\(\int \frac {x^5}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [112]

Optimal result

Integrand size = 40, antiderivative size = 435 \[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 a^5 e^5}{c^5 d^5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (13 c d^2+7 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 e^4}+\frac {2 \left (c^5 d^{10}+3 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^5 d^5 e^4 \left (c d^2-a e^2\right )^2 (d+e x)^2}-\frac {2 \left (10 c^5 d^{10}-15 a c^4 d^8 e^2-3 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^4 d^4 e^4 \left (c d^2-a e^2\right )^3 (d+e x)}+\frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 e^4}+\frac {5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{7/2} d^{7/2} e^{9/2}} \] Output:

2*a^5*e^5/c^5/d^5/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) 
^(1/2)-1/4*(7*a*e^2+13*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/ 
d^3/e^4+2/3*(3*a^5*e^10+c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
c^5/d^5/e^4/(-a*e^2+c*d^2)^2/(e*x+d)^2-2/3*(-3*a^5*e^10-15*a*c^4*d^8*e^2+1 
0*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^4/(-a*e^2+c* 
d^2)^3/(e*x+d)+1/2*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2 
/e^4+5/4*(3*a^2*e^4+6*a*c*d^2*e^2+7*c^2*d^4)*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^(9/2 
)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (-45 a^5 e^9 (d+e x)^2+15 a^4 c d e^7 (2 d-e x) (d+e x)^2+6 a^3 c^2 d^2 e^5 (d+e x)^2 \left (6 d^2+2 d e x+e^2 x^2\right )+c^5 d^8 x \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )-2 a^2 c^3 d^4 e^3 \left (95 d^4+111 d^3 e x-6 d^2 e^2 x^2-9 d e^3 x^3+9 e^4 x^4\right )+a c^4 d^6 e \left (105 d^4-50 d^3 e x-237 d^2 e^2 x^2-48 d e^3 x^3+18 e^4 x^4\right )\right )}{\left (c d^2-a e^2\right )^3}+15 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{12 c^{7/2} d^{7/2} e^{9/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
 

Output:

(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(-45*a^5*e^9*(d + e*x)^2 + 15*a^ 
4*c*d*e^7*(2*d - e*x)*(d + e*x)^2 + 6*a^3*c^2*d^2*e^5*(d + e*x)^2*(6*d^2 + 
 2*d*e*x + e^2*x^2) + c^5*d^8*x*(105*d^3 + 140*d^2*e*x + 21*d*e^2*x^2 - 6* 
e^3*x^3) - 2*a^2*c^3*d^4*e^3*(95*d^4 + 111*d^3*e*x - 6*d^2*e^2*x^2 - 9*d*e 
^3*x^3 + 9*e^4*x^4) + a*c^4*d^6*e*(105*d^4 - 50*d^3*e*x - 237*d^2*e^2*x^2 
- 48*d*e^3*x^3 + 18*e^4*x^4)))/(c*d^2 - a*e^2)^3) + 15*(7*c^2*d^4 + 6*a*c* 
d^2*e^2 + 3*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[c]* 
Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(12*c^(7/2)*d^(7/2)*e 
^(9/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1244, 27, 1233, 27, 1225, 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.

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

\(\Big \downarrow \) 1244

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1233

\(\displaystyle \frac {\frac {2 \int \frac {x \left (4 a d e \left (7 c^2 d^4-12 a c e^2 d^2-3 a^2 e^4\right )+\left (35 c^3 d^6-61 a c^2 e^2 d^4+9 a^2 c e^4 d^2-15 a^3 e^6\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d e \left (c d^2-a e^2\right )^2}-\frac {2 x^2 \left (a d e \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{c d e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 e \left (c d^2-a e^2\right )}-\frac {2 d x^4}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {x \left (4 a d e \left (7 c^2 d^4-12 a c e^2 d^2-3 a^2 e^4\right )+\left (35 c^3 d^6-61 a c^2 e^2 d^4+9 a^2 c e^4 d^2-15 a^3 e^6\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d e \left (c d^2-a e^2\right )^2}-\frac {2 x^2 \left (a d e \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{c d e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 e \left (c d^2-a e^2\right )}-\frac {2 d x^4}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}-\frac {\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{c d e \left (c d^2-a e^2\right )^2}-\frac {2 x^2 \left (a d e \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{c d e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 e \left (c d^2-a e^2\right )}-\frac {2 d x^4}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}-\frac {\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{c d e \left (c d^2-a e^2\right )^2}-\frac {2 x^2 \left (a d e \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{c d e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 e \left (c d^2-a e^2\right )}-\frac {2 d x^4}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{5/2}}-\frac {\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{c d e \left (c d^2-a e^2\right )^2}-\frac {2 x^2 \left (a d e \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{c d e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 e \left (c d^2-a e^2\right )}-\frac {2 d x^4}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
 

Output:

(-2*d*x^4)/(3*e*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) + ((-2*x^2*(a*d*e*(7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2*e^4) + 
(7*c^3*d^6 - 11*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*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]) + (-1/4*((105*c 
^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 45*a^ 
4*e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3* 
e^6)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^2*d^2*e^2) + (15*( 
c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*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])])/(8*c^(5/2)*d^(5/2)*e^(5/2)))/(c*d*e*(c*d^2 - a*e^2)^2)) 
/(3*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 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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[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])
 

rule 1244

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d 
_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(e*f - d*g))*(f + g*x)^(n - 1)*((a + 
b*x + c*x^2)^(p + 1)/(p*(2*c*d - b*e)*(d + e*x))), x] + Simp[1/(p*e^2*(2*c* 
d - b*e))   Int[(f + g*x)^(n - 2)*(a + b*x + c*x^2)^p*Simp[b*e*g*((-e)*f + 
d*g + e*f*n - d*g*n - e*f*p) + c*(d^2*g^2*(n - 1) - d*e*f*g*n + e^2*f^2*(2* 
p + 1)) - e*g*(b*e*g*p - c*(e*f*n - d*g*n + 2*e*f*p))*x, x], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] && LtQ[p, -1] && EqQ[c*d^2 - b*d* 
e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1922\) vs. \(2(401)=802\).

Time = 2.88 (sec) , antiderivative size = 1923, normalized size of antiderivative = 4.42

method	result	size
default	\(\text {Expression too large to display}\)	\(1923\)

Input:

int(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (401) = 802\).

Time = 4.13 (sec) , antiderivative size = 2120, normalized size of antiderivative = 4.87 \[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm 
="fricas")
 

Output:

[1/48*(15*(7*a*c^5*d^12*e - 15*a^2*c^4*d^10*e^3 + 6*a^3*c^3*d^8*e^5 + 2*a^ 
4*c^2*d^6*e^7 + 3*a^5*c*d^4*e^9 - 3*a^6*d^2*e^11 + (7*c^6*d^11*e^2 - 15*a* 
c^5*d^9*e^4 + 6*a^2*c^4*d^7*e^6 + 2*a^3*c^3*d^5*e^8 + 3*a^4*c^2*d^3*e^10 - 
 3*a^5*c*d*e^12)*x^3 + (14*c^6*d^12*e - 23*a*c^5*d^10*e^3 - 3*a^2*c^4*d^8* 
e^5 + 10*a^3*c^3*d^6*e^7 + 8*a^4*c^2*d^4*e^9 - 3*a^5*c*d^2*e^11 - 3*a^6*e^ 
13)*x^2 + (7*c^6*d^13 - a*c^5*d^11*e^2 - 24*a^2*c^4*d^9*e^4 + 14*a^3*c^3*d 
^7*e^6 + 7*a^4*c^2*d^5*e^8 + 3*a^5*c*d^3*e^10 - 6*a^6*d*e^12)*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*(105*a*c^5*d^11*e^2 - 190*a^2*c^4*d^9* 
e^4 + 36*a^3*c^3*d^7*e^6 + 30*a^4*c^2*d^5*e^8 - 45*a^5*c*d^3*e^10 - 6*(c^6 
*d^9*e^4 - 3*a*c^5*d^7*e^6 + 3*a^2*c^4*d^5*e^8 - a^3*c^3*d^3*e^10)*x^4 + 3 
*(7*c^6*d^10*e^3 - 16*a*c^5*d^8*e^5 + 6*a^2*c^4*d^6*e^7 + 8*a^3*c^3*d^4*e^ 
9 - 5*a^4*c^2*d^2*e^11)*x^3 + (140*c^6*d^11*e^2 - 237*a*c^5*d^9*e^4 + 12*a 
^2*c^4*d^7*e^6 + 66*a^3*c^3*d^5*e^8 - 45*a^5*c*d*e^12)*x^2 + (105*c^6*d^12 
*e - 50*a*c^5*d^10*e^3 - 222*a^2*c^4*d^8*e^5 + 84*a^3*c^3*d^6*e^7 + 45*a^4 
*c^2*d^4*e^9 - 90*a^5*c*d^2*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e 
^2)*x))/(a*c^7*d^12*e^6 - 3*a^2*c^6*d^10*e^8 + 3*a^3*c^5*d^8*e^10 - a^4*c^ 
4*d^6*e^12 + (c^8*d^11*e^7 - 3*a*c^7*d^9*e^9 + 3*a^2*c^6*d^7*e^11 - a^3*c^ 
5*d^5*e^13)*x^3 + (2*c^8*d^12*e^6 - 5*a*c^7*d^10*e^8 + 3*a^2*c^6*d^8*e^...
 

Sympy [F]

\[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \] Input:

integrate(x**5/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral(x**5/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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]

\[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm 
="giac")
 

Output:

integrate(x^5/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^5}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int(x^5/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
 

Output:

int(x^5/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {x^5}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}d x \] Input:

int(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

int(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)