\(\int \frac {d+e x}{x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [144]

Optimal result

Integrand size = 38, antiderivative size = 369 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {1}{a e x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 c^2 d^4-3 a^2 e^4+c d e \left (5 c d^2-3 a e^2\right ) x}{3 a^2 d e^2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {15 c^4 d^8-26 a c^3 d^6 e^2-2 a^2 c^2 d^4 e^4+6 a^3 c d^2 e^6-9 a^4 e^8+c d e \left (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) x}{3 a^3 d^2 e^3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{7/2} d^{5/2} e^{7/2}} \] Output:

-1/a/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/3*(5*c^2*d^4-3*a^2*e^4+ 
c*d*e*(-3*a*e^2+5*c*d^2)*x)/a^2/d/e^2/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*e*x^2)^(3/2)-1/3*(15*c^4*d^8-26*a*c^3*d^6*e^2-2*a^2*c^2*d^4*e^4+6*a^ 
3*c*d^2*e^6-9*a^4*e^8+c*d*e*(-9*a^3*e^6+9*a^2*c*d^2*e^4-31*a*c^2*d^4*e^2+1 
5*c^3*d^6)*x)/a^3/d^2/e^3/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 
2)^(1/2)+(3*a*e^2+5*c*d^2)*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^(5/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.91 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x)^2 \left (-15 c^5 d^8 x^2 (d+e x)+3 a^5 e^8 (d+3 e x)-3 a^4 c d e^6 \left (3 d^2+d e x-6 e^2 x^2\right )+a c^4 d^6 e x \left (-20 d^2+11 d e x+31 e^2 x^2\right )-3 a^2 c^3 d^4 e^2 \left (d^3-13 d^2 e x-11 d e^2 x^2+3 e^3 x^3\right )+3 a^3 c^2 d^2 e^4 \left (3 d^3-3 d^2 e x-5 d e^2 x^2+3 e^3 x^3\right )\right )}{\left (-c d^2+a e^2\right )^3 x}+3 \left (5 c d^2+3 a e^2\right ) (a e+c d x)^{5/2} (d+e x)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{3 a^{7/2} d^{5/2} e^{7/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

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

Output:

(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(d + e*x)^2*(-15*c^5*d^8*x^2*(d 
+ e*x) + 3*a^5*e^8*(d + 3*e*x) - 3*a^4*c*d*e^6*(3*d^2 + d*e*x - 6*e^2*x^2) 
 + a*c^4*d^6*e*x*(-20*d^2 + 11*d*e*x + 31*e^2*x^2) - 3*a^2*c^3*d^4*e^2*(d^ 
3 - 13*d^2*e*x - 11*d*e^2*x^2 + 3*e^3*x^3) + 3*a^3*c^2*d^2*e^4*(3*d^3 - 3* 
d^2*e*x - 5*d*e^2*x^2 + 3*e^3*x^3)))/((-(c*d^2) + a*e^2)^3*x)) + 3*(5*c*d^ 
2 + 3*a*e^2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a*e 
 + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(3*a^(7/2)*d^(5/2)*e^(7/2)*(( 
a*e + c*d*x)*(d + e*x))^(5/2))
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1235, 27, 1235, 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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {2 \left (-3 a^3 e^6+c d e x \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )}{a d e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \int -\frac {15 c^3 d^6-31 a c^2 e^2 d^4+9 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4-10 a c e^2 d^2-3 a^2 e^4\right ) x d-9 a^3 e^6}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}}{3 a e \left (c d^2-a e^2\right )}+\frac {2 c d (d+e x)}{3 a e x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {15 c^3 d^6-31 a c^2 e^2 d^4+9 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4-10 a c e^2 d^2-3 a^2 e^4\right ) x d-9 a^3 e^6}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-3 a^3 e^6+c d e x \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )}{a d e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 a e \left (c d^2-a e^2\right )}+\frac {2 c d (d+e x)}{3 a e x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {-\frac {3 \left (3 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-3 a^3 e^6+c d e x \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )}{a d e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 a e \left (c d^2-a e^2\right )}+\frac {2 c d (d+e x)}{3 a e x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {3 \left (c d^2-a e^2\right )^3 \left (3 a e^2+5 c d^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-3 a^3 e^6+c d e x \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )}{a d e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 a e \left (c d^2-a e^2\right )}+\frac {2 c d (d+e x)}{3 a e x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {2 \left (-3 a^3 e^6+c d e x \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )}{a d e x \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\frac {3 \left (c d^2-a e^2\right )^3 \left (3 a e^2+5 c d^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{a d e \left (c d^2-a e^2\right )^2}}{3 a e \left (c d^2-a e^2\right )}+\frac {2 c d (d+e x)}{3 a e x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

