\(\int \frac {x^2 (d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [104]

Optimal result

Integrand size = 40, antiderivative size = 334 \[ \int \frac {x^2 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 a^2 e^2 \left (c d^2-a e^2\right ) (d+e x)}{c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3 c^2 d^4-52 a c d^2 e^2+57 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^4 d^4 e}+\frac {\left (7 c d^2-11 a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {e x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+10 a c d^2 e^2-35 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2} e^{3/2}} \] Output:

-2*a^2*e^2*(-a*e^2+c*d^2)*(e*x+d)/c^4/d^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2 
)^(1/2)+1/24*(57*a^2*e^4-52*a*c*d^2*e^2+3*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(1/2)/c^4/d^4/e+1/12*(-11*a*e^2+7*c*d^2)*x*(a*d*e+(a*e^2+c*d^2) 
*x+c*d*e*x^2)^(1/2)/c^3/d^3+1/3*e*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2)/c^2/d^2-1/8*(-a*e^2+c*d^2)*(-35*a^2*e^4+10*a*c*d^2*e^2+c^2*d^4)*arctan 
h(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^(9/2)/d^(9/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 10.68 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {(d+e x) \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (105 a^3 e^5-5 a^2 c d e^3 (20 d-7 e x)+a c^2 d^2 e \left (3 d^2-38 d e x-14 e^2 x^2\right )+c^3 d^3 x \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )-3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (c^2 d^4+10 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a e+c d x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{24 c^{9/2} d^{9/2} e^{3/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

((d + e*x)*(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]* 
(105*a^3*e^5 - 5*a^2*c*d*e^3*(20*d - 7*e*x) + a*c^2*d^2*e*(3*d^2 - 38*d*e* 
x - 14*e^2*x^2) + c^3*d^3*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2)) - 3*Sqrt[c*d]* 
Sqrt[c*d^2 - a*e^2]*(c^2*d^4 + 10*a*c*d^2*e^2 - 35*a^2*e^4)*Sqrt[a*e + c*d 
*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c* 
d^2 - a*e^2])]))/(24*c^(9/2)*d^(9/2)*e^(3/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - 
 a*e^2)]*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1211, 25, 2192, 27, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

\(\displaystyle -\frac {\frac {\int \frac {6 a c d \left (c d^2-a e^2\right )^2 e^6-c^3 d^3 \left (7 c d^2-11 a e^2\right ) x^2 e^6-2 c^2 d^2 \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x e^5}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {6 a c d \left (c d^2-a e^2\right )^2 e^6-c^3 d^3 \left (7 c d^2-11 a e^2\right ) x^2 e^6-2 c^2 d^2 \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x e^5}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle -\frac {\frac {\frac {\int \frac {c^2 d^2 e^6 \left (2 a e \left (19 c^2 d^4-35 a c e^2 d^2+12 a^2 e^4\right )-c d \left (3 c^2 d^4-52 a c e^2 d^2+57 a^2 e^4\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}-\frac {1}{2} c^2 d^2 e^5 x \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \int \frac {2 a e \left (19 c^2 d^4-35 a c e^2 d^2+12 a^2 e^4\right )-c d \left (3 c^2 d^4-52 a c e^2 d^2+57 a^2 e^4\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {1}{2} c^2 d^2 e^5 x \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-35 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}-\frac {\left (57 a^2 e^4-52 a c d^2 e^2+3 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c^2 d^2 e^5 x \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-35 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}-\frac {\left (57 a^2 e^4-52 a c d^2 e^2+3 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c^2 d^2 e^5 x \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-35 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}-\frac {\left (57 a^2 e^4-52 a c d^2 e^2+3 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}\right )-\frac {1}{2} c^2 d^2 e^5 x \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^4}-\frac {2 a^2 e^2 (d+e x) \left (c d^2-a e^2\right )}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

(-2*a^2*e^2*(c*d^2 - a*e^2)*(d + e*x))/(c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^ 
2)*x + c*d*e*x^2]) - (-1/3*(c^2*d^2*e^5*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2]) + (-1/2*(c^2*d^2*e^5*(7*c*d^2 - 11*a*e^2)*x*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2]) + (c*d*e^5*(-(((3*c^2*d^4 - 52*a*c*d^2*e^2 + 
 57*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/e) + (3*(c*d^2 - 
 a*e^2)*(c^2*d^4 + 10*a*c*d^2*e^2 - 35*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)/(6*c*d*e))/(c^4*d^4*e^4)
 

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. \(2894\) vs. \(2(304)=608\).

