\(\int \frac {d+e x}{x (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [143]

Optimal result

Integrand size = 38, antiderivative size = 255 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 c d (d+e x)}{3 a e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6+c d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right ) x\right )}{3 a^2 d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{5/2} d^{3/2} e^{5/2}} \] Output:

2/3*c*d*(e*x+d)/a/e/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2) 
+2/3*(3*c^3*d^6-7*a*c^2*d^4*e^2-a^2*c*d^2*e^4-3*a^3*e^6+c*d*e*(-3*a*e^2+c* 
d^2)*(a*e^2+3*c*d^2)*x)/a^2/d/e^2/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(1/2)-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^(5/2)/d^(3/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x)^4 \left (-a c^3 d^4 e+\frac {3 a^2 e^5 (a e+c d x)^2}{(d+e x)^2}-\frac {3 c^3 d^5 (a e+c d x)}{d+e x}+\frac {9 a c^2 d^3 e^2 (a e+c d x)}{d+e x}\right )}{\left (-c d^2+a e^2\right )^3}-3 (a e+c d x)^{5/2} (d+e x)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )}{3 a^{5/2} d^{3/2} e^{5/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

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

Output:

(2*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(d + e*x)^4*(-(a*c^3*d^4*e) + ( 
3*a^2*e^5*(a*e + c*d*x)^2)/(d + e*x)^2 - (3*c^3*d^5*(a*e + c*d*x))/(d + e* 
x) + (9*a*c^2*d^3*e^2*(a*e + c*d*x))/(d + e*x)))/(-(c*d^2) + a*e^2)^3 - 3* 
(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(S 
qrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(3*a^(5/2)*d^(3/2)*e^(5/2)*((a*e + c*d*x) 
*(d + e*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1235, 27, 1235, 27, 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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[(d + e*x)/(x*(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)*(a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2)^(3/2)) + ((2*(3*c^3*d^6 - 7*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e 
^6 + c*d*e*(c*d^2 - 3*a*e^2)*(3*c*d^2 + a*e^2)*x))/(a*d*e*(c*d^2 - a*e^2)^ 
2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (3*(c*d^2 - a*e^2)*ArcTan 
h[(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])])/(a^(3/2)*d^(3/2)*e^(3/2)))/(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 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. \(588\) vs. \(2(233)=466\).

Time = 2.20 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.31

method	result	size
default	\(e \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+d \left (\frac {1}{3 a d e {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 a d e}+\frac {\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}}{a d e}\right )\)	\(589\)

Input:

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

Output:

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))+d*(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. 752 vs. \(2 (233) = 466\).

Time = 4.73 (sec) , antiderivative size = 1524, normalized size of antiderivative = 5.98 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6*(3*(a^2*c^3*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 
 + (c^5*d^8*e - 3*a*c^4*d^6*e^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2*d^2*e^7)*x^3 
 + (c^5*d^9 - a*c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^3*e^6 - 2*a^ 
4*c*d*e^8)*x^2 + (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + 
a^4*c*d^2*e^7 - a^5*e^9)*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*(4*a^2*c^3*d^7*e^2 - 9*a^3*c^2*d^5*e^4 - 3*a^5*d*e^8 + (3*a*c^4*d 
^7*e^2 - 8*a^2*c^3*d^5*e^4 - 3*a^3*c^2*d^3*e^6)*x^2 + (3*a*c^4*d^8*e - 4*a 
^2*c^3*d^6*e^3 - 9*a^3*c^2*d^4*e^5 - 6*a^4*c*d^2*e^7)*x)*sqrt(c*d*e*x^2 + 
a*d*e + (c*d^2 + a*e^2)*x))/(a^5*c^3*d^9*e^5 - 3*a^6*c^2*d^7*e^7 + 3*a^7*c 
*d^5*e^9 - a^8*d^3*e^11 + (a^3*c^5*d^10*e^4 - 3*a^4*c^4*d^8*e^6 + 3*a^5*c^ 
3*d^6*e^8 - a^6*c^2*d^4*e^10)*x^3 + (a^3*c^5*d^11*e^3 - a^4*c^4*d^9*e^5 - 
3*a^5*c^3*d^7*e^7 + 5*a^6*c^2*d^5*e^9 - 2*a^7*c*d^3*e^11)*x^2 + (2*a^4*c^4 
*d^10*e^4 - 5*a^5*c^3*d^8*e^6 + 3*a^6*c^2*d^6*e^8 + a^7*c*d^4*e^10 - a^8*d 
^2*e^12)*x), 1/3*(3*(a^2*c^3*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 
 - a^5*d*e^8 + (c^5*d^8*e - 3*a*c^4*d^6*e^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2* 
d^2*e^7)*x^3 + (c^5*d^9 - a*c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^ 
3*e^6 - 2*a^4*c*d*e^8)*x^2 + (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^ 
2*d^4*e^5 + a^4*c*d^2*e^7 - a^5*e^9)*x)*sqrt(-a*d*e)*arctan(1/2*sqrt(c*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)/x/(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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F]

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

integrate((e*x+d)/x/(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), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 3117, normalized size of antiderivative = 12.22 \[ \int \frac {d+e x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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**7 + 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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**4*e**8*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt( 
a*e + c*d*x)*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**5 - 6*sqr 
t(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqr 
t(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**6*x + 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqr 
t(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))*a**3*c*d*e**7*x**2 + 9*sqrt(e)*sqrt(d)*sqrt( 
a)*sqrt(a*e + c*d*x)*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**2*d**5*e 
**3 - 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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)*sq 
rt(d + e*x))*a**2*c**2*d**3*e**5*x**2 - 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e 
 + c*d*x)*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**3*d**7*e + 6*sqrt(e)*s 
qrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2...