

Output:

-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d*x^4) + (-1/3*(((c*d)/( 
a*e) - (7*e)/d)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x^3 - (-1/2*( 
((5*c^2*d^4)/a + 6*c*d^2*e^2 - 35*a*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2])/(d*e*x^2) - (-(((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2* 
e^4 - 105*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x)) 
 + (3*(c*d^2 - a*e^2)*(5*c^3*d^6 + 9*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*a*d*e))/(6*a*d*e))/(8*d)
 

Defintions of rubi rules used

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 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 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 1237

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)/((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*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3470\) vs. \(2(340)=680\).

Time = 3.16 (sec) , antiderivative size = 3471, normalized size of antiderivative = 9.33

method	result	size
default	\(\text {Expression too large to display}\)	\(3471\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/x^5/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 5.25 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx=\left [-\frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} + {\left (15 \, a c^{3} d^{7} e + 17 \, a^{2} c^{2} d^{5} e^{3} + 25 \, a^{3} c d^{3} e^{5} - 105 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} + 6 \, a^{3} c d^{4} e^{4} - 35 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} - 7 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, a^{4} d^{5} e^{4} x^{4}}, -\frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {-a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} + {\left (15 \, a c^{3} d^{7} e + 17 \, a^{2} c^{2} d^{5} e^{3} + 25 \, a^{3} c d^{3} e^{5} - 105 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} + 6 \, a^{3} c d^{4} e^{4} - 35 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} - 7 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, a^{4} d^{5} e^{4} x^{4}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^5/(e*x+d),x, algorithm 
="fricas")
 

Output:

[-1/768*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2 
*e^6 - 35*a^4*e^8)*sqrt(a*d*e)*x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 17*a^2*c^2*d^5*e^3 + 25*a^3*c*d^3* 
e^5 - 105*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 + 6*a^3*c*d^4*e^4 - 35*a^4 
*d^2*e^6)*x^2 + 8*(a^3*c*d^5*e^3 - 7*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d* 
e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^4), -1/384*(3*(5*c^4*d^8 + 4*a*c^3* 
d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*sqrt(-a*d*e)* 
x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 17*a^2*c^2*d^5*e^3 
+ 25*a^3*c*d^3*e^5 - 105*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 + 6*a^3*c*d 
^4*e^4 - 35*a^4*d^2*e^6)*x^2 + 8*(a^3*c*d^5*e^3 - 7*a^4*d^3*e^5)*x)*sqrt(c 
*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^4)]
 

Sympy [F(-1)]

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

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**5/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^5/(e*x+d),x, algorithm 
="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1483 vs. \(2 (340) = 680\).

Time = 0.15 (sec) , antiderivative size = 1483, normalized size of antiderivative = 3.99 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^5/(e*x+d),x, algorithm 
="giac")
 

Output:

-1/64*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 
- 35*a^4*e^8)*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)*a^3*d^4*e^3) + 1/192*(15*(sqrt(c*d*e 
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^3*c^4*d^11*e^3 + 396* 
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c^3*d^9* 
e^5 + 1170*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a 
^5*c^2*d^7*e^7 + 1212*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x 
+ a*d*e))*a^6*c*d^5*e^9 + 279*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + 
a*e^2*x + a*d*e))*a^7*d^3*e^11 + 128*sqrt(c*d*e)*a^5*c^2*d^8*e^6 + 256*sqr 
t(c*d*e)*a^6*c*d^6*e^8 + 384*sqrt(c*d*e)*a^7*d^4*e^10 + 73*(sqrt(c*d*e)*x 
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^2*c^4*d^10*e^2 + 980*(s 
qrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^3*c^3*d^8* 
e^4 + 2238*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3 
*a^4*c^2*d^6*e^6 + 292*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x 
 + a*d*e))^3*a^5*c*d^4*e^8 - 511*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x 
 + a*e^2*x + a*d*e))^3*a^6*d^2*e^10 + 384*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqr 
t(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^3*c^3*d^9*e^3 + 1792*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))^2*a^4*c 
^2*d^7*e^5 + 2432*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))^2*a^5*c*d^5*e^7 - 55*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 ...
 

Mupad [F(-1)]

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

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^5*(d + e*x)),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^5*(d + e*x)), x)
 

Reduce [B] (verification not implemented)

Time = 3.74 (sec) , antiderivative size = 1694, normalized size of antiderivative = 4.55 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^5/(e*x+d),x)
 

Output:

( - 192*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d**4*e**6 + 224*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**5*d**3*e**7*x - 280*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**5*d**2*e**8*x**2 + 420*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d*e**9*x**3 
 - 192*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d**6*e**4 + 192*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**4*c*d**5*e**5*x - 232*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*a**4*c*d**4*e**6*x**2 + 320*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d**3 
*e**7*x**3 - 32*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**7*e**3*x + 88 
*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**6*e**4*x**2 - 168*sqrt(d + e 
*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**5*e**5*x**3 + 40*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a**2*c**3*d**8*e**2*x**2 - 128*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**2*c**3*d**7*e**3*x**3 - 60*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**9*e 
*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 
**5*e**10*x**4 + 90*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*c*d**2*e**8*x**4 - 156*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**3*c**2*d**4*e**6*x**4 - 60*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**3*d**6*e**4*x**4 - 54*sq...