\(\int \frac {x^2 (d+e x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [96]

Optimal result

Integrand size = 40, antiderivative size = 239 \[ \int \frac {x^2 (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 a^2 e^2 (d+e x)}{c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (c d^2-7 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 e}+\frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2}-\frac {\left (c^2 d^4+6 a c d^2 e^2-15 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{7/2} d^{7/2} e^{3/2}} \] Output:

-2*a^2*e^2*(e*x+d)/c^3/d^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/4*(-7 
*a*e^2+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+1/2*x*(a*d 
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2-1/4*(-15*a^2*e^4+6*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^(3/2)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.75 \[ \int \frac {x^2 (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (-15 a^2 e^3+a c d e (d-5 e x)+c^2 d^2 x (d+2 e x)\right )-\left (c^2 d^4+6 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{4 c^{7/2} d^{7/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1211, 25, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {\frac {1}{4} e^3 \left (\frac {\left (-15 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}-\frac {\left (c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 e^3}-\frac {2 a^2 e^2 (d+e x)}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {1}{4} e^3 \left (\frac {\left (-15 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}-\frac {\left (c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 e^3}-\frac {2 a^2 e^2 (d+e x)}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {1}{4} e^3 \left (\frac {\left (-15 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}-\frac {\left (c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c d e^3 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 e^3}-\frac {2 a^2 e^2 (d+e x)}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[(x^2*(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^2*e^2*(d + e*x))/(c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
]) - (-1/2*(c*d*e^3*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (e^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) + (( 
c^2*d^4 + 6*a*c*d^2*e^2 - 15*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^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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. \(1547\) vs. \(2(213)=426\).

Time = 2.69 (sec) , antiderivative size = 1548, normalized size of antiderivative = 6.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1548\)

Input:

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

Output:

d^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))+e^2*(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)))+2*d*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...
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.43 \[ \int \frac {x^2 (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} - 15 \, a^{3} e^{5} + {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (2 \, c^{3} d^{3} e^{2} x^{2} + a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4} + {\left (c^{3} d^{4} e - 5 \, a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (c^{5} d^{5} e^{2} x + a c^{4} d^{4} e^{3}\right )}}, \frac {{\left (a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} - 15 \, a^{3} e^{5} + {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (2 \, c^{3} d^{3} e^{2} x^{2} + a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4} + {\left (c^{3} d^{4} e - 5 \, a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (c^{5} d^{5} e^{2} x + a c^{4} d^{4} e^{3}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Maxima [F(-2)]

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

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

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

int((x^2*(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^2*(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 = 0.25 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.47 \[ \int \frac {x^2 (d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}-6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4}-10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} e^{4}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}-15 \sqrt {e x +d}\, a^{2} c d \,e^{4}+\sqrt {e x +d}\, a \,c^{2} d^{3} e^{2}-5 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{3} x +\sqrt {e x +d}\, c^{3} d^{4} e x +2 \sqrt {e x +d}\, c^{3} d^{3} e^{2} x^{2}}{4 \sqrt {c d x +a e}\, c^{4} d^{4} e^{2}} \] Input:

int(x^2*(e*x+d)^2/(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**2*e**4 - 6*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*c*d**2*e**2 - sqrt 
(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqr 
t(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**4 - 10*sqrt(e)* 
sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*e**4 + 2*sqrt(e)*sqrt(d)*sqrt(c)*sq 
rt(a*e + c*d*x)*a*c*d**2*e**2 - 15*sqrt(d + e*x)*a**2*c*d*e**4 + sqrt(d + 
e*x)*a*c**2*d**3*e**2 - 5*sqrt(d + e*x)*a*c**2*d**2*e**3*x + sqrt(d + e*x) 
*c**3*d**4*e*x + 2*sqrt(d + e*x)*c**3*d**3*e**2*x**2)/(4*sqrt(a*e + c*d*x) 
*c**4*d**4*e**2)