Time = 3.27 (sec) , antiderivative size = 2895, normalized size of antiderivative = 8.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2895\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 754, normalized size of antiderivative = 2.26 \[ \int \frac {x^2 (d+e x)^3}{\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 + 9 \, a^{2} c^{2} d^{4} e^{3} - 45 \, a^{3} c d^{2} e^{5} + 35 \, a^{4} e^{7} + {\left (c^{4} d^{7} + 9 \, a c^{3} d^{5} e^{2} - 45 \, a^{2} c^{2} d^{3} e^{4} + 35 \, 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 (8 \, c^{4} d^{4} e^{3} x^{3} + 3 \, a c^{3} d^{5} e^{2} - 100 \, a^{2} c^{2} d^{3} e^{4} + 105 \, a^{3} c d e^{6} + 14 \, {\left (c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{2} + {\left (3 \, c^{4} d^{6} e - 38 \, a c^{3} d^{4} e^{3} + 35 \, 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}}{96 \, {\left (c^{6} d^{6} e^{2} x + a c^{5} d^{5} e^{3}\right )}}, \frac {3 \, {\left (a c^{3} d^{6} e + 9 \, a^{2} c^{2} d^{4} e^{3} - 45 \, a^{3} c d^{2} e^{5} + 35 \, a^{4} e^{7} + {\left (c^{4} d^{7} + 9 \, a c^{3} d^{5} e^{2} - 45 \, a^{2} c^{2} d^{3} e^{4} + 35 \, 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 (8 \, c^{4} d^{4} e^{3} x^{3} + 3 \, a c^{3} d^{5} e^{2} - 100 \, a^{2} c^{2} d^{3} e^{4} + 105 \, a^{3} c d e^{6} + 14 \, {\left (c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{2} + {\left (3 \, c^{4} d^{6} e - 38 \, a c^{3} d^{4} e^{3} + 35 \, 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}}{48 \, {\left (c^{6} d^{6} e^{2} x + a c^{5} d^{5} e^{3}\right )}}\right ] \] Input:

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

Output:

[1/96*(3*(a*c^3*d^6*e + 9*a^2*c^2*d^4*e^3 - 45*a^3*c*d^2*e^5 + 35*a^4*e^7 
+ (c^4*d^7 + 9*a*c^3*d^5*e^2 - 45*a^2*c^2*d^3*e^4 + 35*a^3*c*d*e^6)*x)*sqr 
t(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*sqr 
t(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*(8*c^4*d^4*e^3*x^3 + 3*a*c^3*d^5 
*e^2 - 100*a^2*c^2*d^3*e^4 + 105*a^3*c*d*e^6 + 14*(c^4*d^5*e^2 - a*c^3*d^3 
*e^4)*x^2 + (3*c^4*d^6*e - 38*a*c^3*d^4*e^3 + 35*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))/(c^6*d^6*e^2*x + a*c^5*d^5*e^3), 1 
/48*(3*(a*c^3*d^6*e + 9*a^2*c^2*d^4*e^3 - 45*a^3*c*d^2*e^5 + 35*a^4*e^7 + 
(c^4*d^7 + 9*a*c^3*d^5*e^2 - 45*a^2*c^2*d^3*e^4 + 35*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*(8*c^4*d^4*e^3*x^3 + 3*a*c^3*d^5*e^2 - 100*a^2*c^2*d^ 
3*e^4 + 105*a^3*c*d*e^6 + 14*(c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^2 + (3*c^4*d^ 
6*e - 38*a*c^3*d^4*e^3 + 35*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))/(c^6*d^6*e^2*x + a*c^5*d^5*e^3)]
 

Sympy [F]

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

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

Maxima [F(-2)]

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

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

integrate(x^2*(e*x+d)^3/(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,[5,5,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,0] 
%%%}+%%%{
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 6.07 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-840 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}+1080 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}-216 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}-24 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}+525 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{3} e^{6}-615 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}+87 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{3} d^{6}+840 \sqrt {e x +d}\, a^{3} c d \,e^{6}-800 \sqrt {e x +d}\, a^{2} c^{2} d^{3} e^{4}+280 \sqrt {e x +d}\, a^{2} c^{2} d^{2} e^{5} x +24 \sqrt {e x +d}\, a \,c^{3} d^{5} e^{2}-304 \sqrt {e x +d}\, a \,c^{3} d^{4} e^{3} x -112 \sqrt {e x +d}\, a \,c^{3} d^{3} e^{4} x^{2}+24 \sqrt {e x +d}\, c^{4} d^{6} e x +112 \sqrt {e x +d}\, c^{4} d^{5} e^{2} x^{2}+64 \sqrt {e x +d}\, c^{4} d^{4} e^{3} x^{3}}{192 \sqrt {c d x +a e}\, c^{5} d^{5} e^{2}} \] Input:

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

Output:

( - 840*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 + 
 1080*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*d**2*e 
**4 - 216*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 - 24*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 + 525*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*e**6 - 615*sqrt(e 
)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**4 + 87*sqrt(e)*sqrt(d)* 
sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*sqr 
t(a*e + c*d*x)*c**3*d**6 + 840*sqrt(d + e*x)*a**3*c*d*e**6 - 800*sqrt(d + 
e*x)*a**2*c**2*d**3*e**4 + 280*sqrt(d + e*x)*a**2*c**2*d**2*e**5*x + 24*sq 
rt(d + e*x)*a*c**3*d**5*e**2 - 304*sqrt(d + e*x)*a*c**3*d**4*e**3*x - 112* 
sqrt(d + e*x)*a*c**3*d**3*e**4*x**2 + 24*sqrt(d + e*x)*c**4*d**6*e*x + 112 
*sqrt(d + e*x)*c**4*d**5*e**2*x**2 + 64*sqrt(d + e*x)*c**4*d**4*e**3*x**3) 
/(192*sqrt(a*e + c*d*x)*c**5*d**5*e**2)