\(\int \frac {x^3 (d+e x)}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [88]

Optimal result

Integrand size = 38, antiderivative size = 257 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 a^3 e^3 (d+e x)}{c^3 d^3 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (3 c d^2+7 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 e^2}+\frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 e}+\frac {3 \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{7/2} d^{7/2} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (-15 a^3 e^5+a^2 c d e^3 (4 d-5 e x)+c^3 d^4 x (3 d-2 e x)+a c^2 d^2 e \left (3 d^2+2 d e x+2 e^2 x^2\right )\right )+3 \left (c^3 d^6+a c^2 d^4 e^2+3 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{4 c^{7/2} d^{7/2} e^{5/2} \left (c d^2-a e^2\right ) \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(x^3*(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]*(d + e*x)*(-15*a^3*e^5 + a^2*c*d*e^3*(4*d - 5*e 
*x) + c^3*d^4*x*(3*d - 2*e*x) + a*c^2*d^2*e*(3*d^2 + 2*d*e*x + 2*e^2*x^2)) 
) + 3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*Sqrt[a*e + c 
*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[ 
a*e + c*d*x])])/(4*c^(7/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)*Sqrt[(a*e + c*d 
*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1211, 2192, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} e^3 \left (\frac {\left (7 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}\right )}{c^3 d^3 e^4}+\frac {2 a^3 e^3 (d+e x)}{c^3 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} e^3 \left (\frac {\left (7 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}\right )}{c^3 d^3 e^4}+\frac {2 a^3 e^3 (d+e x)}{c^3 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 a^3 e^3 (d+e x)}{c^3 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} e^3 \left (\frac {\left (7 a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}\right )}{c^3 d^3 e^4}\)

Input:

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

Output:

(2*a^3*e^3*(d + e*x))/(c^3*d^3*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2]) + ((c*d*e^3*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
])/2 - (e^3*(((3*c*d^2 + 7*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2])/e - (3*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*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])])/(2*Sqrt[c]*Sqrt[d]*e^(3/2))))/4)/(c^3*d^3*e^4)
 

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 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1274\) vs. \(2(231)=462\).

Time = 2.56 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1275\)

Input:

int(x^3*(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:

d*(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)))+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-(...
 

Fricas [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 778, normalized size of antiderivative = 3.03 \[ \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 )^{3/2}} \, dx=\left [\frac {3 \, {\left (a c^{3} d^{6} e + a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - 5 \, a^{4} e^{7} + {\left (c^{4} d^{7} + a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - 5 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (3 \, a c^{3} d^{5} e^{2} + 4 \, a^{2} c^{2} d^{3} e^{4} - 15 \, a^{3} c d e^{6} - 2 \, {\left (c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{2} + {\left (3 \, c^{4} d^{6} e + 2 \, a c^{3} d^{4} e^{3} - 5 \, a^{2} c^{2} d^{2} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (a c^{5} d^{6} e^{4} - a^{2} c^{4} d^{4} e^{6} + {\left (c^{6} d^{7} e^{3} - a c^{5} d^{5} e^{5}\right )} x\right )}}, -\frac {3 \, {\left (a c^{3} d^{6} e + a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - 5 \, a^{4} e^{7} + {\left (c^{4} d^{7} + a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - 5 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (3 \, a c^{3} d^{5} e^{2} + 4 \, a^{2} c^{2} d^{3} e^{4} - 15 \, a^{3} c d e^{6} - 2 \, {\left (c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{2} + {\left (3 \, c^{4} d^{6} e + 2 \, a c^{3} d^{4} e^{3} - 5 \, a^{2} c^{2} d^{2} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (a c^{5} d^{6} e^{4} - a^{2} c^{4} d^{4} e^{6} + {\left (c^{6} d^{7} e^{3} - a c^{5} d^{5} e^{5}\right )} x\right )}}\right ] \] Input:

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

Output:

[1/16*(3*(a*c^3*d^6*e + a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - 5*a^4*e^7 + (c 
^4*d^7 + a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - 5*a^3*c*d*e^6)*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*c^3*d^5*e^2 + 4*a^2*c^2*d^3*e^4 - 1 
5*a^3*c*d*e^6 - 2*(c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^2 + (3*c^4*d^6*e + 2*a*c 
^3*d^4*e^3 - 5*a^2*c^2*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2 
)*x))/(a*c^5*d^6*e^4 - a^2*c^4*d^4*e^6 + (c^6*d^7*e^3 - a*c^5*d^5*e^5)*x), 
 -1/8*(3*(a*c^3*d^6*e + a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - 5*a^4*e^7 + (c 
^4*d^7 + a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - 5*a^3*c*d*e^6)*x)*sqrt(-c*d*e 
)*arctan(1/2*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)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c* 
d*e^3)*x)) + 2*(3*a*c^3*d^5*e^2 + 4*a^2*c^2*d^3*e^4 - 15*a^3*c*d*e^6 - 2*( 
c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^2 + (3*c^4*d^6*e + 2*a*c^3*d^4*e^3 - 5*a^2* 
c^2*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*c^5*d^6*e^ 
4 - a^2*c^4*d^4*e^6 + (c^6*d^7*e^3 - a*c^5*d^5*e^5)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[4,4,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,0] 
%%%}+%%%{
 

Mupad [F(-1)]

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

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

Output:

int((x^3*(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 = 0.25 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.04 \[ \int \frac {x^3 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}-9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}-3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}-3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}-10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{3} e^{6}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{3} d^{6}-15 \sqrt {e x +d}\, a^{3} c d \,e^{6}+4 \sqrt {e x +d}\, a^{2} c^{2} d^{3} e^{4}-5 \sqrt {e x +d}\, a^{2} c^{2} d^{2} e^{5} x +3 \sqrt {e x +d}\, a \,c^{3} d^{5} e^{2}+2 \sqrt {e x +d}\, a \,c^{3} d^{4} e^{3} x +2 \sqrt {e x +d}\, a \,c^{3} d^{3} e^{4} x^{2}+3 \sqrt {e x +d}\, c^{4} d^{6} e x -2 \sqrt {e x +d}\, c^{4} d^{5} e^{2} x^{2}}{4 \sqrt {c d x +a e}\, c^{4} d^{4} e^{3} \left (a \,e^{2}-c \,d^{2}\right )} \] Input:

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

Output:

(15*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*e**6 - 9*s 
qrt(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*d**2*e**4 - 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*c**2*d**4*e**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))*c**3*d**6 - 10*s 
qrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*e**6 + 3*sqrt(e)*sqrt(d)*sqr 
t(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**4 - sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e 
 + c*d*x)*c**3*d**6 - 15*sqrt(d + e*x)*a**3*c*d*e**6 + 4*sqrt(d + e*x)*a** 
2*c**2*d**3*e**4 - 5*sqrt(d + e*x)*a**2*c**2*d**2*e**5*x + 3*sqrt(d + e*x) 
*a*c**3*d**5*e**2 + 2*sqrt(d + e*x)*a*c**3*d**4*e**3*x + 2*sqrt(d + e*x)*a 
*c**3*d**3*e**4*x**2 + 3*sqrt(d + e*x)*c**4*d**6*e*x - 2*sqrt(d + e*x)*c** 
4*d**5*e**2*x**2)/(4*sqrt(a*e + c*d*x)*c**4*d**4*e**3*(a*e**2 - c*d**2))