Input:

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

Output:

(2*c*d*(d + e*x))/(3*a*e*(c*d^2 - a*e^2)*x*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2)^(3/2)) + ((2*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3 
*e^6 + c*d*e*(5*c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e^4)*x))/(a*d*e*(c*d^2 - 
a*e^2)^2*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (-(((15*c^3*d^6 
- 31*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 9*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)^3*(5*c*d^2 + 3*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*a^(3/2)*d^(3/2)*e^(3/2)))/(a*d*e 
*(c*d^2 - a*e^2)^2))/(3*a*e*(c*d^2 - a*e^2))
 

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 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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -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. \(1089\) vs. \(2(345)=690\).

Time = 2.24 (sec) , antiderivative size = 1090, normalized size of antiderivative = 2.95

method	result	size
default	\(\text {Expression too large to display}\)	\(1090\)

Input:

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

Output:

d*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-5/2*(a*e^2+c*d^2)/a/ 
d/e*(1/3/a/d/e/(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*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+( 
a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2) 
^2*(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))+1/a/d/ 
e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2* 
c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*x^2*e)^(1/2)-1/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*(2/3*(2*c*d 
*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c 
*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^2*(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)))+e*(1/3/a/d/e/(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*(2/3*(2*c*d*e*x+a* 
e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^2*(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))+1/a/d/e*(1/a/d/e/(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4 
*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (345) = 690\).

Time = 14.63 (sec) , antiderivative size = 1856, normalized size of antiderivative = 5.03 \[ \int \frac {d+e x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(3*((5*c^6*d^10*e - 12*a*c^5*d^8*e^3 + 6*a^2*c^4*d^6*e^5 + 4*a^3*c^3 
*d^4*e^7 - 3*a^4*c^2*d^2*e^9)*x^4 + (5*c^6*d^11 - 2*a*c^5*d^9*e^2 - 18*a^2 
*c^4*d^7*e^4 + 16*a^3*c^3*d^5*e^6 + 5*a^4*c^2*d^3*e^8 - 6*a^5*c*d*e^10)*x^ 
3 + (10*a*c^5*d^10*e - 19*a^2*c^4*d^8*e^3 + 14*a^4*c^2*d^4*e^7 - 2*a^5*c*d 
^2*e^9 - 3*a^6*e^11)*x^2 + (5*a^2*c^4*d^9*e^2 - 12*a^3*c^3*d^7*e^4 + 6*a^4 
*c^2*d^5*e^6 + 4*a^5*c*d^3*e^8 - 3*a^6*d*e^10)*x)*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*(3*a^3*c^3*d^8*e^3 - 9*a^4*c^2*d^6*e^5 + 9* 
a^5*c*d^4*e^7 - 3*a^6*d^2*e^9 + (15*a*c^5*d^9*e^2 - 31*a^2*c^4*d^7*e^4 + 9 
*a^3*c^3*d^5*e^6 - 9*a^4*c^2*d^3*e^8)*x^3 + (15*a*c^5*d^10*e - 11*a^2*c^4* 
d^8*e^3 - 33*a^3*c^3*d^6*e^5 + 15*a^4*c^2*d^4*e^7 - 18*a^5*c*d^2*e^9)*x^2 
+ (20*a^2*c^4*d^9*e^2 - 39*a^3*c^3*d^7*e^4 + 9*a^4*c^2*d^5*e^6 + 3*a^5*c*d 
^3*e^8 - 9*a^6*d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a 
^4*c^5*d^11*e^5 - 3*a^5*c^4*d^9*e^7 + 3*a^6*c^3*d^7*e^9 - a^7*c^2*d^5*e^11 
)*x^4 + (a^4*c^5*d^12*e^4 - a^5*c^4*d^10*e^6 - 3*a^6*c^3*d^8*e^8 + 5*a^7*c 
^2*d^6*e^10 - 2*a^8*c*d^4*e^12)*x^3 + (2*a^5*c^4*d^11*e^5 - 5*a^6*c^3*d^9* 
e^7 + 3*a^7*c^2*d^7*e^9 + a^8*c*d^5*e^11 - a^9*d^3*e^13)*x^2 + (a^6*c^3*d^ 
10*e^6 - 3*a^7*c^2*d^8*e^8 + 3*a^8*c*d^6*e^10 - a^9*d^4*e^12)*x), -1/6*(3* 
((5*c^6*d^10*e - 12*a*c^5*d^8*e^3 + 6*a^2*c^4*d^6*e^5 + 4*a^3*c^3*d^4*e...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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