\(\int \frac {x^4}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [113]

Optimal result

Integrand size = 40, antiderivative size = 356 \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 a^4 e^4}{c^4 d^4 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 e^3}-\frac {2 \left (c^4 d^8+3 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^4 d^4 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {2 \left (7 c^4 d^8-12 a c^3 d^6 e^2-3 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e^3 \left (c d^2-a e^2\right )^3 (d+e x)}-\frac {\left (5 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2} e^{7/2}} \] Output:

-2*a^4*e^4/c^4/d^4/(-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)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e^3-2/3*(3*a^4*e^8 
+c^4*d^8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^3/(-a*e^2+c*d^ 
2)^2/(e*x+d)^2+2/3*(-3*a^4*e^8-12*a*c^3*d^6*e^2+7*c^4*d^8)*(a*d*e+(a*e^2+c 
*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^3/(-a*e^2+c*d^2)^3/(e*x+d)-(3*a*e^2+5*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^(5/2)/d^(5/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (-9 a^4 e^7 (d+e x)^2+3 a^3 c d e^5 (3 d-e x) (d+e x)^2+c^4 d^7 x \left (15 d^2+20 d e x+3 e^2 x^2\right )+a c^3 d^5 e \left (15 d^3-11 d^2 e x-39 d e^2 x^2-9 e^3 x^3\right )+a^2 c^2 d^3 e^3 \left (-31 d^3-33 d^2 e x+9 d e^2 x^2+9 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^3}-3 \left (5 c d^2+3 a e^2\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{3 c^{5/2} d^{5/2} e^{7/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[x^4/((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)*(-9*a^4*e^7*(d + e*x)^2 + 3*a^3*c* 
d*e^5*(3*d - e*x)*(d + e*x)^2 + c^4*d^7*x*(15*d^2 + 20*d*e*x + 3*e^2*x^2) 
+ a*c^3*d^5*e*(15*d^3 - 11*d^2*e*x - 39*d*e^2*x^2 - 9*e^3*x^3) + a^2*c^2*d 
^3*e^3*(-31*d^3 - 33*d^2*e*x + 9*d*e^2*x^2 + 9*e^3*x^3)))/(c*d^2 - a*e^2)^ 
3 - 3*(5*c*d^2 + 3*a*e^2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqr 
t[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(3*c^(5/2)*d^(5/ 
2)*e^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.21, 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, 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.

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

\(\Big \downarrow \) 1244

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[x^4/((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^3)/(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*(a*d*e*(5*c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e^4) + (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)*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]) + (((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])/(c*d*e) - (3*(c*d^2 - a*e^2)^3*(5*c*d^2 + 3*a*e^2)*ArcT 
anh[(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*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 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])
 

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. \(1111\) vs. \(2(330)=660\).

Time = 2.88 (sec) , antiderivative size = 1112, normalized size of antiderivative = 3.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1112\)

Input:

int(x^4/(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:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (330) = 660\).

Time = 1.59 (sec) , antiderivative size = 1782, normalized size of antiderivative = 5.01 \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \] Input:

integrate(x**4/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral(x**4/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

\[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(x^4/(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^4/((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. \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^4}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int(x^4/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
 

Output:

int(x^4/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.56 \[ \int \frac {x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

( - 72*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*d**2*e* 
*8 - 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**4*d*e** 
9*x - 72*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*e**10 
*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**3*c*d 
**4*e**6 + 192*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*d**3*e**7*x + 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**3*c*d**2*e**8*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**2*c**2*d**6*e**4 + 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**2*c**2*d**5*e**5*x + 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**2*c**2*d**4*e**6*x**2 - 288*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(...