\(\int \frac {(d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^5} \, dx\) [9]

Optimal result

Integrand size = 38, antiderivative size = 287 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+3 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^3 d^2 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 a e x^4}+\frac {\left (5 c d^2-3 a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 a^2 d e^2 x^3}+\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a^{7/2} d^{5/2} e^{7/2}} \] Output:

-1/64*(-a*e^2+c*d^2)*(3*a*e^2+5*c*d^2)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a 
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^2/e^3/x^2-1/4*(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(3/2)/a/e/x^4+1/24*(-3*a*e^2+5*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(3/2)/a^2/d/e^2/x^3+1/64*(-a*e^2+c*d^2)^3*(3*a*e^2+5*c*d^2)*arcta 
nh(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^(5/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (15 c^3 d^6 x^3-a c^2 d^4 e x^2 (10 d+31 e x)+a^2 c d^2 e^2 x \left (8 d^2+20 d e x+9 e^2 x^2\right )+3 a^3 e^3 \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )\right )}{x^4}+\frac {3 \left (c d^2-a e^2\right )^3 \left (5 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 a^{7/2} d^{5/2} e^{7/2}} \] Input:

Integrate[((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x^5,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 + 31*e*x) + a^2*c*d^2*e^2*x*(8*d^2 + 20*d*e*x + 9 
*e^2*x^2) + 3*a^3*e^3*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 - 3*e^3*x^3)))/x^ 
4) + (3*(c*d^2 - a*e^2)^3*(5*c*d^2 + 3*a*e^2)*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] 
)))/(192*a^(7/2)*d^(5/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1237, 27, 1228, 1152, 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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {\left (\frac {5 c^2 d^4}{a}-3 a e^4-2 c d^2 e^2\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x^3}}{8 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 a e x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\left (\frac {5 c^2 d^4}{a}-3 a e^4-2 c d^2 e^2\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x^3}}{8 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 a e x^4}\)

Input:

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

Output:

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

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 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 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. \(2108\) vs. \(2(259)=518\).

Time = 2.66 (sec) , antiderivative size = 2109, normalized size of antiderivative = 7.35

method	result	size
default	\(\text {Expression too large to display}\)	\(2109\)

Input:

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

Output:

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*e)...
 

Fricas [A] (verification not implemented)

Time = 4.65 (sec) , antiderivative size = 702, normalized size of antiderivative = 2.45 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx=\left [-\frac {3 \, {\left (5 \, c^{4} d^{8} - 12 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} - 3 \, 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 - 31 \, a^{2} c^{2} d^{5} e^{3} + 9 \, a^{3} c d^{3} e^{5} - 9 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} - 10 \, a^{3} c d^{4} e^{4} - 3 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} + 9 \, 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^{3} e^{4} x^{4}}, -\frac {3 \, {\left (5 \, c^{4} d^{8} - 12 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} - 3 \, 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 - 31 \, a^{2} c^{2} d^{5} e^{3} + 9 \, a^{3} c d^{3} e^{5} - 9 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} - 10 \, a^{3} c d^{4} e^{4} - 3 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} + 9 \, 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^{3} e^{4} x^{4}}\right ] \] Input:

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

Output:

[-1/768*(3*(5*c^4*d^8 - 12*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2 
*e^6 - 3*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 - 31*a^2*c^2*d^5*e^3 + 9*a^3*c*d^3*e^ 
5 - 9*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 - 10*a^3*c*d^4*e^4 - 3*a^4*d^2 
*e^6)*x^2 + 8*(a^3*c*d^5*e^3 + 9*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^3*e^4*x^4), -1/384*(3*(5*c^4*d^8 - 12*a*c^3*d^6 
*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 - 3*a^4*e^8)*sqrt(-a*d*e)*x^4*a 
rctan(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 - 31*a^2*c^2*d^5*e^3 + 9*a 
^3*c*d^3*e^5 - 9*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 - 10*a^3*c*d^4*e^4 
- 3*a^4*d^2*e^6)*x^2 + 8*(a^3*c*d^5*e^3 + 9*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^3*e^4*x^4)]
 

Sympy [F]

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

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

Output:

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)/x**5, x)
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (259) = 518\).

Time = 0.16 (sec) , antiderivative size = 1618, normalized size of antiderivative = 5.64 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/64*(5*c^4*d^8 - 12*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 
- 3*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^2*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 + 348*( 
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 + 402*(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 + 12*(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 - 9*(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 + 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 + 9 
00*(sqrt(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 + 1854*(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 + 724*(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 + 33*(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 - 
 sqrt(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 + 1024*sqr 
t(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 + 768*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 
+ c*d^2*x + a*e^2*x + a*d*e))^5*a*c^4*d^9*e + 132*(sqrt(c*d*e)*x - sqrt...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 3.35 (sec) , antiderivative size = 1664, normalized size of antiderivative = 5.80 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 192*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d**4*e**6 - 288*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**5*d**3*e**7*x - 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**5*d**2*e**8*x**2 + 36*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 - 320*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**4*c*d**5*e**5*x - 104*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**4*c*d**4*e**6*x**2 - 32*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**7 
*e**3*x - 40*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**6*e**4*x**2 + 88 
*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 + 64*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 + 18*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 - 6*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 - 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) + sq 
rt(d)*sqrt(c)*sqrt(d + e*x))*a**3*c**2*d**4*e**6*x**4 + 36*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 + 4 
2*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c...