\(\int \frac {x (d+e x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [97]

Optimal result

Integrand size = 38, antiderivative size = 160 \[ \int \frac {x (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 a e (d+e x)}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2}+\frac {\left (c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.86 \[ \int \frac {x (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \sqrt {d} \sqrt {e} (3 a e+c d x) (d+e x)+\left (c d^2-3 a e^2\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{c^{5/2} d^{5/2} \sqrt {e} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(x*(d + e*x)^2)/(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]*(3*a*e + c*d*x)*(d + e*x) + (c*d^2 - 3*a*e^2)*Sqr 
t[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])])/(c^(5/2)*d^(5/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + 
e*x)])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1211, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(878\) vs. \(2(142)=284\).

Time = 2.28 (sec) , antiderivative size = 879, normalized size of antiderivative = 5.49

method	result	size
default	\(d^{2} \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+e^{2} \left (\frac {x^{2}}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{d e c \sqrt {d e c}}\right )}{2 d e c}-\frac {2 a \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{c}\right )+2 d e \left (-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{d e c \sqrt {d e c}}\right )\)	\(879\)

Input:

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

Output:

d^2*(-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^2*(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)))+2*d*e*(-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 [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.74 \[ \int \frac {x (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (a c d^{2} e - 3 \, a^{2} e^{3} + {\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (c^{2} d^{2} e x + 3 \, a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left (c^{4} d^{4} e x + a c^{3} d^{3} e^{2}\right )}}, -\frac {{\left (a c d^{2} e - 3 \, a^{2} e^{3} + {\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (c^{2} d^{2} e x + 3 \, a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (c^{4} d^{4} e x + a c^{3} d^{3} e^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*((a*c*d^2*e - 3*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)*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*(c^2*d^2*e*x + 3*a*c*d*e^2)*sqrt(c*d*e*x^2 + 
 a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e*x + a*c^3*d^3*e^2), -1/2*((a*c*d^2 
*e - 3*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)*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*(c^2 
*d^2*e*x + 3*a*c*d*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4* 
d^4*e*x + a*c^3*d^3*e^2)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

integrate(x*(e*x+d)^2/(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%%%{%%{[%%%{2,[3,3,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,2] 
%%%}+%%%{
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.37 \[ \int \frac {x (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{2}+4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c \,d^{2}+12 \sqrt {e x +d}\, a c d \,e^{2}+4 \sqrt {e x +d}\, c^{2} d^{2} e x}{4 \sqrt {c d x +a e}\, c^{3} d^{3} e} \] Input:

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

Output:

( - 12*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*e**2 + 4*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))*c*d**2 + 9*sqrt(e)*s 
qrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*e**2 - sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e 
 + c*d*x)*c*d**2 + 12*sqrt(d + e*x)*a*c*d*e**2 + 4*sqrt(d + e*x)*c**2*d**2 
*e*x)/(4*sqrt(a*e + c*d*x)*c**3*d**3*e)