\(\int \frac {x^3 (d+e x)}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [139]

Optimal result

Integrand size = 38, antiderivative size = 284 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 a e x^2 (d+e x)}{3 c d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-a e^2\right ) \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (c d^2-a e^2\right ) \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c^2 d^2 e \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x)^4 \left (-a^3 c d e^4+\frac {3 c^2 d^5 (a e+c d x)^2}{(d+e x)^2}+\frac {9 a^2 c d^2 e^3 (a e+c d x)}{d+e x}-\frac {3 a^3 e^5 (a e+c d x)}{d+e x}\right )}{\left (c d^2-a e^2\right )^3}+3 (a e+c d x)^{5/2} (d+e x)^{5/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{3 c^{5/2} d^{5/2} e^{3/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1233, 27, 27, 1224, 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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1224

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[(x^3*(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^2*(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*(a*d*e*(3*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (3*c^ 
3*d^6 + a*c^2*d^4*e^2 + 7*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]) - (3*(d/e - (a*e)/(c* 
d))*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])])/(Sqrt[c]*Sqrt[d]*Sqrt[e]))/(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 1224

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( 
b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p 
+ 1)/(c*(p + 1)*(b^2 - 4*a*c))), 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))/(c*(p + 1)*(b^2 - 4*a*c))   Int[(a + 
b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 
1] &&  !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
 

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. \(1776\) vs. \(2(262)=524\).

Time = 2.34 (sec) , antiderivative size = 1777, normalized size of antiderivative = 6.26

method	result	size
default	\(\text {Expression too large to display}\)	\(1777\)

Input:

int(x^3*(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*(-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))))+e*( 
-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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (262) = 524\).

Time = 2.28 (sec) , antiderivative size = 1514, normalized size of antiderivative = 5.33 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(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]

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

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

Mupad [F(-1)]

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

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

Output:

int((x^3*(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 = 0.33 (sec) , antiderivative size = 1395, normalized size of antiderivative = 4.91 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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