\(\int \frac {x^4 (d+e x)}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [138]

Optimal result

Integrand size = 38, antiderivative size = 373 \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 a e x^3 (d+e x)}{3 c d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x \left (a d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right )+\left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6\right ) x\right )}{3 c^2 d^2 e \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e^2 \left (c d^2-a e^2\right )^3}-\frac {\left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{7/2} d^{7/2} e^{5/2}} \] Output:

2/3*a*e*x^3*(e*x+d)/c/d/(-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*x*(a*d*e*(-5*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d^4)+(-5*a^3*e^6+9*a^2* 
c*d^2*e^4+a*c^2*d^4*e^2+3*c^3*d^6)*x)/c^2/d^2/e/(-a*e^2+c*d^2)^3/(a*d*e+(a 
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/3*(-15*a^3*e^6+31*a^2*c*d^2*e^4-9*a*c^2*d 
^4*e^2+9*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^2/(-a* 
e^2+c*d^2)^3-(5*a*e^2+3*c*d^2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a* 
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(7/2)/d^(7/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x)^2 \left (-15 a^5 e^8 (d+e x)+3 c^5 d^8 x^2 (3 d+e x)+a^4 c d e^6 \left (31 d^2+11 d e x-20 e^2 x^2\right )-3 a c^4 d^6 e x \left (-6 d^2+d e x+3 e^2 x^2\right )-3 a^3 c^2 d^2 e^4 \left (3 d^3-11 d^2 e x-13 d e^2 x^2+e^3 x^3\right )+3 a^2 c^3 d^4 e^2 \left (3 d^3-5 d^2 e x-3 d e^2 x^2+3 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^3}-3 \left (3 c d^2+5 a e^2\right ) (a e+c d x)^{5/2} (d+e x)^{5/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{3 c^{7/2} d^{7/2} e^{5/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

Integrate[(x^4*(d + e*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1233, 27, 1233, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[(x^4*(d + e*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
 

Output:

(2*a*e*x^3*(d + e*x))/(3*c*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2)^(3/2)) - ((2*x*(a*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 - 5*a^2*e^4) 
+ (3*c^3*d^6 + a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 5*a^3*e^6)*x))/(c*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]) - (((9*c^3*d^6 
 - 9*a*c^2*d^4*e^2 + 31*a^2*c*d^2*e^4 - 15*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + 
a*e^2)*x + c*d*e*x^2])/(c*d*e) - (3*(c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)* 
ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e 
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^(3/2)*d^(3/2)*e^(3/2)))/(c*d*e*(c 
*d^2 - a*e^2)^2))/(3*c*d*(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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2900\) vs. \(2(347)=694\).

Time = 3.05 (sec) , antiderivative size = 2901, normalized size of antiderivative = 7.78

method	result	size
default	\(\text {Expression too large to display}\)	\(2901\)

Input:

int(x^4*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (347) = 694\).

Time = 2.26 (sec) , antiderivative size = 1826, normalized size of antiderivative = 4.90 \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (d + e x\right )}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**4*(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
 

Output:

Integral(x**4*(d + e*x)/((d + e*x)*(a*e + c*d*x))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4*(e*x+d)/(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]

\[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{4}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(e*x+d)/(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)*x^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (d+e\,x\right )}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int((x^4*(d + e*x))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)
 

Output:

int((x^4*(d + e*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 = 1.14 (sec) , antiderivative size = 2091, normalized size of antiderivative = 5.61 \[ \int \frac {x^4 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^4*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

( - 120*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + 
c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**5*d*e**9 
 - 120*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c 
*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**5*e**10*x 
 + 288*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c 
*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**4*c*d**3* 
e**7 + 168*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e 
 + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**4*c*d 
**2*e**8*x - 120*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sq 
rt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a* 
*4*c*d*e**9*x**2 - 144*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt 
(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d** 
2))*a**3*c**2*d**5*e**5 + 144*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*lo 
g((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 
- c*d**2))*a**3*c**2*d**4*e**6*x + 288*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + 
c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqr 
t(a*e**2 - c*d**2))*a**3*c**2*d**3*e**7*x**2 - 96*sqrt(e)*sqrt(d)*sqrt(c)* 
sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d 
+ e*x))/sqrt(a*e**2 - c*d**2))*a**2*c**3*d**7*e**3 - 240*sqrt(e)*sqrt(d)*s 
qrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(...